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Canonical forms and discrete Liouville–Green asymptotics for second-order linear difference equations

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Abstract

Liouville–Green (WKB) asymptotic approximations are constructed for some classes of linear second-order difference equations. This is done starting from certain “canonical forms” for the three-term linear recurrence. Rigorous explicit bounds are established for the error terms in the asymptotic approximations of recessive as well as dominant solutions. The asymptotics with respect to parameters affecting the equation is also discussed. Several illustrative examples are given.

Introduction

In this paper we first derive certain “canonical forms” for the three-term linear recurrence, and then we use them to obtain asymptotic approximations of the Liouville–Green (LG, for short, or WKB) type for their solutions. More precisely, we establish asymptotic representations for two linearly independent solutions, providing explicit rigorous bounds for the error terms.

A typical feature of the classical LG approximation is its “double asymptotic nature” since it holds when both, the independent variable and some parameter, tend to certain limits [9, 14]. We extend the results achieved in [14] to some other classes of difference equations; see also [8, 19, 21]. In Sect. 2, we derive the aforementioned canonical forms, and in Sect. 3 we construct the corresponding LG approximations, equipped with rigorous bounds for the error terms. Some details, lengthy though rather elementary, are relegated to Appendices. Several examples are given in Sect. 4. It is possible to proceed similarly in case of matrix difference equations or, equivalently, for systems, but we will not do it in this paper; see, e.g., [4, 11, 12]. Finally, we summarize the high points of the paper in the short concluding Sect. 5.

Canonical forms

Consider the three-term linear recurrence

$$ u_{n+2} + A_{n} u_{n+1} + B_{n} u_{n} = 0, \quad n \in {\mathbf{Z}} _{\nu }, $$
(1)

where \(\mathbf{Z}_{\nu } := \{ n \in {\mathbf{Z}}: n \geq \nu \}\) for some fixed \(\nu \in {\mathbf{Z}}\), the sequences \(\{ A_{n} \}\) and \(\{ B_{n} \}\) are scalar (either real- or complex-valued), and \(\Delta u_{n} := u_{n+1} - u_{n}\), \(\Delta ^{2} := \Delta (\Delta )\). We can show that the following forms

  1. (A)
    $$ \Delta ^{2} y_{n} + q_{n} y_{n} = 0, \quad n \in {\mathbf{Z}}_{\nu }, $$
  2. (B)
    $$ \Delta ^{2} y_{n} + r_{n} y_{n+1} = 0, \quad n \in {\mathbf{Z}}_{ \nu }, $$
  3. (C)
    $$ \Delta (c_{n} \Delta y_{n}) + r_{n} y_{n} = 0, \quad n \in {\mathbf{Z}} _{\nu }, $$
  4. (D)
    $$ \Delta (c_{n} \Delta y_{n}) + r_{n} y_{n+1} = 0, \quad n \in {\mathbf{Z}} _{\nu }, $$

are canonical forms for recurrence (1), in the sense that the latter can be recast (with some exceptions) into such forms. Only if \(A_{n} = - 2\) for every n, equation (1) can be immediately rewritten as \(\Delta ^{2} u_{n} + Q_{n} u_{n} = 0\), with \(Q_{n} := A_{n} + B_{n} + 1\).

Note that equations of cases (C) and (D) are in self-adjoint form. Equations (B) and (D) seem to have been preferred in oscillation theory [5, 6, 10]. All the previous forms are also reminiscent of the analogous ones for linear second-order differential equations (sometimes called Sturm–Liouville or Jacobi forms), and hence some inspiration can be drawn from such similarity. Below, we show how a general recurrence like (1) can be taken into the various canonical forms (A)–(D) (with some exceptions).

Case (A). In [14], an equation like that in (A) was obtained setting

$$ u_{n} = \alpha _{n} y_{n} $$
(2)

in (1), and choosing

$$ \alpha _{n} := \alpha _{\nu +1} \prod _{k=\nu }^{n-2} \biggl( - \frac{A _{k}}{2} \biggr) \quad \text{for } n \geq \nu + 2, $$
(3)

with \(\alpha _{\nu }\) and \(\alpha _{\nu +1} \neq 0\) arbitrary constants. The canonical form (A) is thus obtained with

$$ q_{n} := -1 + \frac{4 B_{n}}{A_{n} A_{n-1}} \quad \text{for } n \geq \nu + 1, $$
(4)

while

$$ q_{\nu } = -1 - 2 \frac{B_{\nu }}{A_{\nu }} \frac{\alpha _{\nu }}{ \alpha _{\nu +1}}, $$

provided that \(A_{n} \neq 0\) for all \(n \geq \nu \).

Case (B). The same transformation (2) also allows to take equation (1) into the form (B), choosing \(\alpha _{n}\) and \(r_{n}\) such that

$$ -1 + B_{n} \frac{\alpha _{n}}{\alpha _{n+2}} \equiv 0, \qquad 2 + A_{n} \frac{\alpha _{n+1}}{\alpha _{n+2}} \equiv r_{n} \quad \text{for } n \geq \nu . $$

The first of this, i.e., \(\alpha _{n+2} = B_{n} \alpha _{n}\), can be solved explicitly, but we have to distinguish the four cases according to n and ν being even or odd. We obtain

  1. (i)

    for n and ν both even or both odd,

    $$ \alpha _{n} = B_{n-2} B_{n-4} \cdots B_{\nu +2} B_{\nu } \quad \text{for} \ n \geq \nu $$
    (5)

    and

    $$ r_{n} = 2 + A_{n} \frac{\prod_{k=0}^{(n-2-\nu )/2} B_{n-2k-1}}{ \prod_{k=0}^{(n-\nu )/2} B_{n-2k}} \quad \text{for } n \geq \nu ; $$
    (6)
  2. (ii)

    for n even and ν odd, or n odd and ν even,

    $$ \alpha _{n} = B_{n-2} B_{n-4} \cdots B_{\nu +3} B_{\nu +1} \quad \text{for } n \geq \nu + 1, $$
    (7)

    and

    $$ r_{n} = 2 + A_{n} \frac{\prod_{k=0}^{(n-1-\nu )/2} B_{n-2k-1}}{ \prod_{k=0}^{(n-1-\nu )/2} B_{n-2k}} \quad \text{for } n \geq \nu + 1. $$
    (8)

    All this can be done provided that \(B_{n} \neq 0\) for all \(n \geq \nu \), or at least for n sufficiently large.

Other than cases (A) and (B), given the two coefficients \(A_{n}\) and \(B_{n}\) of equation (1), the latter can immediately be written in the form (C) or (D) merely rearranging its terms, since now there are two “degrees of freedom”, represented by the possibility of choosing arbitrarily the two coefficients \(c_{n}\) and \(r_{n}\). In fact

Case (C). Imposing

$$ -1 - \frac{c_{n}}{c_{n+1}} \equiv A_{n}, \quad \text{and} \quad \frac{c _{n} + r_{n}}{c_{n+1}} \equiv B_{n}, $$

i.e., choosing

$$ c_{n} := c_{\nu } \prod_{k=\nu }^{n-1} \biggl(\frac{-1}{A_{k} + 1} \biggr) \quad \text{for } n \geq \nu + 1, $$
(9)

and

$$ r_{n} := c_{\nu } (A_{n} + B_{n} + 1) \prod_{k=\nu }^{n} \biggl( \frac{-1}{A _{k} + 1} \biggr) \quad \text{for } n \geq \nu , $$
(10)

for any arbitrary constant \(c_{\nu } \neq 0\), we obtain the form C with \(y_{n} = u_{n}\), except when \(A_{n} = -1\) for all n (but it will hold true if \(A_{n} \neq - 1\) for all n sufficiently large values of n).

Case (D). Imposing instead

$$ -1 + \frac{r_{n} - c_{n}}{c_{n+1}} \equiv A_{n}, \quad \text{and} \quad \frac{c_{n}}{c_{n+1}} \equiv B_{n}, $$

i.e.,

$$ c_{n} := c_{\nu } \prod_{k=\nu }^{n-1} \bigl(B_{k}^{-1} \bigr) \quad \text{for } n \geq \nu + 1, $$
(11)

and

$$ r_{n} = c_{\nu } (A_{n} + B_{n} + 1) \prod_{k=\nu }^{n} \bigl(B_{k} ^{-1} \bigr) \quad \text{for } n \geq \nu , $$
(12)

for any arbitrary constant \(c_{\nu } \neq 0\), we obtain the form (D), again with \(y_{n} = u_{n}\), with the exception now that \(B_{n} = 0\) for all (or for sufficiently large) values of n.

Note that we could also use transformation (2) to obtain equations like (C) or (D), having now three degrees of freedom, represented by \(\alpha _{n}\), \(c_{n}\), and \(r_{n}\), for any two given sequences \(A_{n}\) and \(B_{n}\). This requires choosing the former quantities from the conditions

$$\begin{aligned}& \frac{c_{n}}{c_{n+1}} = -1 - A_{n} \frac{\alpha _{n+1}}{\alpha _{n+2}}, \\& r_{n} = c_{n+1} \biggl(B_{n} \frac{\alpha _{n}}{\alpha _{n+2}} - \frac{c _{n}}{c_{n+1}} \biggr) \end{aligned}$$

to obtain the form (C), and

$$\begin{aligned}& \frac{c_{n}}{c_{n+1}} = B_{n} \frac{\alpha _{n}}{\alpha _{n+2}}, \\& r_{n} = 1 + \frac{c_{n}}{c_{n+1}} + A_{n} \frac{\alpha _{n+1}}{\alpha _{n+2}} = 1 + \frac{A_{n} \alpha _{n+1} + B_{n} \alpha _{n}}{\alpha _{n+2}} \end{aligned}$$

to obtain (D).

In the next section, we formulate and prove two theorems, yielding the asymptotic representations of the LG (or WKB) type for a basis of solutions to the model equations (C) and (D). The corresponding results for (A) and (B) are special cases of (C) and (D), respectively; besides, case (A) was already considered in [14].

The main results

In this section, we develop an LG (or WKB) asymptotic approximation theory for the three classes of linear second-order difference equations (B), (C), and (D), along the same lines followed for the form (A) in [14]. We observe that case (A) is a special instance of (C), while case (B) is a special instance of (D), and hence we will consider only cases (C) and (D).

Our spirit here is that of the LG approximation, put on rigorous grounds by F.W.J. Olver for second-order differential equations [9], in that we wish to obtain rigorous and explicit computable error bounds.

Starting from the canonical forms to establish LG asymptotic approximations is convenient, since we are able to exploit their similarity with second-order differential equations with no first-order derivative terms (Jacobi or Sturm–Liouville forms). This approach parallels closely the original one followed by F.W.J. Olver for differential equations [9].

Below, we assume that the coefficients of equations (C) and (D) have always the form

$$ r_{n} = a + g_{n}, \qquad c_{n} = c + h_{n}, $$
(13)

with a real, \(c > 0\), and that the “perturbations” \(g_{n}\) and \(h_{n}\) are in \(\ell ^{1}\). We choose further \(c = 1\), without any loss of generality, to simplify the notation, but confine ourselves to \(a \neq 0\), \(a \neq -1\). In fact, in case (C), e.g., \(a + c = a + 1 = 0\) for \(a = -1\), and this lets the lowest order term of the unperturbed equation vanish. This case is in some sense pathological, since in this occurrence the unperturbed equation is no longer of the second order.

In fact, equation (1) is properly called a second-order difference equation only if \(B_{n} \neq 0\) (for all \(n > \nu \)). It then can be written in the form

$$ \Delta ^{2} u_{n} + a_{n} \Delta u_{n} + b_{n} u_{n} = 0, $$
(14)

where \(a_{n} := A_{n} + 2\) and \(b_{n} := A_{n} + B_{n} + 1\), and conversely equation (14) can be recast into the form of equation (1), setting \(A_{n} = a_{n} - 2\) and \(B_{n} = b _{n} - a_{n} + 1\). But some care should be paid to the form (14) since, other than in the formally analogous case of differential equations, it is not always guaranteed that equation (14) is of the second order, that is, the vector space of all solutions has dimension 2. However, the formal similarity of equation (14) with the linear second-order differential equation suggests the possibility to remove the term containing the first difference from (14) using transformation (2), as it can be done for differential equations. Indeed, we end up with the difference equation in (A).

When \(a = 0\), the unperturbed equation (31) falls in the case of “finite moments perturbations” [13, 15, 17]. Some other cases, e.g., a complex in equation (A), have been considered elsewhere using the canonical form (A), see, e.g., [16, 18, 20].

LG asymptotics for case (C)

We define the basic quantities

$$\begin{aligned}& V_{n} := \sum_{r=n}^{\infty } \vert g_{r} \vert , \qquad W_{n} := \sum _{r=n}^{ \infty } \vert h_{r} \vert , \end{aligned}$$
(15)
$$\begin{aligned}& Z_{n}^{\pm } := \vert \eta _{\pm } \vert ^{-n} \sum_{r=n}^{\infty } \vert \eta _{\pm } \vert ^{r} \vert g_{r} \vert , \qquad U_{n}^{\pm } := \vert \eta _{\pm } \vert ^{-n} \sum _{r=n}^{\infty } \vert \eta _{\pm } \vert ^{r} \vert h_{r} \vert , \end{aligned}$$
(16)

where

$$ \eta _{\pm } := \frac{\lambda _{\pm }}{\lambda _{\mp }} = \frac{1 \pm \sqrt{ \vert a \vert }}{1 \mp \sqrt{ \vert a \vert }}, $$
(17)

and

$$ \lambda _{\pm } := 1 \pm \sqrt{- a} $$
(18)

are the characteristic roots of the unperturbed difference equation associated to (C), that is, \(\lambda ^{2} - 2 \lambda + (1 + a) = 0\). We begin with the case of equation (C), stating the following.

Theorem 3.1

Let equation (C) be given for \(n \geq \nu \), for some fixed \(\nu \in {\mathbf{Z}}\), with the coefficients as in (13), with a real, \(a \neq 0, -1\), and \(c=1\), \(g_{n}\) and \(h_{n}\) real or complex. Then,

  1. (i)

    if \(a > 0\), and assuming that

    $$ V_{n} < \infty , \qquad W_{n} < \infty , $$
    (19)

    two linearly independent solutions of the form

    $$ y_{n}^{\pm } = (\lambda _{\pm })^{n} \bigl(1 + \varepsilon _{n}^{\pm }\bigr), \quad n \geq n_{0}, $$
    (20)

    exist for some integer \(n_{0} \geq \nu \), and the error terms \(\varepsilon _{n}^{\pm }\) can be estimated as

    $$ \bigl\vert \varepsilon _{n}^{\pm } \bigr\vert < \mathcal{V}_{n}, $$
    (21)

    the error control function \(\mathcal{V}_{n}\) being defined as

    $$ {\mathcal{V}}_{n} := \frac{1}{\sqrt{a(a + 1)}} \bigl\{ V_{n} + (\sqrt{a + 1} + 1) [\sqrt{a + 1} W_{n+1} + W_{n+2} ] \bigr\} $$
    (22)

    for all \(n \geq n_{0}\), where

    $$ n_{0} := \min \{n \geq \nu : \mathcal{V}_{n} < 1 \}. $$
    (23)
  2. (ii)

    If \(a < 0\) (but \(a \neq -1\)), a solution \(y_{n}^{-}\) (as in (20)) exists under conditions (19) for any fixed \(n \geq n_{0}^{-}\), and the corresponding error term is estimated as

    $$ \bigl\vert \varepsilon _{n}^{-} \bigr\vert < \mathcal{V}_{n}^{-}, $$
    (24)

    with

    $$ {\mathcal{V}}_{n}^{-} := \textstyle\begin{cases} \frac{1}{2 \sqrt{ \vert a \vert } (1 - \sqrt{ \vert a \vert })} [V_{n} + (2 - \sqrt{ \vert a \vert }) W_{n} + (1 - \sqrt{ \vert a \vert })(2 - \sqrt{ \vert a \vert }) W _{n+1} ] \\ \quad \textit{if } \vert a \vert < 1, \\ \frac{1}{\sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } - 1)} [V_{n} + W_{n} + ( \vert a \vert - 1) W_{n+1} + (\sqrt{ \vert a \vert } - 1) \vert h_{n} \vert ] \\ \quad \textit{if } \vert a \vert > 1, \end{cases} $$
    (25)

    for all \(n \geq n_{0}^{-}\), where we set

    $$ n_{0}^{\pm } := \min \bigl\{ n \geq \nu : \mathcal{V}_{n}^{-} < 1 \bigr\} , $$
    (26)

    while, under the convergence conditions

    $$ Z_{n}^{+} < +\infty , \qquad U_{n}^{+} < +\infty , $$
    (27)

    for any fixed \(n \geq n_{0}^{+}\), stronger than those in (19) (since \(|\eta _{+}| > 1\)), a second, linearly independent solution, \(y_{n}^{+}\) (as in (20)) exists for \(n \geq n_{0}^{+}\), and the corresponding error term, when \(|a| < 1\), is estimated as

    $$ \bigl\vert \varepsilon _{n}^{+} \bigr\vert < \mathcal{V}_{n}^{+}, $$
    (28)

    with

    $$\begin{aligned} \mathcal{V}_{n}^{+} := &\frac{1}{2 \sqrt{ \vert a \vert } (1 + \sqrt{ \vert a \vert })} \bigl\{ V_{n} + W_{n} + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \bigl(3 + \sqrt{ \vert a \vert }\bigr) W_{n+1} \\ &{}+ \bigl(1 + \sqrt{ \vert a \vert }\bigr) \bigl(1 + \sqrt{ \vert a \vert } + \eta _{+}\bigr) \eta _{+} U _{n+1}^{+} + (\eta _{+})^{2} \bigl(1 + \sqrt{ \vert a \vert } + \eta _{+}\bigr) U_{n+2}^{+} \\ &{}+ (\eta _{+})^{3} Z_{n+2}^{+} + \eta _{+} \bigl(1 + \sqrt{ \vert a \vert } + \eta _{+}\bigr) \vert h_{n+1} \vert \\ &{}+ \bigl[\bigl(1 + \sqrt{ \vert a \vert }\bigr) (1 + \eta _{+}) + \eta _{+} \bigr] \vert h_{n+1} \vert + (\eta _{+})^{2} \vert g_{n+1} \vert + \eta _{+} \vert g_{n} \vert \bigr\} . \end{aligned}$$
    (29)

    When \(|a| > 1\), instead, we have for the second solution the estimate

    $$\begin{aligned} \mathcal{V}_{n}^{+} :=& \frac{1}{2 \sqrt{ \vert a \vert } (1 + \sqrt{ \vert a \vert })} \bigl\{ V_{n+2} + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \bigl(2 + \sqrt{ \vert a \vert }\bigr) W_{n+1} +\bigl(2 + \sqrt{ \vert a \vert }\bigr) W _{n+2} \\ & {}+ \bigl(1 + \sqrt{ \vert a \vert }\bigr) \bigl[1 + \sqrt{ \vert a \vert } + \vert \eta _{+} \vert \bigr] \vert \eta _{+} \vert U_{n+1}^{+} + \vert \eta _{+} \vert ^{2} \bigl[1 + \sqrt{ \vert a \vert } + \vert \eta _{+} \vert \bigr] U_{n+2}^{+} \\ & {}+ \vert \eta _{+} \vert ^{3} Z_{n+2}^{+} + \bigl(1 + \vert \eta _{+} \vert \bigr) \bigl(2 + \sqrt{ \vert a \vert }\bigr) \vert h_{n} \vert + \bigl[1 + \sqrt{ \vert a \vert } \\ & {} + \vert \eta _{+} \vert \bigl(1 + \sqrt{ \vert a \vert } + \vert \eta _{+} \vert \bigr) \bigr] \vert h_{n+1} \vert + \bigl(1 + \vert \eta _{+} \vert ^{2}\bigr) \vert g_{n+1} \vert + \bigl(1 + \vert \eta _{+} \vert \bigr) \vert g_{n} \vert \bigr\} . \end{aligned}$$
    (30)

Proof of Theorem 3.1

The proof follows closely that made for the case of equation (A) in Theorem 2.1 of [14], that is, it is based on transforming the difference equation for the error term into a “discrete integral” equation for it. Considering first the “unperturbed” equation associated to it, namely that obtained setting \(h_{n} \equiv 0\), \(g_{n} \equiv 0\) in (13),

$$ \Delta ^{2} y_{n} + a y_{n} = 0, \quad a \in {\mathbf{R}} \setminus \{ 0, -1 \}, $$
(31)

the corresponding “characteristic equation” (obtained looking for solutions to (31) of the form \(y_{n} = \lambda ^{n}\),

$$ \lambda ^{2} - 2 \lambda + (1 + a) = 0, $$
(32)

has the “characteristic roots” \(\lambda _{\pm } := 1 \pm \sqrt{-a}\), also written as

$$ \lambda _{\pm } = 1 \pm i \sqrt{a} \quad \text{when } a > 0. $$
(33)

When \(a = 0\), the characteristic equation has the double root \(\lambda = 1\), and hence two linearly independent solutions to (31) are 1 and n, but we will not consider this case here.

Looking now for solutions to (C) of the form \(y_{n} = \lambda ^{n} (1 + \varepsilon _{n})\), with either \(\lambda = \lambda _{+}\) or \(\lambda = \lambda _{-}\), we obtain, after a little algebra, the error equation

$$ \lambda ^{2} \varepsilon _{n+2} - 2 \lambda \varepsilon _{n+1} + (1 + a) \varepsilon _{n} = \chi _{n}, $$
(34)

where we set

$$ \chi _{n} := - \lambda ^{2} h_{n+1} (1 + \varepsilon _{n+2}) + \lambda (h _{n} + h_{n+1}) (1 + \varepsilon _{n+1}) - (h_{n} + g_{n}) (1 + \varepsilon _{n}). $$
(35)

Note that \(\chi _{n}\) also depends on the choice of either \(\lambda _{+}\) or \(\lambda _{-}\) (as it happens for \(\varepsilon _{n}\)).

We can “solve” equation (34) treating \(\chi _{n}\) as known, constructing a discrete integral equation (also called summary difference equation) for \(\varepsilon _{n}\), rather than an explicit solution. We can proceed following the discrete analogue of the method of variation of parameters. To this purpose, we first set \(\chi _{n} \equiv 0\) in (34), obtaining

$$ \lambda ^{2} \Delta ^{2} \varepsilon _{n} + \beta \Delta \varepsilon _{n} = 0, $$
(36)

where relation (32) has been used, having set

$$ \beta := 2 \biggl(1 - \frac{1}{\lambda } \biggr). $$
(37)

Equation (36) can also be written as

$$ \Delta [\Delta \varepsilon _{n} + \beta \varepsilon _{n} ] = 0, $$

which suggests that two linearly independent solutions can be promptly obtained, one given by \(\varepsilon _{n} \equiv 1\), and the other constructed solving

$$ \Delta \varepsilon _{n} + \beta \varepsilon _{n} = 0, $$

which yields

$$ \varepsilon _{n} = \eta ^{n}, \quad \eta := 1 - \beta = \frac{2}{\lambda } - 1. $$

Note that here

$$ \eta \equiv \eta _{\mp } = \frac{2 - \lambda _{\pm }}{\lambda _{\pm }} = \frac{1 \mp \sqrt{-a}}{1 \pm \sqrt{-a}} = \frac{\lambda _{\mp }}{ \lambda _{\pm }} $$
(38)

for either \(a > 0\) or \(a < 0\).

The general solution to (36) has the form

$$ \varepsilon _{n} = C + D \eta ^{n}, $$
(39)

where C and D are arbitrary constants. To obtain a representation for the solutions to the full equation (34), we look for a solution of the form

$$ \varepsilon _{n} = C_{n} + D_{n} \eta ^{n}, $$
(40)

where \(C_{n}\) and \(D_{n}\) are two sequences to be determined. We evaluate first

$$ \Delta \varepsilon _{n} = \Delta C_{n} + D_{n+1} \eta ^{n+1} - D_{n} \eta ^{n} = \Delta C_{n} + \eta ^{n+1} \Delta D_{n} + (\eta - 1) \eta ^{n} D_{n}, $$

and claim that

$$ \Delta C_{n} + \eta ^{n+1} \Delta D_{n} \equiv 0, $$
(41)

which is the same result that we would obtain if the sequences \(C_{n}\) and \(D_{n}\) would be constant with n (note the strict analogy with the method of variation of parameters for differential equations). Thus, we are left with

$$ \Delta \varepsilon _{n} = - \beta \eta ^{n} D_{n}, $$
(42)

wherefrom we evaluate

$$ \Delta ^{2} \varepsilon _{n} = - \beta \eta ^{n} (\eta D_{n+1} - D _{n} ), $$
(43)

and hence, inserting (42) and (43) in (34) (rewritten in terms of Δ operators), we obtain by a little algebra and using again (32)

$$ - \lambda ^{2} \beta \eta ^{n+1} \Delta D_{n} = \chi _{n}, $$

i.e.,

$$ \Delta D_{n} = - \frac{\eta ^{-n-1}}{\lambda ^{2} \beta } \chi _{n}. $$
(44)

Note that

$$ \lambda ^{2} \beta = 2 \lambda (\lambda - 1). $$
(45)

The next step is to “integrate” (44), i.e., to sum up both sides from \(k=n\) to ∞, assuming (as a “constant of integration”) that \(D_{n} \to 0\) as \(n \to \infty \). We obtain

$$ \sum_{r=n}^{\infty } \Delta D_{r} = - D_{n} = - \frac{1}{\lambda ^{2} \beta } \sum _{r=n}^{\infty } \eta ^{-r-1} \chi _{r}, $$

and hence

$$ D_{n} = \frac{1}{\lambda ^{2} \beta } \sum_{r=n}^{\infty } \eta ^{-r-1} \chi _{r}. $$
(46)

Finally, we determine \(C_{n}\) from (41). We have first

$$ \Delta C_{n} = - \eta ^{n+1} \Delta D_{n} = \frac{\chi _{n}}{\lambda ^{2} \beta }, $$

and then, assuming \(C_{n} \to 0\) as \(n \to \infty \),

$$ \sum_{r=n}^{\infty } \Delta C_{r} = - C_{n} = \frac{1}{\lambda ^{2} \beta } \sum _{r=n}^{\infty } \chi _{r}. $$

Hence,

$$ C_{n} = - \frac{1}{\lambda ^{2} \beta } \sum_{r=n}^{\infty } \chi _{r}. $$
(47)

To be more precise, reintroducing \(\lambda = \lambda _{\pm }\), we have

$$ C_{n} \equiv C_{n}^{\pm } = - \frac{1}{\lambda _{\pm }^{2} \beta _{ \pm }} \sum_{r=n}^{\infty } \chi _{r}^{\pm }, $$
(48)

and thus, from (39),

$$ \varepsilon ^{\pm }_{n} = C^{\pm }_{n} + \eta _{\pm }^{n} D^{\pm }_{n} = - \frac{1}{2 \lambda _{\pm } (\lambda _{\pm } - 1)} \sum_{r=n}^{\infty } \bigl[1 - (\eta _{\pm })^{r-n+1} \bigr] \chi _{r}^{\pm }. $$
(49)

We conclude with the summary equation of the Volterra type

$$ \varepsilon ^{\pm }_{n} = c^{\pm } \sum _{r=n}^{\infty } K_{n,r}^{ \pm } \bigl(1 + \varepsilon ^{\pm }_{r} \bigr), $$
(50)

where we set

$$ c^{\pm } := - \frac{1}{2 \lambda _{\pm } (\lambda _{\pm } - 1)} = \mp \frac{1}{2 i \sqrt{a} (1 \pm i \sqrt{a})}, $$
(51)

and

$$ K_{n,r}^{\pm } := \textstyle\begin{cases} \lambda _{\pm }^{2} a_{n,r-2}^{\pm } h_{r-1} - \lambda _{\pm } a_{n,r-1} ^{\pm } (h_{r-1} + h_{r}) + a_{n,r}^{\pm } (g_{r} + h_{r}), \quad r \geq n+2, \\ - \lambda _{\pm } a_{n,n}^{\pm } (h_{n} + h_{n+1}) + a_{n,n+1}^{\pm } (g _{n+1} + h_{n+1}) \quad \text{for } r = n+1, \\ a_{n,n}^{\pm } (g_{n} + h_{n}) \quad \text{for } r = n, \end{cases} $$
(52)

with

$$ a_{n,r}^{\pm } := 1 - (\eta _{\pm })^{r-n+1} \quad \text{for } r \geq n. $$
(53)

We now prove that (50) has two linearly independent solutions and establish some estimates for them. Writing for convenience \(\varepsilon ^{\pm }(n)\) in place of \(\varepsilon _{n}^{\pm }\), equation (50) becomes

$$ \varepsilon ^{\pm }(n) = c^{\pm } \sum _{r=n}^{\infty } K_{n,r}^{\pm } \bigl(1 + \varepsilon ^{\pm }(r)\bigr). $$
(54)

Defining recursively the s-labeled sequence (of sequences) \(\{\varepsilon _{s}^{\pm }(n)\}_{s=0}^{\infty }\) (for every fixed n, \(n \geq \nu \)) as

$$ \varepsilon _{0}^{\pm }(n) \equiv 0, \qquad \varepsilon _{s+1}^{\pm }(n) := c^{\pm } \sum _{r=n}^{\infty } K_{n,r}^{\pm } \bigl(1 + \varepsilon _{s} ^{\pm }(r)\bigr), \quad s = 0,1,2,\ldots , $$
(55)

we can show by induction (on s) that all these series converge for every fixed \(n \geq \nu \). In fact, considering

$$ \varepsilon _{s+1}^{\pm }(n) - \varepsilon _{s}^{\pm }(n) = c^{\pm } \sum _{r=n}^{\infty } K_{n,r}^{\pm } \bigl[\varepsilon _{s}^{\pm }(r) - \varepsilon _{s-1}^{\pm }(r)\bigr] $$

for \(s \geq 1\), we can show inductively on s that

$$ \bigl\vert \varepsilon _{s+1}^{\pm }(n) - \varepsilon _{s}^{\pm }(n) \bigr\vert \leq ( \mathcal{V}_{n})^{s+1} $$
(56)

for \(s = 0,1,2,\ldots \) , \(n \geq \nu \), where \(\mathcal{V}_{n}\) is defined in (22) for \(a > 0\) and is replaced by \(\mathcal{V}_{n}^{\pm }\), defined by (25), (29), when \(a < 0\). In fact, we have first that

$$ \varepsilon _{1}^{\pm }(n) = c^{\pm } \sum _{r=n}^{\infty } K_{n,r}^{ \pm }, $$

hence

$$ \bigl\vert \varepsilon _{1}^{\pm }(n) \bigr\vert \leq \sum_{r=n}^{\infty } \bigl\vert c^{\pm } K_{n,r} ^{\pm } \bigr\vert \leq { \mathcal{V}}_{n}, $$
(57)

and then, assuming that (56) is satisfied with s replaced by \(s-1\), we have

$$\begin{aligned} \bigl\vert \varepsilon _{s+1}^{\pm }(n) - \varepsilon _{s}^{\pm }(n) \bigr\vert \leq& \sum _{r=n}^{\infty } \bigl\vert c^{\pm } K_{n,r}^{\pm } \bigr\vert \bigl\vert \varepsilon _{s}^{ \pm }(r) - \varepsilon _{s-1}^{\pm }(r) \bigr\vert \leq \sum_{r=n}^{\infty } \bigl\vert c ^{\pm } K_{n,r}^{\pm } \bigr\vert \mathcal{V}_{r}^{s} \\ \leq &{\mathcal{V}}_{n}^{s} \sum _{r=n}^{\infty } \bigl\vert c^{\pm } K_{n,r}^{ \pm } \bigr\vert \leq {\mathcal{V}}_{n}^{s+1}. \end{aligned}$$
(58)

In fact, \(\mathcal{V}_{n}\) decreases monotonically (to zero) as n grows (to ∞). Introducing the sequence

$$ \varepsilon ^{\pm }(n) := \sum_{s=0}^{\infty } \bigl[\varepsilon _{s+1}^{ \pm }(n) - \varepsilon _{s}^{\pm }(n)\bigr], $$
(59)

which is well defined in view of (58) whenever \(\mathcal{V} _{n} < 1\), a condition satisfied for all \(n \geq n_{0}\), since \(\mathcal{V}_{n}\) decreases, where \(n_{0}\) is the smallest index not less than ν such that \(\mathcal{V}_{n} < 1\). Similarly, when \(\mathcal{V}_{n}^{\pm }\) replaces \(\mathcal{V}_{n}\), it is understood that \(n \geq n_{0}^{\pm }\).

Moreover, it follows from (58) and (59) that

$$ \bigl\vert \varepsilon ^{\pm }(n) \bigr\vert \leq \frac{\mathcal{V}_{n}}{1 - \mathcal{V} _{n}}, \quad n \geq n_{0}. $$
(60)

Finally, we prove that the sequence \(\varepsilon ^{\pm }(n)\) solves the summary Volterra equation (54). In fact,

$$\begin{aligned} \varepsilon ^{\pm }(n) =& \varepsilon _{1}^{\pm }(n) + \sum_{s=1}^{ \infty } \bigl[\varepsilon _{s+1}^{\pm }(n) - \varepsilon _{s}^{\pm }(n) \bigr] \\ =& \sum_{r=n}^{\infty } c^{\pm } K_{n,r}^{\pm } + \sum_{s=1}^{\infty } \sum_{r=n}^{\infty } c^{\pm } K_{n,r}^{\pm } \bigl[\varepsilon _{s}^{\pm }(r) - \varepsilon _{s-1}^{\pm }(r)\bigr], \end{aligned}$$

and the proof follows immediately if we can show that it is permissible to interchange the order of summation in the previous relation. This is indeed the case in view of the Lebesgue dominated convergence theorem, since

$$ \Biggl\vert \sum_{s=1}^{S} c^{\pm } K_{n,r}^{\pm } \bigl[\varepsilon _{s}^{ \pm }(n) - \varepsilon _{s-1}^{\pm }(n) \bigr] \Biggr\vert \leq \bigl\vert c^{\pm } K_{n,r} ^{\pm } \bigr\vert \frac{\mathcal{V}_{n_{0}}}{1 - \mathcal{V}_{n_{0}}} $$

for any \(S > 1\) and \(r > n_{0}\), and using the summability condition (19) (or (27)).

To conclude the proof, the key is establishing in all cases an estimate for \(\sum_{r=n}^{\infty } |c^{\pm } K_{n,r}^{\pm }|\) as in (57), with a suitable \(\mathcal{V}_{n}\) [or \(\mathcal{V} _{n}^{\pm }\)]. We have first from (52), (53), for \(r \geq n+2\),

$$\begin{aligned}& \bigl\vert K_{n,r}^{\pm } \bigr\vert \leq \vert \lambda _{\pm } \vert ^{2} \bigl\vert a_{r-2}^{\pm } \bigr\vert \vert h_{r-1} \vert + \vert \lambda _{\pm } \vert \bigl\vert a_{r-1}^{\pm } \bigr\vert \bigl( \vert h_{r-1} \vert + \vert h_{r} \vert \bigr) + \bigl\vert a_{r} ^{\pm } \bigr\vert \bigl( \vert g_{r} \vert + \vert h_{r} \vert \bigr), \\& \bigl\vert K_{n,n+1}^{\pm } \bigr\vert \leq \vert \lambda _{\pm } \vert \bigl\vert a_{n,n}^{\pm } \bigr\vert \bigl( \vert h_{n} \vert + \vert h _{n+1} \vert \bigr) + \vert a_{n,n+1} \vert ^{\pm } \bigl( \vert g_{n+1} \vert + \vert h_{n+1} \vert \bigr), \\& \bigl\vert K_{n,n}^{\pm } \bigr\vert \leq \bigl\vert a_{n,n}^{\pm } \bigr\vert \bigl( \vert g_{n} \vert + \vert h_{n} \vert \bigr), \end{aligned}$$

and then

$$ \sum_{r=n}^{S} \bigl\vert c^{\pm } K_{n,r}^{\pm } \bigr\vert = \bigl\vert c^{\pm } \bigr\vert \sum_{r=n+2} ^{S} \bigl\vert K_{n,r}^{\pm } \bigr\vert + \bigl\vert c^{\pm } K_{n,n+1}^{\pm } \bigr\vert + \bigl\vert c^{\pm } K_{n,n} ^{\pm } \bigr\vert . $$

Finally, an estimate like

$$ \sum_{r=n}^{S} \bigl\vert c^{\pm } K_{n,r}^{\pm } \bigr\vert \leq { \mathcal{V}}_{n} \quad \bigl[\text{or } \mathcal{V}_{n}^{\pm } \bigr] $$

with \(\mathcal{V}_{n}\) [or \(\mathcal{V}_{n}^{\pm }\)] given by (22) [or (25), (29)] can be obtained upon lengthy though rather elementary calculations. Details are given in Appendix 1, where the various cases of a real, \(a \neq 0\), \(a \neq -1\) (i.e., \(a > 0\), \(-1 < a < 0\), and \(a > -1\)) are covered. □

Remark 3.1

Note that, \(W_{n}\) being decreasing, \(W_{n} \leq W_{n-1}\), in (22) we have also

$$ {\mathcal{V}}_{n} \leq \frac{1}{\sqrt{a(a + 1)}} \bigl(V_{n} + (\sqrt{a + 1} + 1)^{2} W_{n} \bigr). $$
(61)

In any case, \(\mathcal{V}_{n} = \mathcal{O}(V_{n}) + \mathcal{O}(W _{n-1})\). A similar observation can be made also in the case \(a < 0\).

The LG asymptotics for case (D)

Here we will use definitions (15), (16), but now with

$$ \eta _{\pm } := \frac{b \pm \sqrt{b^{2} - 4}}{b \mp \sqrt{b^{2} - 4}}, \quad b = a - 2, $$
(62)

instead of (17), and

$$ \lambda _{\pm } := - \frac{1}{2} \bigl(b \pm \sqrt{b^{2} - 4}\bigr), $$
(63)

which are the characteristic roots of the unperturbed equation associated to (D), that is, \(\lambda ^{2} + b \lambda + 1 = 0\) instead of (17) and (18). We can proceed in a strictly similar way for case (D). We report all steps in Appendix 2 for the reader’s convenience. We have thus the following.

Theorem 3.2

Let equation (D) be given for \(n \geq \nu \), for some fixed \(\nu \in {\mathbf{Z}}\), with the coefficients as in (13), with a real, \(a \neq 0, -1\), and \(c=1\), \(g_{n}\) and \(h_{n}\) real or complex. Then, under the conditions in (19) and, when needed, (27),

  1. (i)

    If \(0 < a < 4\) (i.e., \(b^{2} < 4\)), two linearly independent solutions of the form (20) exist for all \(n \geq n_{0}\), for some \(n_{0} \geq \nu \), with

    $$ \lambda _{\pm } = \frac{1}{2} \bigl[-b \pm i \sqrt{4 - b^{2}}\bigr], \qquad \lambda _{+} = 1/\lambda _{-} = \overline{\lambda }_{-}, \qquad \vert \lambda _{\pm } \vert = 1, \qquad \vert \eta _{\pm } \vert = \biggl\vert \frac{\lambda _{ \pm }}{\lambda _{\mp }} \biggr\vert = 1, $$

    hence

    $$ \bigl\vert a_{n,r}^{\pm } \bigr\vert = \bigl\vert 1 - (\eta _{\pm })^{r-n+1} \bigr\vert \leq 2, $$
    (64)

    and

    $$ \bigl\vert K_{n,r}^{\pm } \bigr\vert \leq 2 \vert h_{r-1} \vert + 2 \vert h_{r} \vert + \vert g_{r-1} \vert $$

    for \(r \geq n+2\), while

    $$ \bigl\vert K_{n,n+1}^{\pm } \bigr\vert \leq 2 \vert h_{n+1} \vert + \vert h_{n} \vert + \vert g_{n} \vert , \quad \textit{and} \quad \bigl\vert K_{n,n}^{\pm } \bigr\vert \leq \vert h_{n} \vert . $$

    Then

    $$ \sum_{r=n}^{\infty } \bigl\vert c^{\pm } K^{\pm }_{n,r} \bigr\vert \leq \frac{2}{\sqrt{a(4 - a)}} (V_{n} + 2 W_{n} + 2 W_{n+1} ) =: \mathcal{V} ^{\pm }_{n} \equiv { \mathcal{V}}_{n}. $$
    (65)

    Details are given in Appendix 3.

    When \(b^{2} > 4\), we have the two cases, \(a > 4\) (\(b > 2\)), and \(a < 0\) (\(b < -2\)).

  2. (ii)

    For \(a > 4\) (i.e., \(b > 2\)), \(|\lambda _{+}/\lambda _{-}| < 1\) (hence \(|\lambda _{-}/\lambda _{+}| > 1\)), and thus

    $$ \vert \eta _{+} \vert = \biggl\vert \frac{\lambda _{+}}{\lambda _{-}} \biggr\vert = \biggl\vert \frac{-b + \sqrt{b^{2} - 4}}{-b - \sqrt{b^{2} - 4}} \biggr\vert < 1, \qquad \vert \eta _{-} \vert = \frac{1}{ \vert \eta _{+} \vert }, $$

    and hence

    $$\begin{aligned}& \bigl\vert a_{n,r}^{+} \bigr\vert = \bigl\vert 1 - (\eta _{+})^{r-n+1} \bigr\vert = 1 - (\eta _{+})^{r-n+1} < 1, \\& \bigl\vert a_{n,r}^{-} \bigr\vert = \bigl\vert 1 - (\eta _{-})^{r-n+1} \bigr\vert \leq 1 + \vert \eta _{-} \vert ^{r-n+1}. \end{aligned}$$

    Therefore,

    $$\begin{aligned}& \bigl\vert K_{n,r}^{-} \bigr\vert \leq \lambda _{-} \bigl\vert ^{2} \bigl[1 + \vert \eta _{-} \vert ^{n-r-1}\bigr] \bigr\vert h_{r-1} \vert + |\lambda _{-}| \bigl[1 + \vert \eta _{-} \vert ^{n-r} \vert \bigr] \bigl( \vert g_{r-1} \vert + \vert h_{r-1} \vert + \vert h _{r} \vert \bigr) \\& \hphantom{\bigl\vert K_{n,r}^{-} \bigr\vert \leq}{}+ \bigl[1 + \vert \eta _{-} \vert ^{n-r+1}\bigr] \vert h_{r} \vert , \\& \bigl\vert K_{n,n+1}^{-} \bigr\vert \leq \vert \lambda _{-} \vert \bigl[1 + \vert \eta _{-} \vert \bigr] \bigl( \vert g_{n} \vert + \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) + \bigl[1 + \vert \eta _{-} \vert ^{2}\bigr] \vert h_{n+1} \vert , \\& \bigl\vert K_{n,n}^{-} \bigr\vert \leq \bigl(1 + \vert \eta _{-} \vert \bigr) \vert h_{n} \vert . \end{aligned}$$

    Collecting all these and recalling the definitions of \(V_{n}\), \(W_{n}\), \(Z^{-}_{n}\), and \(U^{-}_{n}\) (given in (19), (27)), we obtain upon lengthy calculations

    $$ \bigl\vert \varepsilon _{n}^{-} \bigr\vert \leq { \mathcal{V}}_{n}^{-}, $$

    with

    $$\begin{aligned}& \mathcal{V}_{n}^{-} := \bigl\vert c^{-} \bigr\vert \bigl[ \vert \lambda _{-} \vert V_{n+1} + \bigl( \vert \lambda _{-} \vert + 1\bigr) W_{n} + \vert \lambda _{-} \vert \bigl( \vert \lambda _{-} \vert + 1\bigr) W _{n+1} \\& \hphantom{\mathcal{V}_{n}^{-} :=}{}+ \vert \lambda _{-} \vert \vert \eta _{-} \vert \bigl( \vert \lambda _{-} \vert + \vert \eta _{-} \vert \bigr) U_{n+1} ^{-} \\& \hphantom{\mathcal{V}_{n}^{-} :=}{}+ \vert \eta _{-} \vert ^{2} \bigl( \vert \lambda _{-} \vert + \vert \eta _{-} \vert \bigr) U_{n+2}^{-} + \vert \lambda _{-} \vert \vert \eta _{-} \vert ^{2} Z^{-}_{n+1} \\& \hphantom{\mathcal{V}_{n}^{-} :=}{} + \vert \lambda _{-} \vert \bigl(1 + \vert \eta _{-} \vert \bigr) \vert g_{n} \vert + \bigl( \vert \lambda _{-} \vert + 1\bigr) \vert \eta _{-} \vert ) \vert h_{n} \vert + \vert \lambda _{-} \vert \bigl( \vert \lambda _{-} \vert + \vert \eta _{-} \vert \bigr) \vert h_{n+1} \vert \bigr]. \end{aligned}$$
    (66)

    The second solution is estimated simply as

    $$ \bigl\vert \varepsilon _{n}^{+} \bigr\vert \leq { \mathcal{V}}_{n}^{+} $$

    with

    $$ {\mathcal{V}}_{n}^{+} := \bigl\vert c^{+} \bigr\vert \bigl(V_{n} + 2 W_{n} + W_{n+1} + \vert h _{n+1} \vert \bigr). $$
    (67)

    Details are reported in Appendix 3.

  3. (iii)

    For \(a < 0\) [\(b < -2\)], we obtain upon some algebra

    $$ {\mathcal{V}}_{n}^{-} := \bigl\vert c^{-} \bigr\vert (V_{n} + 2 W_{n} + 2 W_{n+1} ), $$
    (68)

    while

    $$\begin{aligned}& \mathcal{V}^{+}_{n} := \vert c_{+} \vert \bigl[ \vert \lambda _{+} \vert V_{n} + \bigl(1 + \vert \lambda _{+} \vert \bigr) W_{n} + \vert \lambda _{+} \vert \bigl(1 + \vert \lambda _{+} \vert \bigr) W_{n+1} \\& \hphantom{\mathcal{V}^{+}_{n} :=}{}+ \eta _{+} \vert \lambda _{+} \vert \bigl(\eta _{+} + \vert \lambda _{+} \vert \bigr) U_{n+1}^{+} + ( \eta _{+})^{2} \bigl(\eta _{+} + \vert \lambda _{+} \vert \bigr) U_{n+2}^{+} \\& \hphantom{\mathcal{V}^{+}_{n} :=}{} + (\eta _{+})^{2} \vert \lambda _{+} \vert Z_{n+1}^{+} + \eta _{+} \bigl(1 + \vert \lambda _{+} \vert \bigr) \vert h_{n} \vert \\& \hphantom{\mathcal{V}^{+}_{n} :=}{} + \eta _{+} \bigl(\eta _{+} + \vert \lambda _{+} \vert \bigr) \vert h_{n+1} \vert + \eta _{+} \vert \lambda _{+} \vert \vert g_{n} \vert \bigr]. \end{aligned}$$
    (69)

Proof of Theorem 3.2

Referring to the detailed derivation reported in Appendix 3, we end up with the summary equation of Volterra type

$$ \varepsilon _{n}^{\pm } = c^{\pm } \sum _{r=n}^{\infty } K_{n,r}^{ \pm } \bigl(1 + \varepsilon _{r}^{\pm } \bigr), $$
(70)

where

$$ c^{\pm } := \mp \frac{\lambda _{\mp }}{\sqrt{b^{2} - 4}}, \qquad \lambda _{\pm } = - \frac{1}{2} \bigl(b \pm \sqrt{b^{2} - 4} \bigr), $$
(71)

and

$$ K_{n,r}^{\pm } := \textstyle\begin{cases} \lambda _{\pm }^{2} a_{n,r-2}^{\pm } h_{r-1} + \lambda _{\pm } a_{n,r-1} ^{\pm } (g_{r-1} - h_{r-1} - h_{r}) + a_{n,r}^{\pm } h_{r}, \quad r \geq n+2, \\ \lambda _{\pm } a_{n,n}^{\pm } (g_{n} - h_{n} - h_{n+1}) + a_{n,n+1} ^{\pm } h_{n+1} \quad \text{for } r = n+1, \\ a_{n,n}^{\pm } h_{n} \quad \text{for } r = n, \end{cases} $$
(72)

with \(a_{n,r}^{\pm } := 1 - (\eta _{\pm })^{r-n+1}\) for \(r \geq n\) (as in (53)), being

$$ \vert \eta _{\pm } \vert = \biggl\vert \frac{\lambda _{\pm }}{\lambda _{\mp }} \biggr\vert = \biggl\vert \frac{b \pm \sqrt{b^{2} - 4}}{b \mp \sqrt{b^{2} - 4}} \biggr\vert , $$
(73)

so that \(|\eta _{+}| > 1\) (and \(|\eta _{-}| < 1\)) when \(a > 0\), and \(|\eta _{-}| > 1\) (and \(|\eta _{+}| < 1\) when \(a < 0\).

Similarly to case (C), we can show also for case (D) that two linearly independent solutions \(\varepsilon _{n} ^{\pm }\) to (70) exist and can be estimated in a similar way.

The procedure is standard and follows exactly the same lines of the proof of Theorem 3.1, using now the appropriate values of \(\mathcal{V}_{n}\), \(\mathcal{V}_{n}^{\pm }\). □

Remark 3.2

Clearly, when \(h_{n} \equiv 0\), all formulae and error estimates greatly simplify. Summarizing, the definitions of \(\mathcal{V}_{n}\), \(\mathcal{V}_{n}^{\pm }\) given in (22), (25), (29), (65), (66), and (69) reduce, respectively, to the following.

For case (C).

  1. (i)

    Case (C), \(a > 0\):

    $$ {\mathcal{V}}_{n} := \frac{1}{\sqrt{a (a + 1)}} V_{n}. $$
    (74)
  2. (ii)

    Case (C), \(a < 0\) (with \(a \neq -1\)), first solution:

    $$ {\mathcal{V}}_{n}^{-} := \textstyle\begin{cases} \frac{1}{2 \sqrt{ \vert a \vert } (1 - \sqrt{ \vert a \vert })} (1 - \sqrt{ \vert a \vert }) V_{n}, \\ \quad \text{if } \vert a \vert < 1, \\ \frac{1}{\sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } - 1)} V_{n}, \quad \text{if } \vert a \vert > 1. \end{cases} $$
    (75)
  3. (iii)

    Case (C), \(a < 0\) (with \(a \neq -1\)), second solution:

    $$ {\mathcal{V}}_{n}^{+} := \frac{1}{2 \sqrt{ \vert a \vert } (1 + \sqrt{ \vert a \vert })} \bigl(V_{n} + \vert \eta _{+} \vert Z_{n}^{+} \bigr). $$
    (76)

    In particular, setting \(h_{n} \equiv 0\), we recover the estimate obtained in [14] (as it must be).

For case (D).

  1. (i)

    Case (D), \(0 < a < 4\) [i.e., \(b^{2} < 4\)]:

    $$ {\mathcal{V}}^{\pm }_{n} \equiv {\mathcal{V}}_{n} = \frac{2}{\sqrt{a(4 - a)}} V_{n}. $$
    (77)
  2. (ii)

    Case (D), \(a > 4\) [i.e., \(b > 2\)], first solution:

    $$ {\mathcal{V}}^{-}_{n} = \bigl\vert c^{-} \bigr\vert \bigl( \vert \lambda _{-} \vert \bigl(1 + \vert \eta _{-} \vert \bigr) V_{n} + \vert \lambda _{-} \vert \vert \eta _{-} \vert ^{3} Z^{-}_{n+1} \bigr), $$
    (78)

    where \(c^{-}\), \(\lambda _{-}\), and \(|\eta _{-}|\) are defined in (71) and (73).

  3. (iii)

    Case (D), \(a > 4\) [i.e., \(b > 2\)], second solution:

    $$ {\mathcal{V}}^{+}_{n} = \bigl\vert c^{+} \bigr\vert V_{n}, $$
    (79)

    where \(c^{+}\) is defined in (71);

  4. (iv)

    Case (D), \(a < 0\) [i.e., \(b < -2\)], first solution:

    $$ {\mathcal{V}}^{-}_{n} = \vert c_{-} \vert (V_{n} + 2 W_{n} + 2 W_{n+1}). $$
    (80)
  5. (v)

    Case (D), \(a < 0\) [i.e., \(b < -2\)], second solution (dominant solution):

    $$\begin{aligned} \mathcal{V}^{+}_{n} =& \vert c_{+} \vert \bigl\{ \vert \lambda _{+} \vert (1 + \eta _{+}) V _{n} + \bigl(1 + \vert \lambda _{+} \vert \bigr) (1 + \eta _{+}) W_{n} \bigl[ \vert \lambda _{+} \vert ^{2} + \vert \lambda _{+} \vert (1 + \eta _{+}) \\ &{} + \bigl(1 + (\eta _{+})^{2}\bigr)\bigr] W_{n+1} + W_{n+2} + \vert \lambda _{+} \vert \eta _{+} \bigl( \vert \lambda _{+} \vert + \eta _{+}\bigr) U^{+}_{n+1} \\ &{} + (\eta _{+})^{2} \bigl( \vert \lambda _{+} \vert + \eta _{+}\bigr) U^{+}_{n+2} + \vert \lambda _{+} \vert (\eta _{+})^{2} Z^{+}_{n+1} \bigr\} . \end{aligned}$$
    (81)

In the next section, we present a few examples to illustrate the results provided by Theorems 3.1 and 3.2.

Examples

The validity of Theorems 3.1 and 3.2, generally speaking, rests on the convergence of the series \(\sum_{r=n}^{\infty } |g_{r}|\), \(\sum_{r=n} ^{\infty } |h_{r}|\) for at least one of the basis’ solutions, the recessive solution, while the stronger conditions

$$ \vert \eta _{+} \vert ^{-n} \sum _{r=n}^{\infty } \vert \eta _{+} \vert ^{r} \vert g_{r} \vert < +\infty , \qquad \vert \eta _{+} \vert ^{-n} \sum _{r=n}^{\infty } \vert \eta _{+} \vert ^{r} \vert h_{r} \vert < +\infty , $$

where \(|\eta _{+}| > 1\), are required, correspondingly to the dominant solution. Recall that, if there exist two linearly independent solutions, say \(u_{n}^{-}\) and \(u_{n}^{+}\), of a linear second-order difference equation like (1) such that \(u_{n}^{-}/u_{n}^{+} \to 0\) as \(n \to \infty \), then \(u_{n}^{-}\) is called recessive and \(u_{n}^{+}\) is called dominant solution of (1), see Sect. 6.3, Definition 6.3.1, in [3, Ch. 6, p. 342].

In most examples below, we take either the same sequences \(g_{n} = h _{n} = \frac{1}{n(n+1)}\) or \(g_{n} = h_{n} = \frac{A^{-n}}{n^{2}}\) with suitable \(A > 1\). The purpose being of illustration, these choices are made just to allow simple explicit evaluation of all the needed quantities.

Example 4.1

A case of type (C) with \(a > 0\).

Consider

$$ \Delta \biggl( \biggl(1 + \frac{1}{n(n+1)} \biggr) \Delta y_{n} \biggr) + \biggl(1 + \frac{1}{n(n+1)} \biggr) y_{n} = 0, \quad n \in {\mathbf{Z}} _{1}, $$

that is, equations (C)–(13), with \(a=c=1\), \(g_{n} = h_{n} = \frac{1}{n(n+1)}\), \(\nu = 1\). We have from (15) \(V_{n} = W_{n} = \frac{1}{n}\), and hence, by (22), \(\mathcal{V}_{n} = \mathcal{O} (\frac{1}{n} )\). More precisely,

$$ \mathcal{V}_{n} = \frac{1}{\sqrt{2}} \biggl\{ \frac{1}{n} + ( \sqrt{2} + 1) \biggl[\sqrt{2} \frac{1}{n+1} + \frac{1}{n+2} \biggr] \biggr\} , $$

hence, e.g., \(\mathcal{V}_{10} \approx 0.42\), reduced to \(\mathcal{V} _{100} \approx 0.047\) for \(n=100\). Then, using (33) and (20), we have

$$ y_{n}^{\pm } = (1 \pm i)^{n} \bigl(1 + \varepsilon _{n}^{\pm }\bigr), $$

with the error estimates \(|\varepsilon _{n}^{\pm }| \lesssim 0.42\) for all \(n \geq 10\) and \(|\varepsilon _{n}^{\pm }| \lesssim 0.047\) for all \(n \geq 100\). Note that being \(\mathcal{V}_{3} \approx 1.18\) and \(\mathcal{V}_{4} \approx 0.94\), we have \(n_{0} = 4\), see (23).

Clearly, the smallness of the errors depends on the speed of convergence to zero of the series in (19). In fact, if, for instance, \(g_{n} = h_{n} = \frac{1}{n^{2}(n^{2}+1)}\), we have \(V_{n} = W_{n} = 1/n ^{2}\) and \(\mathcal{V}_{10} \lesssim 0.038\), \(\mathcal{V}_{100} \lesssim 3 \times 10^{-4}\).

Example 4.2

A case of type (C) with \(a < 0\) (\(a \neq -1\)).

Consider

$$ \Delta \biggl( \biggl(1 + \frac{A^{-n}}{n^{2}} \biggr) \Delta y_{n} \biggr) + \biggl(-2 + \frac{A^{-n}}{n^{2}} \biggr) y_{n} = 0, \quad n\in {\mathbf{Z}}_{1}, $$

that is, equations (C)–(13) with \(a=-2\), and again \(c=1\), \(g_{n} = h_{n} = \frac{A^{-n}}{n^{2}}\), where \(A = 2 |\eta _{+}| = 2 (3 + 2 \sqrt{2}) \approx 11.6568\). Thus, there are two linearly independent solutions \(y_{n}^{\pm }\) with

$$ y_{n}^{+} = (1 + \sqrt{2})^{n} \bigl(1 + \varepsilon _{n}^{+}\bigr), \qquad y _{n}^{-} = (1 - \sqrt{2})^{n} \bigl(1 + \varepsilon _{n}^{-} \bigr) $$

for \(n \geq n_{0}^{+}\) and \(n \geq n_{0}^{-}\), respectively, with \(n_{0}^{\pm }\) given by (26), with error terms estimated by \(|\varepsilon _{n}| \leq {\mathcal{V}}_{n}^{\pm }\), according to (24), (25) and (28), (29). Note that, being

$$ \biggl\vert \frac{y_{n}^{-}}{y_{n}^{+}} \biggr\vert \sim \biggl\vert \frac{1- \sqrt{2}}{1+\sqrt{2}} \biggr\vert ^{n} \to 0, \quad \text{as } n \to \infty , $$

the solution \(y_{n}^{-}\) is recessive and \(y_{n}^{+}\) is dominant. Now,

$$ V_{n} = W_{n} \leq 2 \frac{(2 \vert \eta _{+} \vert )^{-n}}{n} $$

and

$$\begin{aligned} Z_{n}^{+} =& U_{n}^{+} = \vert \eta _{+} \vert ^{-n} \sum _{r=n}^{\infty } \vert \eta _{+} \vert ^{r} \frac{(2 \vert \eta _{+} \vert )^{-r}}{r^{2}} \\ < & \vert \eta _{+} \vert ^{-n} \sum _{r=n}^{\infty } \frac{1}{2^{r} r^{2}} \leq \vert \eta _{+} \vert ^{-n} Li_{2}(1/2) \approx 0.5822 \times (5.8284)^{-n}, \end{aligned}$$

where we used the dilogarithm

$$ Li_{2}(x) := \sum_{r=1}^{\infty } \frac{x^{r}}{r^{2}} $$

[7, § 25.15], [2, § 27.7]. It is known that \(Li_{2}(1/2) = \pi ^{2}/12-(\ln ^{2} 2)/2 \approx 0.5822\) [1]. Therefore, we obtain from (30), by rather tedious but elementary computations, \(\mathcal{V}_{3}^{+} \lesssim 0.0296\) and \(\mathcal{V}_{10}^{+} \lesssim 0.01280 \times 10^{-6}\).

As for the second solution, we have from (25) \(\mathcal{V} _{3}^{-} \approx 0.15 \times 10^{-2}\). The smallness achieved already for \(n=3\) is due to the fast decay of \(g_{n}\) and \(h_{n}\). Indeed, if, instead, \(g_{n} = h_{n} = 1/n^{2}\), and hence \(V_{n} = W_{n} < 2/n\), we obtain \(\mathcal{V}_{3}^{-} \lesssim 3.208\) (that is useless, see (23)), while \(\mathcal{V}_{10}^{-} \lesssim 0.5859\), but the corresponding estimates for \(\mathcal{V}_{n}^{+}\) blow up.

Example 4.3

A case of type (D) with \(0 < a < 4\) (i.e., \(b^{2} < 4\)).

Consider

$$ \Delta \biggl( \biggl(1 + \frac{1}{n(n+1)} \biggr) \Delta y_{n} \biggr) + \biggl(2 + \frac{1}{n \ln ^{2} n} \biggr) y_{n+1} = 0, \quad n\in {\mathbf{Z}}_{2}, $$

that is, equations (D)–(13), with \(\nu =2\), \(a=2\), \(c=1\), \(g_{n} = \frac{1}{n \ln ^{2} n}\), \(h_{n} = \frac{1}{n(n+1)}\). We have

$$ y_{n}^{\pm } = (\pm i)^{n} \bigl(1 + \varepsilon _{n}^{\pm }\bigr), $$

where the error terms can be bounded through (100),

$$ \mathcal{V}_{n}^{\pm } \equiv {\mathcal{V}}_{n} = V_{n} + 2 W_{n} + 2 W_{n+1} < \frac{1}{\ln n} + \frac{2}{n} + \frac{2}{n+1}, $$

since we can estimate \(V_{n}\) with \(V_{n} < \int _{n}^{+\infty } \frac{dx}{x \ln ^{2} x} = \frac{1}{\ln n}\) for every \(n \geq 1\). We then have \(\mathcal{V}_{10} \lesssim 0.816\), \(\mathcal{V}_{20} \lesssim 0.529\), \(\mathcal{V}_{30} \lesssim 0.4251\), \(\mathcal{V}_{50} \lesssim 0.3348\), \(\mathcal{V}_{100} \lesssim 0.2569\).

Example 4.4

A case of type (D) with \(a > 4\) (i.e., \(b > 2\)).

Consider

$$ \Delta \biggl( \biggl(1 + \frac{1}{n(n+1)} \biggr) \Delta y_{n} \biggr) + \biggl(4.1 + \frac{1}{n(n+1)} \biggr) y_{n+1} = 0, \quad n\in {\mathbf{Z}} _{1}, $$

that is, equations (D)–(13), with \(\nu =1\), \(a=4.1\) (hence \(b = 2.1\)), \(c=1\), \(g_{n} = h_{n} = \frac{1}{n(n+1)}\). Therefore, we have

$$ y_{n}^{+} = (-0.1266\ldots)^{n} \bigl(1 + \varepsilon _{n}^{+}\bigr), \qquad y_{n} ^{-} = (-1.9733\ldots)^{n} \bigl(1 + \varepsilon _{n}^{-}\bigr). $$

These solutions are recessive and dominant, respectively. We have \(V_{n} = W_{n} = \frac{1}{n}\), and from (101), (103), \(|c^{+}| \approx 1.1398\), \(|c^{-}| \approx 2.1398\), \(|\lambda _{-}| \approx 1.8262\), \(|\eta _{-}| \approx 6.6698\). We have from (104) \(\mathcal{V}_{10}^{+} \approx 0.4541\), \(\mathcal{V}_{100}^{+} \approx 0.0455\), while the estimate for \(\mathcal{V}_{n}^{-}\) blows up.

Taking instead \(g_{n} = h_{n} = \frac{(2 |\eta _{-}|)^{-n}}{n^{2}} \approx \frac{1}{n^{2}(13.3396)^{n}}\), so that \(Z_{n}^{-} = U_{n}^{-} = |\eta _{-}|^{-n} \sum_{r=n}^{\infty } \frac{1}{2^{r} r^{2}} \approx 0.5822 \times (6.6698)^{-n}\) (using the dilogarithm), we obtain from (104) \(\mathcal{V}_{3}^{+} \lesssim 0.96 \times 10^{-5}\), already very small, and from (102) \(\mathcal{V}_{3}^{-} \lesssim 0.1561\).

Example 4.5

A case of type (D) with \(a < 0\) (i.e., \(b < -2\)).

Consider finally

$$ \Delta \biggl( \biggl(1 + \frac{1}{n(n+1)} \biggr) \Delta y_{n} \biggr) + \biggl(-2 + \frac{1}{n(n+1)} \biggr) y_{n+1} = 0, \quad n\in {\mathbf{Z}} _{1}, $$

that is, equations (D)–(13), with \(\nu =1\), \(a = -2\) (hence \(b = -4\)), \(c=1\), and compute from (105), (107) \(|c^{-}| = \sqrt{3}/3 - 1/2 \approx 0.0773\), \(|c^{+}| = \sqrt{3}/3 + 1/2 \approx 1.0773\), \(|\lambda _{+}| = 2 + \sqrt{3} \approx 3.7320\), \(|\eta _{+}| = (2 + \sqrt{3})/(2 - \sqrt{3}) \approx 13.9282\). Here we have

$$ y_{n}^{+} = (3.732\ldots)^{n} \bigl(1 + \varepsilon _{n}^{+}\bigr), \quad y_{n}^{-} = (0.02679\ldots)^{n} \bigl(1 + \varepsilon _{n}^{-} \bigr). $$

Here again we can see that these solutions are dominant and recessive, respectively. Taking \(g_{n} = h_{n} = \frac{1}{n(n+1)}\), using (108) and (106), we can evaluate \(\mathcal{V} _{10}^{-} \lesssim 0.0373\), but the estimate for \(\mathcal{V}_{n}^{+}\) blows up.

Taking \(g_{n} = h_{n} = \frac{(2 |\eta _{+}|)^{-n}}{n^{2}} \approx \frac{1}{n ^{2} (27.8564)^{n}}\) instead, we have \(V_{n} = W_{n} \approx \sum_{r=n} ^{\infty } \frac{1}{r^{2} (27.8564)^{r}} \leq \frac{2}{n (27.8564)^{n}}\), \(Z_{n}^{+} = U_{n}^{+} \approx 0.5855 \times |\eta _{+}|^{n} \approx 0.5822 \times (13.9282)^{n}\). We can then evaluate by (108) and (106) \(\mathcal{V}_{3} ^{-} \lesssim 0.7 \times 10^{-5}\), while \(\mathcal{V}_{3}^{+} \lesssim 0.0324\).

Remark 4.1

Note that in all estimates established in Theorems 3.1 and 3.2, we have \(|g_{n}| = o(V_{n})\), etc., hence the terms like \(V_{p}\) dominate over those containing \(g_{q}\), and \(W_{p}\) dominate over those with \(h_{q}\). Also, \(V_{n}\) and \(W_{n}\) being monotonic decreasing functions of n, we have, e.g., \(V_{n+1} < V_{n}\), etc., hence, \(V_{n}\) dominates over \(V_{n+1}\). Note that also the sequences \(Z_{n}^{\pm }\) and \(U_{n}^{\pm }\) decrease monotonically with n.

Remark 4.2

The “double asymptotic nature” of the classical LG approximations [9] can be extended to the present discrete version, as it was done in [14] for equation (A), but not in all cases, in particular when there are perturbations of the coefficient \(c_{n}\). Indeed, we can see that, in case (C), \(\mathcal{V}_{n} = \mathcal{O}(a^{-1})\) as \(a \to +\infty \) (with fixed n), but only if \(h_{n} \equiv 0\), otherwise we have merely \(\mathcal{V}_{n} = \mathcal{O}(1)\). The same results are obtained for \(\mathcal{V}_{n}^{\pm }\), when, still in case (C), \(a \to -\infty \).

In case (D), we have, as \(a \to +\infty \), \(\mathcal{V}_{n} ^{-} = \mathcal{O}(a^{-1})\) and \(\mathcal{V}_{n}^{+} = \mathcal{O}(a ^{-2})\), while for \(a \to -\infty \), \(\mathcal{V}_{n} = \mathcal{O}(|a|^{-1})\) and \(\mathcal{V}_{n} = \mathcal{O}(a^{-2})\).

Summary

A Liouville–Green (LG, or WKB) asymptotic theory has been developed for some classes of linear second-order difference equations on the basis of certain canonical forms, into which rather general linear three-term recurrences can be transformed. Both recessive and dominant solutions are included. Rigorous and computable bounds have been derived for the error terms appearing in the asymptotic representations of solutions. These bounds may also be useful for the numerical evaluation of solutions. The results are similar, in general, to those obtained in the analogous cases of differential equations.

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Appendices

Appendix 1

In this appendix we give details, rather elementary but lengthy, pertaining to the proof of Theorem 3.1, that is, to case (C) of equation \(\Delta (c_{n} \Delta y_{n} ) + r_{n} y_{n} = 0\). We have

  1. (i)

    for \(a > 0\),

    $$ \bigl\vert c^{\pm } \bigr\vert = \frac{1}{2 \sqrt{a (a + 1)}}, \qquad \vert \lambda _{\pm } \vert = \sqrt{a + 1}, \qquad \vert \eta _{\pm } \vert = 1, $$

    and hence

    $$ \bigl\vert a^{\pm }_{n,r} \bigr\vert \equiv 1 - (\eta _{\pm } )^{r-n+1} \leq 2. $$

    Thus, for \(r \geq n+2\),

    $$\begin{aligned}& \bigl\vert c^{\pm } K_{n,r}^{\pm } \bigr\vert \\& \quad \leq\frac{1}{2 \sqrt{a(a + 1)}} \bigl\{ \vert \lambda _{\pm } \vert ^{2} |a^{ \pm }_{n,r-2} \vert h_{r-1} \vert + |\lambda _{\pm }| \bigl|a^{\pm }_{n,r-1}\bigr| \bigl( \vert h_{r-1} \vert + \vert h_{r} \vert \bigr) \\& \qquad {}+ \bigl|a^{\pm}_{n,r}\bigr| \bigl( \vert g_{r} \vert + \vert h_{r} \vert \bigr) \bigr\} \\& \quad \leq \frac{1}{\sqrt{a(a + 1)}} \bigl\{ (a + 1) \vert h_{r-1} \vert + \sqrt{a + 1} \bigl( \vert h_{r-1} \vert + \vert h_{r} \vert \bigr) + \vert g_{r} \vert + \vert h_{r} \vert \bigr\} ; \\& \bigl\vert c^{\pm } K_{n,n+1}^{\pm } \bigr\vert \leq \frac{1}{2 \sqrt{a(a + 1)}} \bigl\{ \vert \lambda _{\pm } \vert \bigl\vert a^{\pm }_{n,n} \bigr\vert \bigl( \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) + \bigl\vert a ^{\pm }_{n,n+1} \bigr\vert \bigl( \vert g_{n+1} \vert + \vert h_{n+1} \vert \bigr) \bigr\} \\& \hphantom{\bigl\vert c^{\pm } K_{n,n+1}^{\pm } \bigr\vert } \leq \frac{1}{\sqrt{a(a + 1)}} \bigl\{ \sqrt{a + 1} \bigl( \vert h_{|} + |h _{n+1} \vert \bigr) + \vert g_{n+1} \vert + \vert h_{n+1} \vert \bigr\} ; \\& \bigl\vert c^{\pm } K_{n,n}^{\pm } \bigr\vert \leq \frac{1}{2 \sqrt{a(a + 1)}} \bigl\vert a ^{\pm }_{n,n} \bigr\vert \bigl( \vert g_{n+1} \vert + \vert h_{n+1} \vert \bigr) \leq \frac{1}{ \sqrt{a(a + 1)}} \bigl( \vert g_{n+1} \vert + \vert h_{n+1} \vert \bigr). \end{aligned}$$

    Collecting all the previous terms, upon some little algebra and recalling the definitions of \(V_{n}\) and \(W_{n}\), we have

    $$\begin{aligned}& \sum_{r=n}^{\infty } \bigl\vert c^{\pm } K^{\pm }_{n,r} \bigr\vert \\& \quad = \sum _{r=n+2}^{\infty } \bigl\vert c^{\pm } K^{\pm }_{n,r} \bigr\vert + \bigl\vert c^{\pm } K^{\pm }_{n,n+1} \bigr\vert + \bigl\vert c^{ \pm } K^{\pm }_{n,n} \bigr\vert \\& \quad \leq \frac{1}{\sqrt{a(a + 1)}} \Biggl[(a+1) \sum _{r=n+2}^{\infty } \vert h_{r-1} \vert + \sqrt{a + 1} \Biggl(\sum_{r=n+2}^{\infty } \vert h_{r-1} \vert + \sum_{r=n+2}^{\infty } \vert h_{r} \vert \Biggr) \\& \qquad {}+ \sum_{r=n+2}^{\infty } \vert g_{r} \vert + \sum_{r=n+2}^{\infty } \vert h_{r} \vert + \sqrt{a + 1} \bigl( \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) \\& \qquad {} + \vert g_{n+1} \vert + \vert h _{n+1} \vert + \vert g_{n} \vert + \vert h_{n} \vert \Biggr] \\& \quad = \frac{1}{\sqrt{a(a + 1)}} \bigl\{ V_{n} + (\sqrt{a + 1} + 1) [\sqrt{a + 1} W_{n+1} + W_{n+2} ] \bigr\} =: \mathcal{V}^{\pm }_{n} \equiv {\mathcal{V}}_{n}. \end{aligned}$$
    (82)

    Here we used the fact that \(\sum_{r=n+2}^{\infty } |h_{r-1}| = \sum_{j=n+1}^{\infty } |h_{j}| = W_{n+1}\), etc., and that \(V_{n+2} + |g _{n+1}| + |g_{n}| = V_{n}\) and \(W_{n+2} + |h_{n+1}| + |h_{n}| = W_{n}\). It is worth noting that \(W_{n}\) decreases as n grows, hence, e.g., \(W_{n+1}\) could be estimated by \(W_{n}\).

  2. (ii)

    for \(-1 < a < 0\) (i.e., \(a < 0\), \(|a| < 1\)),

    $$ \bigl\vert c^{-} \bigr\vert = \frac{1}{2 \sqrt{ \vert a \vert } (1 - \sqrt{ \vert a \vert })}, \qquad \vert \lambda _{-} \vert = 1 - \sqrt{ \vert a \vert }, \qquad \vert \eta _{-} \vert < 1 \quad (\eta _{-} > 0), $$

    and hence

    $$ \bigl\vert a^{-}_{n,r} \bigr\vert < 1. $$

    Thus, using (52), we have, for \(r \geq n+2\),

    $$\begin{aligned}& \bigl\vert c^{-} K_{n,r}^{-} \bigr\vert \\& \quad < \frac{1}{2 \sqrt{ \vert a \vert } (1 - \sqrt{ \vert a \vert })} \bigl[\bigl(1 - \sqrt{ \vert a \vert } \bigr)^{2} \vert h_{r-1} \vert + \bigl(1 - \sqrt{ \vert a \vert }\bigr) \bigl( \vert h_{r-1} \vert + \vert h_{r} \vert \bigr) + \vert g_{r} \vert + \vert h_{r} \vert \bigr]; \\& \bigl\vert c^{-} K_{n,n+1}^{-} \bigr\vert < \frac{1}{2 \sqrt{ \vert a \vert } (1 - \sqrt{ \vert a \vert })} \bigl[\bigl(1 - \sqrt{ \vert a \vert }\bigr) \bigl( \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) + \vert g_{n+1} \vert + \vert h_{n+1} \vert \bigr]; \\& \bigl\vert c^{-} K_{n,n}^{-} \bigr\vert < \frac{1}{2 \sqrt{ \vert a \vert } (1 - \sqrt{ \vert a \vert })} \bigl( \vert g_{n} \vert + \vert h_{n} \vert \bigr), \end{aligned}$$

    and then

    $$\begin{aligned}& \sum_{r=n}^{\infty } \bigl\vert c^{-} K^{-}_{n,r} \bigr\vert \\& \quad < \frac{1}{2 \sqrt{ \vert a \vert } (1 - \sqrt{ \vert a \vert })} \Biggl[\bigl(1 - \sqrt{ \vert a \vert } \bigr)^{2} \sum_{r=n+2}^{ \infty } \vert h_{r-1} \vert \\& \qquad {}+ \bigl(1 - \sqrt{ \vert a \vert }\bigr) \Biggl(\sum _{r=n+2}^{\infty } \vert h_{r-1} \vert + \sum_{r=n+2}^{\infty } \vert h_{r} \vert \Biggr) + \sum_{r=n+2}^{\infty } \vert g _{r} \vert + \sum_{r=n+2}^{\infty } \vert h_{r} \vert \\& \qquad {} + \bigl(1 - \sqrt{ \vert a \vert }\bigr) \bigl( \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) + \vert g_{n+1} \vert + \vert h _{n+1} \vert + \vert g_{n} \vert + \vert h_{n} \vert \Biggr] \\& \quad = \frac{1}{2 \sqrt{ \vert a \vert } (1 - \sqrt{ \vert a \vert })} \bigl[V_{n} + \bigl(2 - \sqrt{ \vert a \vert }\bigr) W_{n} + \bigl(1 - \sqrt{ \vert a \vert }\bigr) \bigl(2 - \sqrt{ \vert a \vert }\bigr) W _{n+1} \bigr] \\& \quad=: \mathcal{V}^{-}_{n}. \end{aligned}$$
    (83)

    Similarly (still for \(-1 < a < 0\)),

    $$\begin{aligned}& \bigl\vert c^{+} \bigr\vert = \frac{1}{2 \sqrt{ \vert a \vert } (1 + \sqrt{ \vert a \vert })}, \qquad \vert \lambda _{+} \vert = 1 + \sqrt{ \vert a \vert } > 1,\\& \vert \eta _{+} \vert = \biggl\vert \frac{1 + \sqrt{ \vert a \vert }}{1 - \sqrt{ \vert a \vert }} \biggr\vert > 1 \quad (\eta _{+} > 0), \end{aligned}$$

    and hence

    $$ \bigl\vert a^{+}_{n,r} \bigr\vert < 1 + (\eta _{+})^{r-n+1}. $$

    (Actually, we could do better since \(|a_{n,r}^{+}| = (\eta _{+})^{r-n+1} - 1\). We leave this possible improvement to the reader.) Then, for \(r \geq n+2\),

    $$\begin{aligned}& \bigl\vert c^{+} K_{n,r}^{+} \bigr\vert \\& \quad \leq \frac{1}{2 \sqrt{ \vert a \vert } (1 + \sqrt{ \vert a \vert })} \\& \qquad {}\times \bigl[\bigl(1 + \sqrt{ \vert a \vert }\bigr)^{2} \bigl(1 + (\eta _{+})^{r-n-1}\bigr) \vert h_{r-1} \vert + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \bigl(1 + (\eta _{+})^{r-n}\bigr) \bigl( \vert h_{r-1} \vert + \vert h_{r} \vert \bigr) \\& \qquad {} + \bigl(1 + (\eta _{+})^{r-n+1}\bigr) \bigl( \vert g_{r} \vert + \vert h_{r} \vert \bigr) \bigr]; \\& \bigl\vert c^{+} K_{n,n+1}^{+} \bigr\vert \\& \quad \leq \frac{1}{2 \sqrt{ \vert a \vert } (1 + \sqrt{ \vert a \vert })} \\& \qquad {}\times \bigl[\bigl(1 + \sqrt{ \vert a \vert }\bigr) (1 + \eta _{+}) \bigl( \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) + \bigl(1 + (\eta _{+})^{2} \bigr) ( \vert g_{n+1} \vert + \vert h_{n+1} \vert \bigr]; \\& \bigl\vert c^{+} K_{n,n}^{+} \bigr\vert \leq \frac{1}{2 \sqrt{ \vert a \vert } (1 + \sqrt{ \vert a \vert })} \bigl[(1 + \eta _{+}) ( \vert g _{n} \vert + \vert h_{n} \vert \bigr]. \end{aligned}$$

    After rather lengthy though elementary calculations, we have

    $$\begin{aligned}& \sum_{r=n}^{\infty } \bigl\vert c^{+} K^{+}_{n,r} \bigr\vert \\& \quad \leq \frac{1}{2 \sqrt{ \vert a \vert } (1 + \sqrt{ \vert a \vert })} \Biggl[\bigl(1 + \sqrt{ \vert a \vert } \bigr)^{2} \sum_{r=n+2}^{ \infty } \vert h_{r-1} \vert \\& \qquad {}+ \bigl(1 + \sqrt{ \vert a \vert }\bigr)^{2} (\eta _{+})^{-n-1} \sum_{r=n+2}^{\infty } ( \eta _{+})^{r} \vert h_{r-1} \vert + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \sum_{r=n+2}^{\infty } \vert h _{r-1} \vert \\& \qquad {}+ \bigl(1 + \sqrt{ \vert a \vert }\bigr) \sum_{r=n+2}^{\infty } \vert h_{r} \vert + \bigl(1 + \sqrt{ \vert a \vert }\bigr) (\eta _{+})^{-n} \sum_{r=n+2}^{\infty } (\eta _{+})^{r} \vert h_{r-1} \vert \\& \qquad {} + \bigl(1 + \sqrt{ \vert a \vert }\bigr) (\eta _{+})^{-n} \sum_{r=n+2}^{\infty } (\eta _{+})^{r} \vert h _{r} \vert + \sum_{r=n+2}^{\infty } \vert g_{r} \vert \\& \qquad {} + \sum_{r=n+2}^{\infty } \vert h_{r} \vert + ( \eta _{+})^{-n} \sum_{r=n+2}^{\infty } (\eta _{+})^{r} \vert g_{r} \vert + (\eta _{+})^{-n-1} \sum_{r=n+2}^{\infty } (\eta _{+})^{r} \vert h_{r} \vert \\& \qquad {}+ \bigl(1 + \sqrt{ \vert a \vert }\bigr) \vert h_{n} \vert + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \eta _{+} \vert h_{n} \vert + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \vert h_{n+1} \vert \\& \qquad {}+ \bigl(1 + \sqrt{ \vert a \vert }\bigr) \eta _{+} \vert h_{n+1} \vert + \vert g_{n+1} \vert + \vert h_{n+1} \vert + (\eta _{+})^{2} \vert g_{n+1} \vert + (\eta _{+})^{2} \vert h_{n+1} \vert \\& \qquad {}+ \vert g_{n} \vert + \vert h_{n} \vert + \eta _{+} \vert g_{n} \vert + \eta _{+} \vert h_{n} \vert \Biggr] \\& \quad = \frac{1}{2 \sqrt{ \vert a \vert } (1 + \sqrt{ \vert a \vert })} \bigl\{ \bigl[\bigl(1 + \sqrt{ \vert a \vert } \bigr)^{2} + 2 \bigl(1 + \sqrt{ \vert a \vert }\bigr)\bigr] W_{n+1} + W_{n} + V_{n} \\& \qquad {}+ \bigl(1 + \sqrt{ \vert a \vert }\bigr) \eta _{+} \vert h_{n+1} \vert + (\eta _{+})^{2} \vert h_{n+1} \vert + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \vert h_{n} \vert + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \eta _{+} \vert h_{n} \vert \\& \qquad {} + \eta _{+} \vert h_{n} \vert + \bigl[\bigl(1 + \sqrt{ \vert a \vert }\bigr)^{2} \eta _{+} + \bigl(1 + \sqrt{ \vert a \vert }\bigr) (\eta _{+})^{2}\bigr] U_{n+1}^{+} \\& \qquad {}+ \bigl[\bigl(1 + \sqrt{ \vert a \vert }\bigr) (\eta _{+})^{2} + ) \bigl(\eta ^{3}\bigr)\bigr] U_{n+2}^{+} + (\eta _{+})^{3} Z_{n+2}^{+} + (\eta _{+})^{2} \vert g_{n+1} \vert + \eta _{+} \vert g_{n} \vert \bigr\} \\& \quad = \frac{1}{2 \sqrt{ \vert a \vert } (1 + \sqrt{ \vert a \vert })} \bigl\{ V_{n} + W _{n} + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \bigl(3 + \sqrt{ \vert a \vert } \bigr) W_{n+1} \\& \qquad {} + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \bigl(1 + \sqrt{ \vert a \vert } + \eta _{+}\bigr) \eta _{+} U ^{+}_{n+1} + (\eta _{+})^{2} \bigl(1 + \sqrt{ \vert a \vert } + \eta _{+}\bigr) U^{+} _{n+2} \\& \qquad {}+ (\eta _{+})^{3} Z^{+}_{n+2} + \eta _{+} \bigl(1 + \sqrt{ \vert a \vert } + \eta _{+}\bigr) \vert h_{n+1} \vert + \bigl[\bigl(1 + \sqrt{ \vert a \vert }\bigr) (1 + \eta _{+}) + \eta _{+}\bigr] \vert h_{n+1} \vert \\& \qquad {}+ (\eta _{+})^{2} \vert g _{n+1} \vert + \eta _{+} \vert g_{n} \vert \bigr\} \\& \quad =: \mathcal{V}^{+}_{n}. \end{aligned}$$
    (84)

    Here above we used the definitions of \(V_{n}\), \(W_{n}\), \(U_{n}^{-}\), and \(Z_{n}^{-}\) given by (15), (16).

  3. (iii)

    for \(a < -1\) (i.e., \(a < 0\), \(|a| > 1\)),

    $$ \bigl\vert c^{-} \bigr\vert = \biggl\vert \frac{1}{2 \sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } - 1)}, \qquad |\lambda _{-}| = \sqrt{ \vert a \vert } - 1, \quad \vert \eta _{-} \vert < 1 \quad ( \eta _{-} < 0), $$

    and hence

    $$ \bigl\vert a^{-}_{n,r} \bigr\vert = \bigl\vert 1 - (\eta _{-})^{r-n+1} \bigr\vert \leq 1 + \vert \eta _{-} \vert ^{r-n+1} < 2. $$

    Thus, for \(r \geq n+2\),

    $$\begin{aligned}& \bigl\vert c^{-} K_{n,r}^{-} \bigr\vert < \frac{1}{\sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } - 1)} \\& \hphantom{\bigl\vert c^{-} K_{n,r}^{-} \bigr\vert < }{}\times \bigl[\bigl(\sqrt{ \vert a \vert } - 1\bigr)^{2} \vert h_{r-1} \vert + \bigl(\sqrt{ \vert a \vert } - 1\bigr) \bigl( \vert h _{r-1} \vert + \vert h_{r} \vert \bigr) + \vert g_{r} \vert + \vert h_{r} \vert \bigr]; \\& \bigl\vert c^{-} K_{n,n+1}^{-} \bigr\vert < \frac{1}{\sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } - 1)} \bigl[\bigl(\sqrt{ \vert a \vert } - 1\bigr) \bigl( \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) + \vert g_{n+1} \vert + \vert h _{n+1} \vert \bigr]; \\& \bigl\vert c^{-} K_{n,n}^{-} \bigr\vert < \frac{1}{\sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } - 1)} \bigl( \vert g_{n} \vert + \vert h_{n} \vert \bigr). \end{aligned}$$

    Summing up, we have

    $$\begin{aligned}& \sum_{r=n}^{\infty } \bigl\vert c^{-} K^{-}_{n,r} \bigr\vert \\& \quad < \frac{1}{\sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } - 1)} \Biggl[\bigl(\sqrt{ \vert a \vert } - 1 \bigr)^{2} \sum_{r=n+2}^{\infty } \vert h_{r-1} \vert \\& \qquad {}+ \bigl(\sqrt{ \vert a \vert } - 1\bigr) \Biggl(\sum_{r=n+2}^{\infty } \vert h_{r-1} \vert + \sum_{r=n+2}^{\infty } \vert h_{r} \vert \Biggr) + \sum_{r=n+2}^{\infty } \vert g_{r} \vert + \sum_{r=n+2}^{\infty } \vert h _{r} \vert \\& \qquad {} + \bigl(\sqrt{ \vert a \vert } - 1\bigr) \bigl( \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) + \vert g_{n+1} \vert + \vert h_{n+1} \vert + \vert g_{n} \vert + \vert h_{n} \vert \Biggr] \\& \quad = \frac{1}{\sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } - 1)} \bigl[\bigl(\sqrt{ \vert a \vert } - 1 \bigr)^{2} W_{n+1} + \bigl(\sqrt{ \vert a \vert } - 1 \bigr) (W_{n+1} + W_{n+2}) + V_{n+2} \\& \qquad {} + W _{n+2} + \bigl(\sqrt{ \vert a \vert } - 1\bigr) \vert h_{n} \vert + \bigl(\sqrt{ \vert a \vert } - 1\bigr) \vert h_{n+1} \vert + \vert g_{n+1} \vert + \vert h_{n+1} \vert + \vert g_{n} \vert + \vert h_{n} \vert \bigr] \\& \quad = \frac{1}{\sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } - 1)} \bigl[V_{n} + W_{n} + \bigl( \vert a \vert - 1\bigr) W_{n+1} + \bigl(\sqrt{ \vert a \vert } - 1\bigr) \vert h_{n} \vert \bigr] =: \mathcal{V}^{-}_{n}. \end{aligned}$$
    (85)

    Here above we used the fact that \(V_{n+2} + |g_{n+1}| + |g_{n}| = V _{n}\), \(W_{n+2} + |h_{n+1}| = W_{n+1}\), \(W_{n+2} + |h_{n+1}| + |h_{n}| = W_{n}\), and \((\sqrt{|a|} - 1) (W_{n+1} + W_{n+2}) + W_{n+2} + ( \sqrt{|a|} - 1) |h_{n+1}| + |h_{n+1}| + |h_{n}| = 2 (\sqrt{|a|} - 1) W_{n+1} + W_{n}\).

As for the other solution (still pertaining to the case \(a < -1\)), we have

$$ \bigl\vert c^{+} \bigr\vert = \biggl\vert \frac{1}{2 \sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } + 1)}, \qquad |\lambda _{+} | = \sqrt{ \vert a \vert } + 1, \qquad \vert \eta _{+} \vert > 1 \quad ( \eta _{+} < 0), $$

and hence

$$ \bigl\vert a^{+}_{n,r} \bigr\vert \leq 1 + \vert \eta _{+} \vert ^{r-n+1}. $$

Thus, for \(r \geq n+2\),

$$\begin{aligned}& \bigl\vert c^{+} K_{n,r}^{+} \bigr\vert \leq \frac{1}{2 \sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } + 1)} \bigl[\bigl(1 + \sqrt{ \vert a \vert } \bigr)^{2} \bigl(1 + \vert \eta _{+} \vert ^{r-n-1}\bigr) \vert h_{r-1} \vert \\& \hphantom{\bigl\vert c^{+} K_{n,r}^{+} \bigr\vert }{} + \bigl(\sqrt{ \vert a \vert } + 1\bigr) \bigl(1 + \vert \eta _{+} \vert ^{r-n}\bigr) \bigl( \vert h_{r-1} \vert + \vert h_{r} \vert \bigr) + \bigl(1 + \vert \eta _{+} \vert ^{r-n+1}\bigr) \bigl( \vert g_{r} \vert + \vert h_{r} \vert \bigr) \bigr]; \\& \bigl\vert c^{+} K_{n,n+1}^{+} \bigr\vert \leq \frac{1}{2 \sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } + 1)} \bigl[\bigl(\sqrt{ \vert a \vert } + 1 \bigr) \bigl(1 + \vert \eta _{+} \vert \bigr) \bigl( \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) \\& \hphantom{\bigl\vert c^{+} K_{n,n+1}^{+} \bigr\vert }{} + \bigl(1 + \vert \eta _{+} \vert ^{2} \bigr) \bigl( \vert g_{n+1} \vert + \vert h_{n+1} \vert \bigr) \bigr]; \\& \bigl\vert c^{+} K_{n,n}^{+} \bigr\vert \leq \frac{1}{2 \sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } + 1)} \bigl(1 + \vert \eta _{+} \vert \bigr) \bigl( \vert g_{n} \vert + \vert h_{n} \vert \bigr), \end{aligned}$$

and finally,

$$\begin{aligned}& \sum_{r=n}^{\infty } \bigl\vert c^{+} K^{+}_{n,r} \bigr\vert \\& \quad \leq \frac{1}{2 \sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } + 1)} \Biggl\{ \bigl(1 + \sqrt{ \vert a \vert } \bigr)^{2} \Biggl[\sum_{r=n+2} ^{\infty } \vert h_{r-1} \vert + \sum_{r=n+2}^{\infty } \vert \eta _{+} \vert ^{r-n-1} \vert h_{r-1} \vert \Biggr] \\& \qquad {} + \bigl(\sqrt{ \vert a \vert } + 1\bigr) \Biggl[\sum _{r=n+2}^{\infty } \vert h_{r-1} \vert + \sum_{r=n+2}^{\infty } \vert h_{r} \vert + \sum_{r=n+2}^{\infty } \vert \eta _{+} \vert ^{r-n} \vert h_{r-1} \vert + \sum_{r=n+2}^{\infty } \vert \eta _{+} \vert ^{r-n} \vert h_{r} \vert \Biggr] \\& \qquad {} + \sum_{r=n+2} ^{\infty } \vert g_{r} \vert + \sum_{r=n+2}^{\infty } \vert h_{r} \vert + \sum_{r=n+2}^{\infty } \vert \eta _{+} \vert ^{r-n+1} \vert g_{r} \vert + \sum_{r=n+2} ^{\infty } \vert \eta _{+} \vert ^{r-n+1} \vert h_{r} \vert \\& \qquad {} + \bigl(\sqrt{ \vert a \vert } + 1\bigr) \bigl(1 + \vert \eta _{+} \vert \bigr) \bigl( \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) \\& \qquad {} + \bigl(1 + \vert \eta _{+} \vert ^{2} \bigr) \bigl( \vert g_{n+1} \vert + \vert h_{n+1} \vert \bigr) + \bigl(1 + \vert \eta _{+} \vert \bigr) \bigl( \vert g_{n} \vert + \vert h_{n} \vert \bigr) \Biggr\} \\& \quad = \frac{1}{2 \sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } + 1)} \bigl\{ \bigl(1 + \sqrt{ \vert a \vert } \bigr)^{2} \bigl[W_{n+1} + \vert \eta _{+} \vert U_{n+1}^{+} \bigr] + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \bigl[W_{n+1} + W_{n+2} \\& \qquad {} + \vert \eta _{+} \vert ^{2} U_{n+1}^{+} + \vert \eta _{+} \vert ^{2} U_{n+2}^{+} \bigr] + V_{n+2} + W_{n+2} + \vert \eta _{+} \vert ^{3} Z_{n+2}^{+} + \vert \eta _{+} \vert ^{3} U_{n+2}^{+} \\& \qquad {} + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \bigl(1 + \vert \eta _{+} \vert \bigr) \bigl( \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) + \bigl(1 + \vert \eta _{+} \vert ^{2}\bigr) \bigl( \vert g_{n+1} \vert + \vert h_{n+1} \vert \bigr) \\& \qquad {} + \bigl(1 + \vert \eta _{+} \vert \bigr) \bigl( \vert g_{n} \vert + \vert h_{n} \vert \bigr) \bigr\} \\& \quad = \frac{1}{2 \sqrt{ \vert a \vert } (\sqrt{ \vert a \vert } + 1)} \bigl\{ V_{n+2} + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \bigl(2 + \sqrt{ \vert a \vert }\bigr) W_{n+1} + \bigl(2 + \sqrt{ \vert a \vert }\bigr) W_{n+2} \\& \qquad {} + \bigl(1 + \sqrt{ \vert a \vert }\bigr) \bigl[1 + \sqrt{ \vert a \vert } + \vert \eta _{+} \vert \bigr] \vert \eta _{+} \vert U_{n+1}^{+} + \vert \eta _{+} \vert ^{2} \bigl[1 + \sqrt{ \vert a \vert } + \vert \eta _{+} \vert \bigr] U^{+}_{n+2} + \vert \eta _{+} \vert ^{3} Z^{+}_{n+2} \\& \qquad {} + \bigl(1 + \vert \eta _{+} \vert \bigr) \bigl(2 + \sqrt{ \vert a \vert }\bigr) \vert h_{n} \vert + \bigl[1 + \sqrt{ \vert a \vert } + \vert \eta _{+} \vert \bigl(1 + \sqrt{ \vert a \vert } + \vert \eta _{+} \vert \bigr) \bigr] \vert h _{n+1} \vert \\& \qquad {} + \bigl(1 + \vert \eta _{+} \vert ^{2} \bigr) \vert g_{n+1} \vert + \bigl(1 + \vert \eta _{+} \vert \bigr) \vert g_{n} \vert \bigr\} =: \mathcal{V}^{+}_{n}. \end{aligned}$$
(86)

Here we used the definitions of \(V_{n}\), \(W_{n}\), \(U_{n}^{-}\), and \(Z_{n}^{-}\) given in (15), (16).

Appendix 2

In this appendix, we report all steps of the proof of Theorem 3.2, that is, for case ((D). They follow closely the corresponding ones for case (C).

The characteristic equation of the associated unperturbed equation is now

$$ \lambda ^{2} + b \lambda + 1 = 0, \quad b := a - 2 $$
(87)

(which coincides with that of (B)), so that we have the characteristic roots

$$ \lambda _{\pm } := - \frac{1}{2} \bigl(b \pm \sqrt{b^{2} - 4} \bigr). $$
(88)

Hence, we have, for \(a > 0\) [\(b > -2\)],

$$ \vert \eta _{+} \vert \equiv \biggl\vert \frac{\lambda _{+}}{\lambda _{-}} \biggr\vert = \biggl\vert \frac{b + \sqrt{b^{2} - 4}}{b - \sqrt{b^{2} - 4}} \biggr\vert > 1, $$

and for \(a < 0\) [\(b < -2\)],

$$ \vert \eta _{-} \vert \equiv \biggl\vert \frac{\lambda _{-}}{\lambda _{+}} \biggr\vert > 1. $$

The error equation is

$$ \lambda ^{2} \varepsilon _{n+2} + b \lambda \varepsilon _{n+1} + \varepsilon _{n} = \chi _{n}, $$
(89)

where

$$ \chi _{n} := - \bigl\{ \lambda ^{2} h_{n+1} (1 + \varepsilon _{n+2}) + \lambda (g_{n} - h_{n} - h_{n+1}) (1 + \varepsilon _{n+1}) + h_{n} (1 + \varepsilon _{n}) \bigr\} . $$
(90)

Here above relation (87) has been used. Note that \(\chi _{n}\) also depends on the choice of either \(\lambda _{+}\) or \(\lambda _{-}\).

Setting first \(\chi _{n} \equiv 0\) in (89), we obtain

$$ \lambda \Delta ^{2} \varepsilon _{n} + (2 \lambda + b) \Delta \varepsilon _{n} = 0, $$
(91)

which can be written as

$$ \Delta \bigl[\lambda \Delta \varepsilon _{n} + (2 \lambda + b) \varepsilon _{n} \bigr] = 0. $$

This suggests that two linearly independent solutions are given by

$$ \varepsilon _{n} \equiv 1, \quad \text{and} \quad \varepsilon _{n} = \eta ^{n}, \qquad \eta := -1 - \frac{b}{\lambda }, $$
(92)

where the latter was obtained by solving

$$ \lambda \Delta \varepsilon _{n} + (2 \lambda + b) \varepsilon _{n} = 0. $$

Note that, for \(\lambda = \lambda _{+}\), we have \(\eta = \eta _{-} = \lambda _{-}/\lambda _{+}\), while for \(\lambda = \lambda _{-}\), we have \(\eta = \eta _{+} = \lambda _{+}/\lambda _{-}\), and that \(\lambda _{+} = 1/ \lambda _{-}\) for any value of b (i.e., for any a), see (87).

The general solution to (91) has the form \(\varepsilon _{n} = C + D \eta ^{n}\), where C and D are arbitrary constants as long as \(\lambda _{+} \neq \lambda _{-}\), that is, for any \(b \neq \pm 2\) (i.e., \(a \neq 0\), \(a \neq 4\)). [If \(a = 4\), then \(\lambda _{+} = \lambda _{-}\), \(\eta = 1\), and the general solution should be \(\varepsilon _{n} = C + D n\).]

To obtain a representation for the solutions to the full equation (89), following the method of variation of parameters, we look for a solution of the form

$$ \varepsilon _{n} = C_{n} + D_{n} \eta ^{n}, $$
(93)

where \(C_{n}\) and \(D_{n}\) are two sequences to be determined. We evaluate first

$$ \Delta \varepsilon _{n} = \Delta C_{n} + \eta ^{n+1} \Delta D_{n} + ( \eta - 1) \eta ^{n} D_{n}, $$

and claim that

$$ \Delta C_{n} + \eta ^{n+1} \Delta D_{n} \equiv 0, $$
(94)

as it would be if the sequences \(C_{n}\) and \(D_{n}\) would be constant with n. Thus, it remains

$$ \Delta \varepsilon _{n} = (\eta - 1) \eta ^{n} D_{n}, $$
(95)

wherefrom

$$ \Delta ^{2} \varepsilon _{n} = (\eta - 1) \bigl( \eta ^{n+1} D_{n+1} - \eta ^{n} D_{n} \bigr), $$
(96)

and hence, from (95), (96), and (89), rewritten in terms of Δ operators, we obtain using (87)

$$ \lambda ^{2} (\eta - 1) \eta ^{n+1} \Delta D_{n} = \chi _{n}, $$

i.e.,

$$ \Delta D_{n}^{\pm } = \mp \frac{\chi _{n}^{\pm }}{\sqrt{b^{2} - 4}} \lambda _{\pm } (\eta _{\pm })^{n}, $$
(97)

since \(\lambda ^{2} \eta = 1\). In fact, \(\lambda ^{2} \eta \equiv \lambda _{\pm }^{2} \eta _{\pm } = \lambda _{\pm }^{2} \lambda _{\mp }/ \lambda _{\pm } = \lambda _{\pm } \lambda _{\mp } = 1\).

“Integrating” (97), i.e., summing up both sides from \(k=n\) to ∞, assuming that \(D_{n} \to 0\) as \(n \to \infty \), we obtain

$$ \sum_{r=n}^{\infty } \Delta D_{r}^{\pm } = - D_{n}^{\pm } = \mp \frac{ \lambda _{\pm }}{\sqrt{b^{2} - 4}} (\eta _{\pm })^{n} \chi _{n}^{ \pm }, $$

and finally

$$ D_{n}^{\pm } = \pm \frac{\lambda _{\pm }}{\sqrt{b^{2} - 4}} \sum _{r=n} ^{\infty } (\eta _{\pm })^{r} \chi _{r}^{\pm }. $$
(98)

We can now determine \(C_{n}\) from (94). We have first

$$ \Delta C_{n}^{\pm } = - \eta _{\pm }^{n+1} \Delta D_{n}^{\pm } = \pm \frac{\lambda _{\mp }}{\sqrt{b^{2} - 4}} \chi _{n}^{\pm }, $$

and then, assuming \(C_{n}^{\pm } \to 0\) as \(n \to \infty \),

$$ \sum_{r=n}^{\infty } \Delta C_{k}^{\pm } = - C_{n}^{\pm } = \pm \frac{ \lambda _{\mp }}{\sqrt{b^{2} - 4}} \sum_{r=n}^{\infty } \chi _{r}^{ \pm }. $$

Hence,

$$ C_{n}^{\pm } = \pm \frac{\lambda _{\mp }}{\sqrt{b^{2} - 4}} \sum _{r=n} ^{\infty } \chi _{r}^{\pm }. $$
(99)

Appendix 3

In this appendix, we provide details for the proof of Theorem 3.2, which concerns case (D) of equation \(\Delta (c_{n} \Delta y_{n}) + r_{n} y_{n+1} = 0\). Recalling that \(b = a - 2\), we have

  1. (i)

    for \(0 < a < 4\) [\(b^{2} < 4\)],

    $$ \bigl\vert c^{\pm } \bigr\vert = \frac{1}{\sqrt{4 - b^{2}}} = \frac{1}{\sqrt{a(4-a)}}, \qquad \vert \lambda _{\pm } \vert = 1, \qquad \vert \eta _{\pm } \vert = 1, $$

    and hence

    $$ \bigl\vert a^{\pm }_{n,r} \bigr\vert \leq \bigl\vert 1 - (\eta _{\pm })^{r-n+1} \bigr\vert \leq 2. $$

    Then (72) yields

    $$\begin{aligned}& \bigl\vert c^{\pm } K_{n,r}^{\pm } \bigr\vert \leq \frac{2}{\sqrt{a(4 - a)}} \bigl(2 \vert h_{r-1} \vert + 2 \vert h_{r} \vert + \vert g_{r-1} \vert \bigr) \quad \mbox{for } r \geq n+2; \\& \bigl\vert c^{\pm } K_{n,n+1}^{\pm } \bigr\vert \leq \frac{2}{\sqrt{a(4 - a)}} \bigl( \vert g _{n} \vert + \vert h_{n} \vert + 2 \vert h_{n+1} \vert \bigr); \\& \bigl\vert c^{\pm } K_{n,n}^{\pm } \bigr\vert \leq \frac{2}{\sqrt{a(4 - a)}} \vert h_{n} \vert . \end{aligned}$$

    Therefore,

    $$\begin{aligned} \sum_{r=n}^{\infty } \bigl\vert c^{\pm } K^{\pm }_{n,r} \bigr\vert \leq&\frac{2}{\sqrt{a(4 - a)}} \Biggl[2 \sum_{r=n+2}^{\infty } \vert h_{r-1} \vert + 2 \sum_{r=n+2}^{\infty } \vert h_{r} \vert + \sum_{r=n+2}^{\infty } \vert g_{r-1} \vert \\ & {}+ \vert g_{n} \vert + \vert h_{n} \vert + 2 \vert h_{n+1} \vert + \vert h_{n} \vert \Biggr] \\ =& \frac{2}{\sqrt{a(4 - a)}} \bigl(V_{n} + W_{n} + 2 W_{n+1} + W _{n+2} + \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) \\ =& \frac{2}{\sqrt{a(4 - a)}} (V_{n} + 2 W_{n} + 2 W_{n+1} ) =: \mathcal{V}^{\pm }_{n} \equiv { \mathcal{V}}_{n}. \end{aligned}$$
    (100)

    Here we recalled in particular that \(W_{n} = W_{n+2} + |h_{n+1}| + |h _{n}|\).

  2. (ii)

    for \(a > 4\) [\(b > 2\)],

    $$\begin{aligned}& \bigl\vert c^{-} \bigr\vert = \frac{(a - 2) + \sqrt{a (a - 4)}}{2 \sqrt{a(a - 4)}}, \\& \vert \lambda _{-} \vert = \biggl\vert \frac{1}{2} \bigl(-b - \sqrt{b^{2} - 4}\bigr) \biggr\vert = \frac{1}{2} \bigl(b + \sqrt{b^{2} - 4}\bigr) > 1, \\& \vert \eta _{-} \vert = \frac{b + \sqrt{b^{2} - 4}}{b - \sqrt{b^{2} - 4}} > 1 \quad (\eta _{-} < 0), \end{aligned}$$
    (101)

    and hence

    $$ \bigl\vert a^{-}_{n,r} \bigr\vert \leq 1 + \vert \eta _{-} \vert ^{r-n+1}. $$

    Then we have

    $$\begin{aligned}& \bigl\vert c^{-} K_{n,r}^{-} \bigr\vert \\& \quad \leq \bigl\vert c^{-} \bigr\vert \bigl[ \vert \lambda _{-} \vert ^{2} \bigl\vert a_{n,r-2} ^{-} \bigr\vert \vert h_{r-1} \vert + \vert \lambda _{-} \vert \bigl\vert a_{n,r-1}^{-} \bigr\vert \bigl( \vert g_{r-1} \vert + \vert h_{r-1} \vert + \vert h_{r} \vert \bigr) + \bigl\vert a_{n,r}^{-} \bigr\vert \vert h_{r} \vert \bigr] \\& \quad \leq \bigl\vert c^{-} \bigr\vert \bigl\{ \vert \lambda _{-} \vert ^{2} \bigl[1 + \vert \eta _{-} \vert ^{n-r-1}\bigr] \vert h _{r-1} \vert + \vert \lambda _{-} \vert \bigl[1 + \vert \eta _{-} \vert ^{n-r} \vert \bigr] \bigl( \vert g_{r-1} \vert + \vert h_{r-1} \vert + \vert h_{r} \vert \bigr) \\& \qquad {} + \bigl[1 + \vert \eta _{-} \vert ^{n-r+1} \bigr] \vert h_{r} \vert \bigr\} \quad \mbox{for }r \geq n+2; \\& \bigl\vert c^{-} K_{n,n+1}^{-} \bigr\vert \leq \bigl\vert c^{-} \bigr\vert \bigl( \vert \lambda _{-} \vert \bigl\vert a_{n,n}^{-} \bigr\vert \bigl( \vert g_{n} \vert + \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) + \bigl\vert a_{n,n+1}^{-} \bigr\vert \vert h_{n+1} \vert \bigr) \\& \hphantom{\bigl\vert c^{-} K_{n,n+1}^{-} \bigr\vert }\leq \bigl\vert c^{-} \bigr\vert \bigl[ \vert \lambda _{-} \vert \bigl(1 + \vert \eta _{-} \vert \bigr) \bigl( \vert g_{n} \vert + \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) + \bigl(1 + \vert \eta _{-} \vert ^{2}\bigr) \vert h_{n+1} \vert \bigr]; \\& \bigl\vert c^{-} K_{n,n}^{-} \bigr\vert \leq \bigl\vert c^{-} \bigr\vert \bigl\vert a_{n,n}^{-} \bigr\vert \vert h_{n} \vert \leq \bigl\vert c^{-} \bigr\vert \bigl(1 + \vert \eta _{-} \vert \bigr) \vert h_{n} \vert . \end{aligned}$$

    Collecting all these and recalling the definitions of \(V_{n}\), \(W_{n}\), \(Z^{-}_{n}\), and \(U^{-}_{n}\) (given in (15), (16)), we obtain

    $$\begin{aligned}& \sum_{r=n}^{\infty } \bigl\vert c^{-} K_{n,r}^{-} \bigr\vert \\& \quad = \sum _{r=n+2}^{\infty } \bigl\vert c ^{-} K_{n,r}^{-} \bigr\vert + \bigl\vert c^{-} K_{n,n+1}^{-} \bigr\vert + \bigl\vert c^{-} K_{n,n}^{-} \bigr\vert \\& \quad \leq \bigl\vert c^{-} \bigr\vert \Biggl[ \vert \lambda _{-} \vert ^{2} \Biggl(\sum _{r=n+2}^{\infty } \vert h _{r-1} \vert + \sum_{r=n+2}^{\infty } \vert \eta _{-} \vert ^{r-n-1} \vert h_{r-1} \vert \Biggr) \\& \qquad{} + \vert \lambda _{-} \vert \Biggl(\sum _{r=n+2}^{\infty } \vert g_{r-1} \vert + \sum_{r=n+2} ^{\infty } \vert h_{r-1} \vert + \sum_{r=n+2}^{\infty } \vert h_{r} \vert + \sum_{r=n+2} ^{\infty } \vert \eta _{-} \vert ^{r-n} \vert g_{r-1} \vert \\& \qquad {} + \sum_{r=n+2}^{\infty } \vert \eta _{-} \vert ^{r-n} \vert h_{r-1} \vert + \sum_{r=n+2}^{\infty } \vert \eta _{-} \vert ^{r-n} \vert h_{r} \vert \Biggr) \\& \qquad{}+ \sum_{r=n+2}^{\infty } \vert h_{r} \vert + \sum_{r=n+2}^{\infty } \vert \eta _{-} \vert ^{r-n+1} \vert h_{r} \vert + \vert \lambda _{-} \vert \bigl(1 + \vert \eta _{-} \vert \bigr) \bigl( \vert g_{n} \vert + \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) \\& \qquad{} + \bigl(1 + \vert \eta _{-} \vert ^{2} \bigr) \vert h_{n+1} \vert + \bigl(1 + \vert \eta _{-} \vert \bigr) \vert h_{n} \vert \Biggr] \\& \quad = \bigl\vert c^{-} \bigr\vert \bigl\{ \vert \lambda _{-} \vert ^{2} \bigl(W_{n+1} + \vert \eta _{-} \vert U^{-} _{n+1} \bigr) + \vert \lambda _{-} \vert \bigl[V_{n+1} + W_{n+1} + W_{n+2} + \vert \eta _{-} \vert ^{2} Z^{-}_{n+1} \\& \qquad {} + \vert \eta _{-} \vert ^{2} U^{-}_{n+1} + \vert \eta _{-} \vert ^{2} U^{-}_{n+2} \bigr] + W_{n+2} + \vert \eta _{-} \vert ^{3} U^{-}_{n+2} \\& \qquad {} + \vert \lambda _{-} \vert \bigl(1 + \vert \eta _{-} \vert \bigr) \bigl( \vert g_{n} \vert + \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) + \bigl(1 + \vert \eta _{-} \vert ^{2} \bigr) \vert h_{n+1} \vert + \bigl(1 + \vert \eta _{-} \vert \bigr) \vert h_{n} \vert \bigr\} \\& \quad = \bigl\vert c^{-} \bigr\vert \bigl[ \vert \lambda _{-} \vert V_{n+1} + \bigl( \vert \lambda _{-} \vert + 1\bigr) W _{n} + \vert \lambda _{-} \vert \bigl( \vert \lambda _{-} \vert + 1 \bigr) W_{n+1} \\& \qquad {}+ \vert \lambda _{-} \vert \vert \eta _{-} \vert \bigl( \vert \lambda _{-} \vert + \vert \eta _{-} \vert \bigr) U_{n+1}^{-}+ \vert \eta _{-} \vert ^{2} \bigl( \vert \lambda _{-} \vert + \vert \eta _{-} \vert \bigr) U_{n+2}^{-} + \vert \lambda _{-} \vert \vert \eta _{-} \vert ^{2} Z_{n+1}^{-} \\& \qquad {}+ \vert \lambda _{-} \vert \bigl(1 + \vert \eta _{-} \vert \bigr) \vert g_{n} \vert + \bigl( \vert \lambda _{-} \vert + 1\bigr) \vert \eta _{-} \vert \vert h_{n} \vert + \vert \eta _{-} \vert \bigl( \vert \lambda _{-} \vert + \vert \eta _{-} \vert \bigr) \vert h _{n+1} \vert \bigr] \\& \quad =: \mathcal{V}^{-}_{n}. \end{aligned}$$
    (102)

    As for the other solution (still pertaining to the case \(a > 4\)), we have

    $$\begin{aligned}& \begin{gathered} \bigl\vert c^{+} \bigr\vert = \frac{(a - 2) - \sqrt{a (a - 4)}}{2 \sqrt{a(a - 4)}}, \\ \vert \lambda _{+} \vert = \frac{1}{2} \bigl(b - \sqrt{b^{2} - 4}\bigr) < 1, \\ \vert \eta _{+} \vert = \frac{b - \sqrt{b^{2} - 4}}{b + \sqrt{b^{2} - 4}} < 1 \quad (\eta _{+} > 0), \end{gathered} \end{aligned}$$
    (103)

    and hence

    $$ \bigl\vert a^{+}_{n,r} \bigr\vert \leq 1 + \vert \eta _{-} \vert ^{r-n+1} < 2. $$

    Then we have

    $$\begin{aligned}& \sum_{r=n}^{\infty } \bigl\vert c^{+} K_{n,r}^{+} \bigr\vert \\& \quad < \bigl\vert c^{+} \bigr\vert \Biggl(2 \sum _{r=n+2} ^{\infty } \vert h_{r-1} \vert + \sum_{r=n+2}^{\infty } \vert h_{r} \vert + \sum_{r=n+2} ^{\infty } \vert g_{r-1} \vert + \vert g_{n} \vert + \vert h_{n} \vert + 2 \vert h_{n+1} \vert + \vert h_{n} \vert \Biggr) \\& \quad = \bigl\vert c^{+} \bigr\vert \bigl(2 W_{n+1} + W_{n+2} + V_{n+1} + \vert g_{n} \vert + 2 \vert h_{n} \vert + 2 \vert h_{n+1} \vert \bigr) \\& \quad = \bigl\vert c^{+} \bigr\vert \bigl(V_{n} + 2 W_{n} + W_{n+1} + \vert h_{n+1} \vert \bigr) =: \mathcal{V}^{+}_{n}. \end{aligned}$$
    (104)
  3. (iii)

    for \(a < 0\) [\(b < -2\)],

    $$\begin{aligned}& \begin{gathered} \bigl\vert c^{-} \bigr\vert = \frac{ \vert a \vert + 2 - \sqrt{a(a - 4)}}{2 \sqrt{a(a - 4)}}, \\ \vert \lambda _{-} \vert = \frac{1}{2} \bigl( \vert b \vert - \sqrt{b^{2} - 4}\bigr) < 1, \\ \vert \eta _{-} \vert = \frac{ \vert b \vert - \sqrt{b^{2} - 4}}{ \vert b \vert + \sqrt{b^{2} - 4}} < 1 \quad (\eta _{-} > 0), \end{gathered} \end{aligned}$$
    (105)

    and hence

    $$ \bigl\vert a^{-}_{n,r} \bigr\vert = 1 - (\eta _{+})^{r-n+1} < 1. $$

    Then

    $$\begin{aligned}& \sum_{r=n}^{\infty } \bigl\vert c^{-} K_{n,r}^{-} \bigr\vert \\& \quad = \sum _{r=n+2}^{\infty } \bigl\vert c ^{-} K_{n,r}^{-} \bigr\vert + \bigl\vert c^{-} K_{n,n+1}^{-} \bigr\vert + \bigl\vert c^{-} K_{n,n}^{-} \bigr\vert \\& \quad \leq \bigl\vert c^{-} \bigr\vert \Biggl(2 \sum _{r=n+2}^{\infty } \vert h_{r-1} \vert + 2 \sum_{r=n+2} ^{\infty } \vert h_{r} \vert + \sum_{r=n+2}^{\infty } \vert g_{r-1} \vert + \vert g_{n} \vert + 2 \vert h _{n} \vert + 2 \vert h_{n+1} \vert \Biggr) \\& \quad = \bigl\vert c^{-} \bigr\vert (V_{n} + 2 W_{n} + 2 W_{n+1} )=: \mathcal{V} _{n}^{-}. \end{aligned}$$
    (106)

    As for the other solution,

    $$\begin{aligned}& \begin{gathered} \bigl\vert c^{+} \bigr\vert = \frac{ \vert a \vert + 2 + \sqrt{a(a - 4)}}{2 \sqrt{a(a - 4)}}, \\ \vert \lambda _{+} \vert = \frac{1}{2} \bigl( \vert b \vert + \sqrt{b^{2} - 4}\bigr) > 1, \\ \vert \eta _{+} \vert = \frac{ \vert b \vert + \sqrt{b^{2} - 4}}{ \vert b \vert - \sqrt{b^{2} - 4}} > 1 \quad (\eta _{+} > 0), \end{gathered} \end{aligned}$$
    (107)

    and hence

    $$ \bigl\vert a^{+}_{n,r} \bigr\vert \leq 1 + (\eta _{+})^{r-n+1}. $$

    Then we have

    $$\begin{aligned}& \bigl\vert c^{+} K_{n,r}^{+} \bigr\vert \\& \quad \leq \bigl\vert c^{+} \bigr\vert \bigl\{ \vert \lambda _{+} \vert ^{2} \bigl[1 + (\eta _{+})^{r-n-1} \bigr] \vert h_{r-1} \vert + \vert \lambda _{+} \vert \bigl[1 + (\eta _{+})^{r-n}\bigr] \bigl( \vert g _{r-1} \vert + \vert h_{r-1} \vert + \vert h_{r} \vert \bigr) \\& \qquad {} + \bigl[1 + (\eta _{+})^{r-n+1}\bigr] \vert h_{r} \vert \bigr\} \quad \mbox{for }r \geq n+2; \\& \bigl\vert c^{+} K_{n,n+1}^{+} \bigr\vert \leq \bigl\vert c^{+} \bigr\vert \bigl\{ \vert \lambda _{+} \vert [1 + \eta _{+}] \bigl( \vert g_{n} \vert + \vert h_{n} \vert + \vert h_{n+1} \vert \bigr) + \bigl[1 + (\eta _{+})^{2} \bigr] \vert h_{n+1} \vert \bigr\} ; \\& \bigl\vert c^{+} K_{n,n}^{+} \bigr\vert \leq \bigl\vert c^{+} \bigr\vert (1 + \eta _{+}) \vert h_{n} \vert . \end{aligned}$$

    Collecting all these and recalling the usual definitions of \(V_{n}\), \(W_{n}\), \(Z^{+}_{n}\), and \(U^{+}_{n}\), we obtain upon some algebra

    $$\begin{aligned}& \sum_{r=n}^{\infty } \bigl\vert c^{+} K_{n,r}^{+} \bigr\vert \\& \quad \leq \vert c_{+} \vert \Biggl\{ \vert \lambda _{+} \vert ^{2} \sum_{r=n+2}^{\infty } \vert h_{r-1} \vert + \vert \lambda _{+} \vert ^{2} \sum_{r=n+2}^{\infty } (\eta _{+})^{r-n-1} \vert h_{r-1} \vert \\& \qquad {}+ \vert \lambda _{+} \vert \Biggl(\sum _{r=n+2}^{\infty } \vert g_{r-1} \vert + \sum_{r=n+2} ^{\infty } \vert h_{r-1} \vert + \sum_{r=n+2}^{\infty } \vert h_{r} \vert \Biggr) \\& \qquad {}+ \vert \lambda _{+} \vert \Biggl[\sum _{r=n+2}^{\infty } (\eta _{+})^{r-n} \vert g_{r-1} \vert + \sum_{r=n+2}^{\infty } (\eta _{+})^{r-n} \vert h_{r-1} \vert + \sum_{r=n+2} ^{\infty } (\eta _{+})^{r-n} \vert h_{r} \vert \Biggr] \\& \qquad {}+ \sum_{r=n+2}^{\infty } \vert h_{r} \vert + \sum_{r=n+2}^{\infty } (\eta _{+})^{r-n+1} \vert h_{r} \vert + \vert \lambda _{+} \vert (1 + \eta _{+}) \bigl( \vert g_{n} \vert + \vert h _{n} \vert + \vert h_{n+1} \vert \bigr) \\& \qquad {} + \bigl(1 + (\eta _{+})^{2}\bigr) \vert h_{n+1} \vert + (1 + \eta _{+}) \vert h_{n} \vert \Biggr\} \\& \quad = \bigl\vert c^{+} \bigr\vert \bigl[ \vert \lambda _{+} \vert V_{n} + \bigl(1 + \vert \lambda _{+} \vert \bigr) W_{n} + \vert \lambda _{+} \vert \bigl(1 + \vert \lambda _{+} \vert \bigr) W_{n+1} + \eta _{+} \vert \lambda _{+} \vert \bigl(\eta _{+} + \vert \lambda _{+} \vert \bigr) U^{+}_{n+1} \\& \qquad {}+ (\eta _{+})^{2} \bigl(\eta _{+} + \vert \lambda _{+} \vert \bigr) U^{+}_{n+2} + (\eta _{+})^{2} \vert \lambda _{+} \vert Z_{n+1}^{+} + \eta _{+} \bigl(1 + \vert \lambda _{+} \vert \bigr) \vert h _{n} \vert \\& \qquad {}+ \eta _{+} \bigl(\eta _{+} + \vert \lambda _{+} \vert \bigr) \vert h_{n+1} \vert + \eta _{+} \vert \lambda _{+} \vert \vert g_{n} \vert \bigr] =: \mathcal{V}^{+}_{n}. \end{aligned}$$
    (108)

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MSC

  • 39A10
  • 39A06
  • 39A99
  • 41A60

Keywords

  • Difference equations
  • Canonical forms
  • Liouville–Green (WKB) asymptotics
  • Error bounds