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Odd and even Lidstonetype polynomial sequences. Part 1: basic topics
Advances in Difference Equations volume 2018, Article number: 299 (2018)
Abstract
Two new general classes of polynomial sequences called respectively odd and even Lidstonetype polynomials are considered. These classes include classic Lidstone polynomials of first and second kind. Some characterizations of the two classes are given, including matrix form, conjugate sequences, generating function, recurrence relations, and determinant forms. Some examples are presented and some applications are sketched.
Introduction
“Je m’occupe, dans ce Memoir, de certains polynômes en x formant une suite \(A_{0}, A_{1}, \ldots, A_{n}, \ldots\) dont le terme \(A_{n}\) est un polynôme de degré n et dans laquelle deux termes consécutifs sont liés par la relation \(\frac{d A_{n}}{dx}=nA_{n1}\)…”. [9]
By paraphrasing Appell [9], we will consider a sequence of polynomials
such that two consecutive terms satisfy the differential relation
We call this sequence Lidstonetype polynomial sequence (LPS).
Really Lidstone [27] extended an Aitken’s theorem on general linear interpolation [8] to Everett type I and Everett type II interpolatory formulas [32]. As an example, he considered the \(\frac{d^{2}}{dx^{2}}\) operator and expansions of polynomials of odd and even degree involving two points, which are expressed in terms of a basis of polynomials satisfying (1).
The Lidstone interpolation theorem can be formulated as follows.
Theorem 1
Let \(P_{k}(x)\) be a polynomial of odd degree \(\le2k+1\). The following expansion holds:
where \(\varLambda _{i}(x)\) are polynomials of degree \(2i+1\), \(i=0,1,\ldots\) , which satisfy
Analogously, a polynomial \(Q_{k}(x)\) of even degree at most 2k, can be expanded as
where \(v_{i} (x )\) are polynomials of degree 2i, \(i=0,1,\ldots \) , satisfying
Afterwards, attention was focused on the socalled Lidstone series
Several authors, including Boas [11, 12], Poritsky [28], Schoenberg [31], Whittaker [36], and Widder [37], characterized the convergence of the series (6) in terms of completely continuous functions. Lately, a new and simplified proof of the convergence of this series was given in [19].
In the recent years, Agarwal et al. [1–7], Costabile et al. [14, 17, 20–22] connected the Lidstone expansion to interpolatory problem for regular functions and applied it to high order boundary value problems.
More precisely, they considered the BVPs
and
The polynomials (2) and (4) have played a key role in the theoretical and computational studies of the above BVPs. Some applications to quadrature formulas have been given too.
Moreover, Costabile et al. [23] introduced an algebraic approach to Lidstone polynomials \(\{\varLambda _{k} \}_{k}\) and a first form of the generalization of this sequence was proposed in [20, 21].
In this paper, inspired by the last part of the book [15], we will present a systematic study of two general classes of polynomial sequences including respectively the Lidstone polynomials of type I (defined in (3)) and of type II (defined in (5)). As an application, for the sake of brevity, we only mention an interpolation problem. The solution to this problem will be examined in detail in a future paper (Part 2), in which we will present other applications of the two polynomial classes, including to the operators approximation theory.
To the best of our knowledge, the main issues that will be dealt with have not appeared before.
The outline of the paper is as follows: In Sect. 2 we consider the class of odd Lidstone polynomial sequences, and we study some properties of this type of polynomials, particularly, their matrix form, the conjugate sequences related to them, some recurrence relations and determinant forms, the generating function, and the relationship with Appell polynomials. Some examples are given in Sect. 2.9. Then, in Sect. 3, in analogy with the odd Lidstone sequences, we consider the class of even Lidstone polynomial sequences, and we give the main properties. Some examples are presented in Sect. 3.6. Finally, we hint at a new interpolatory problem (Sect. 4) and report some conclusions (Sect. 5).
Odd Lidstonetype polynomial sequences
Let be \(\widetilde{\mathcal{P}} = \operatorname{span} \{x^{2i+1} \mid i=0,1,\ldots \}\). We denote by OLS (odd Lidstone sequences) the set of polynomial sequences which satisfy
From (7) it easily follows

(a)
\(p_{k} (x )\), \(k=0,1,2,\ldots\) , are polynomials of degree \(\le2k+1\);

(b)
\(p_{0} (x )=\alpha_{0} x\), \(\alpha_{0}\in \mathbb{R}\), \(\alpha_{0}\ne0\).
Now we will give a more complete characterization of the set OLS.
Proposition 1
Let \(\{p_{k} \}_{k}\) be a polynomial sequence of Lidstone type, that is, \(p_{k}:=p_{k}(x)\) is a polynomial which satisfies (1), \(\forall k\in \mathbb{N}\). It is an element of OLS iff there exists a numerical sequence \((\alpha_{2j} )_{j\ge0}\), \(\alpha_{0}\ne0\), \(\alpha_{j}\in \mathbb{R}\) such that
Proof
Vice versa, let \(\{p_{k} \}_{k}\in \mathit{OLS}\). Then there exists a constant \(\alpha_{0}\ne0\) such that \(p_{0} (x )=\alpha_{0} x\). Moreover, for \(k=1,2,\ldots\) , we set \(p_{k}' (0 )=\alpha _{2k}\), \(\alpha_{2k}\in \mathbb{R}\). From the differential relation in (7), by two integrations, we get
Hence the result follows from (9) by induction. □
Remark 1
From Proposition 1, \(\mathit{OLS}\subset\widetilde {\mathcal{P}}\).
The following proposition contains some important properties on the elements of OLS.
Proposition 2
For each polynomial sequence \(\{p_{k} \}_{k} \in \mathit{OLS}\), the following statements hold:
Matrix form
Given a numerical sequence \((\alpha_{2k} )_{k}\), \(\alpha _{0}\ne0\), relation (8) suggests considering the infinite lower triangular matrix \(A_{\infty}= (a_{i,j} )_{i,j\ge0}\) [39], with \(a_{i,j}\) defined as
We call \(A_{\infty}\) an odd Lidstonetype matrix.
Let \(\widetilde{X}_{\infty}\) and \(P_{\infty}\) be the vectors with infinite components defined as
Then (8) can be written in a matrix form as \(P_{\infty}=A_{\infty}\widetilde{X}_{\infty}\) or, for simplicity,
where, of course, \(P=P_{\infty}\), \(A=A_{\infty}\), \(\widetilde {X}=\widetilde{X}_{\infty}\).
If in (10) we consider \(i=0,1,\ldots,n\), \(j=0,1,\ldots,i\), \(\forall n\in \mathbb{N}\), we have the matrices \(A_{n}\) which are the principal submatrices of order n of A. Analogously, we consider the vectors of order \(n\ge0\)
From (11) we have
In the following proposition we analyze the structure of the matrix A.
Proposition 3
The infinite lower triangular matrix \(A= (a_{i,j} )_{i,j\ge 0}\) defined in (10) can be factorized as
where
and \(T_{\alpha}\) is the lower triangular Toeplitz matrix with entries \(t_{i,j}^{\alpha}=\frac{\alpha_{2(ij)}}{(2(ij)+1)!}\).
Proof
The proof is trivial by taking into account that the product between infinite lower triangular matrices is well defined [34, 39]. □
Proposition 4
The matrix A, defined in (10), is invertible and, if \(B:=A^{1}\), it results
where D is the diagonal matrix defined in (14), \(T_{\beta}\) is the lower triangular Toeplitz matrix with entries \(t_{i,j}^{\beta }=\frac{\beta_{2(ij)}}{(2(ij)+1)!}\), the numerical sequence \((\beta_{2i} )_{i}\) being implicitly defined by
with \(\delta_{ij}\) the Kronecker symbol.
Proof
The proof easily follows from Proposition 3 and the wellknown results about triangular Toeplitz matrices [34]. □
Remark 2
Let \((\alpha_{2k} )_{k}\), \(\alpha_{0}\ne0\), be a given numerical sequence. Equation (15) can be considered as an infinite linear system which determines the numerical sequence \((\beta_{2k} )_{k}\). By means of Cramer’s rule, the first \(n+1\) equations in (15) give
By symmetry, the coefficients \(\alpha_{2i}\), \(i=1,\ldots ,n\), have an expression similar to (16) by exchanging \(\alpha _{2i}\) with \(\beta_{2i}\), \(i=1,\ldots,n\), in (16).
Conjugate odd Lidstone polynomials
Let \((\alpha_{2k} )_{k}\), \(\alpha_{0}\ne0\) be a given numerical sequence and \((\beta_{2k} )_{k}\) the related sequence defined in (16). \(\forall k\in \mathbb{N}\), we can consider the polynomials
From Proposition 1, the sequence \(\{\widehat {p}_{k} \}_{k}\) defined in (17) is an odd Lidstonetype polynomial sequence, i.e., an element of OLS. Then the two sequences of OLS \(\{p_{k} \}_{k}\) and \(\{\widehat{p}_{k} \} _{k}\), related to the numerical sequences \((\alpha_{2k} )_{k}\) and \((\beta_{2k} )_{k}\) satisfying (15), are called conjugate odd Lidstonetype sequences.
Proposition 4 allows us to get the matrix form of the odd Lidstone sequence \(\{\widehat{p}_{k} \}_{k}\). We set \(\widehat {P}= [\widehat{p}_{0},\widehat{p}_{1},\ldots,\widehat{p}_{k},\ldots ]^{T}\), \(B= (b_{ij} )\) with
From (17) we have
and, \(\forall n\in \mathbb{N}\),
Proposition 5
With the previous notations and hypothesis, the sequences \(\{ p_{k} \}_{k}\) and \(\{\widehat{p}_{k} \}_{k}\) are conjugate odd Lidstonetype sequences iff
Proof
The result follows with easy calculations from Proposition 4 and relations (11), (13), (18), (19). □
Corollary 1
With the previous notations and hypothesis, we can write
where \(a_{k,j}^{*}\) and \(b_{k,j}^{*}\), \(j=0,\ldots, k\), are the elements of the matrices \(A^{2}\) and \(B^{2}\), respectively.
First recurrence relation and determinant form
A sequence of odd Lidstone polynomials satisfies some recurrence relations. Moreover, it can be represented as the determinant of a suitable matrix.
In order to obtain a recurrence relation, from (11) we get \(\widetilde{X}=A^{1}P=BP\).
Hence, for each \(k\in \mathbb{N}\),
where \(\widetilde{X}_{k}\) and \(P_{k}\) are defined as in (12). From (20), for \(i=0,\ldots,k\), we obtain
Theorem 2
(First recurrence relation)
Let \(\{p_{k} \}_{k}\) be an element of \(\widetilde{\mathcal {P}}\). \(\{p_{k} \}_{k}\in \mathit{OLS}\) iff there exists a numerical sequence \((\alpha_{2k} )_{k}\), \(\alpha_{0}\ne0\), and hence the numerical sequence \((\beta_{2k} )_{k}\) defined as in (15) such that the following recursive relation holds:
Proof
Let \(\{p_{k} \}_{k}\) be the odd Lidstone sequence related to the numerical sequence \((\alpha_{2k} )_{k}\). Then relation (21) holds, and from (21) we get (22).
Vice versa, if (22) holds, then relation (21) follows. Hence we obtain (20), (13) and then (8). □
The recurrence relation (22) is equivalent to a determinant form.
Theorem 3
(First determinant form)
A sequence \(\{p_{k} \}_{k}\subset\widetilde{\mathcal{P}}\) is the odd Lidstone polynomial sequence, that is, \(\{p_{k} \} _{k}\in \mathit{OLS}\), related to the numerical sequence \((\alpha_{2k} )_{k}\), \(\alpha_{0}\ne0\), iff the following representation holds:
where \((\beta_{2k} )_{k}\) are defined as in (16).
Proof
Let \(\{p_{k} \}_{k}\) be the odd Lidstone polynomial sequence related to \((\alpha_{2k} )_{k}\). If \((\beta _{2k} )_{k}\) is as in (16), then identity (21) holds. Relation (21) can be considered as an infinite linear system in the unknowns \(p_{k}(x)\). By solving the first \(k+1\) equations of this system by Cramer’s rule, we get (23).
Vice versa, if (23) holds, by expanding the determinant with respect to the last column [16], we have (22) and then the result follows from Theorem 2. □
Remark 3
Expanding the determinant in (23) with respect to the first row, we get (8).
By the same technique used in the proof of Theorems 2 and 3, we can prove the following recurrence relation and determinant form for the conjugate sequence \(\{\widehat{p}_{k} \}_{k}\).
Theorem 4
Given a polynomial sequence \(\{p_{k}\}_{k}\), the conjugate odd polynomial sequence of Lidstone type \(\{\widehat{p}_{k}\}_{k}\) related to the numerical sequence \(\{\alpha_{2k}\}_{k}\) can be written as
Moreover, \(\widehat{p}_{k} (x )\) has a determinant representation similar to (23) with \(\alpha_{2i}\) (defined as in Remark 2) instead of \(\beta_{2i}\), \(i=0,\ldots, k\).
The linear space OLS̃
Theorem 2 suggests extending the classic umbral composition [30] in the set of odd Lidstone polynomial sequences.
Definition 1
Let \(\{r_{k} \}_{k}\) and \(\{s_{k} \}_{k}\) be the odd Lidstone polynomial sequences related respectively to the numerical sequences \((\rho_{2k} )_{k}\) and \((\sigma_{2k} )_{k}\), that is, \(\forall k\in \mathbb{N}\)
The umbral composition of \(r_{k}(x)\) and \(s_{k}(x)\) is defined as
Theorem 5
If + is the usual addition in OLS and ∘ is defined as in (24), then the algebraic structure \(\widetilde {\mathit{OLS}}:= (\mathit{OLS},+,\circ )\) is a linear space, also called an algebra.
Proof
The sequence \(i_{k}= \{x^{2k+1} \}_{k}\) is an element of OLS, and \(\forall \{p_{k} \}_{k} \in \mathit{OLS}\) it results \(p_{k}\circ i_{k}=p_{k}\). Moreover, if \(\{p_{k} \}_{k}\) and \(\{\widehat{p}_{k} \}_{k}\) are conjugate sequences, then \(p_{k}\circ\widehat{p}_{k} =i_{k}\).
Hence it easily follows that OLS̃ is a linear space. □
Let \({\mathcal {L}}\) be the set of odd Lidstonetype matrices, that is, the matrices defined as in (10), and let \(\widetilde {{\mathcal {L}}}:= ({\mathcal {L}},+,\cdot )\), where + is the usual matrix sum and ⋅ is the product between lower triangular infinite matrices.
Theorem 6
The algebraic structures OLS̃ and \(\widetilde{{\mathcal {L}}}\) are isomorphic linear spaces.
Proof
The isomorphism is given by the onetoone correspondence between odd Lidstonetype matrices and odd Lidstone polynomial sequences
□
Second recurrence relation and determinant form
In order to determine a second recurrence relation, we consider the production matrix [24] (also called Stieltjes matrix) of an infinite lower triangular matrix.
Definition 2
The production matrix of a nonsingular infinite lower triangular matrix A is defined by
where A̅ is the matrix A with its first row removed.
We note that, if \({\mathcal{D}} = ( \delta_{i+1,j} )_{i,j\ge0}\) with \(\delta_{i,j}\) the Kronecker symbol, the production matrix of A can be written as \(\tilde{\varPi }=A^{1} {\mathcal {D}} A\) and the production matrix of \(A^{1}\) is
Remark 4
The production matrix is a Hessenberg matrix.
Proposition 6
Let \(A= (a_{i,j} )_{i,j\ge0}\) be an odd Lidstonetype matrix, and let \(B= (b_{i,j} )_{i,j\ge0}\) be the inverse matrix. Then the elements \(\pi_{i,j}\), \(i,j\ge0\), of the production matrix Π of B are given by
where \((\alpha_{2k} )_{k}\) and \((\beta_{2k} )_{k}\) are given in Remark 2.
Proof
The proof follows from (25) and Proposition 4. □
Theorem 7
Let \(\{p_{k} \}_{k}\in \mathit{OLS}\) with the related matrix A as in (10). Then, for all \(k\ge1\), \(p_{k}(x)\) satisfies the following recurrence relation:
where \(\pi_{i,j}\), \(i,j\ge0\), are the elements of the production matrix Π of \(A^{1}\).
Proof
From (25) \(\varPi A=A {\mathcal {D}}\). Multiplying both sides by X̃, we get \(\varPi A \widetilde{X}=A {\mathcal {D}}\widetilde{X}\). Since \(A \widetilde{X}=P\) and \({\mathcal {D}}\widetilde{X}= [x^{3},x^{5},\ldots ]^{T}=x^{2}\widetilde{X} \), we obtain
By considering the \((k+1)\)th equation of (28), we have \(\sum_{j=0}^{k+1}\pi_{k,j}p_{j} (x )=x^{2}p_{k} (x ) \), and hence relation (27) follows after easy calculations. □
Theorem 8
If \(\{p_{k} \}_{k}\) is an odd Lidstone polynomial sequence, then, for \(k\ge0\), the following determinant representation holds:
where \(\pi_{k,j}\) are defined as in (26) and \(p_{0}(x)=\frac{1}{\beta_{0}} x\).
Proof
The infinite linear system (28) in the unknowns \(p_{k}(x)\) can be written as
The solution of the first \(k+1\) equations of this system, obtained by Cramer’s rule, gives (29). □
Remark 5
From Theorem 3, \(p_{k}(x)\) is the determinant of a suitable upper Hessenberg matrix, and it is known that for its calculation the Gauss method without pivoting is stable [25]. From Theorem 8, \(p_{k}(\sqrt{x})\) can be considered the characteristic polynomial of a lower Hessenberg matrix.
Remark 6
Let \(\varPi _{k}\) be the principal submatrix of order k of Π. If \(x_{s}\), \(s=1,\ldots, 2k\), are the nonzero zeros of \(p_{k}(x)\) (symmetric two by two with respect to the origin or complex conjugates), let us indicate by \(x_{\overline{s}}^{2}\), \(\overline{s}=1,\ldots, k\), the k distinct square values of \(x_{s}\). Then, from Theorem 8, \(x_{\overline{s}}^{2}\) are the eigenvalues of \(\varPi _{k}\).
From the Gershgorin theorem we have that \(x_{\overline{s}}^{2}\) satisfies at least one of the following inequalities:
Derivation matrix and differential equations for Lidstonetype polynomials
Let \(\{L_{j} \}_{j}\) be a polynomial sequence and let
It is known that [29] the derivation matrix for \(\overline{L}_{k} (x )\) is the matrix \({\mathcal {D}}\) such that
In recent years, the derivation matrices have been employed to solve many engineering and physical problems [10, 29, 35]. Furthermore, they are used in several areas of numerical analysis, such as differential equations [38], integral equations [33], etc.
Analogously, in the following, we consider the derivation matrix for a Lidstonetype polynomial sequence.
Let \(\{p_{k} \}_{k}\) be a Lidstonetype polynomial sequence. For \(k>0\), let us consider \(P_{k}\) as in (12) and let \(P''_{k}= [p''_{0} (x ),p''_{1} (x ),\ldots ,p''_{k} (x ) ]^{T}\).
The derivation matrix for \(P_{k}\) is the matrix \({\mathcal {D}}\) such that
Proposition 7
The derivation matrix \({\mathcal {D}}= (d_{ij} ) \in{ \mathbb{R}}^{(k+1)\times(k+1)}\) for \(P_{k}\) is defined as
Proof
The result follows by observing that, if \(p_{k} (x )\) is an odd Lidstone polynomial, from (7) we have \(p_{k}'' (x )=2k (2k+1 )p_{k1} (x )\).
The derivation matrix will be used, in the future, for the construction of operational methods, in particular methods for the solution of high order differential equations. □
We observe that the recurrence relations (22) and (27) allow us to prove that the elements of \(OLP\) satisfy some particular linear differential equations.
Theorem 9
If \(p_{k} (x )\) is an odd Lidstone polynomial, it satisfies the following linear differential equations of order 2k:
and
Proof
From Proposition 2 we have \(p_{k}^{(2j)} (x )=\frac{ (2k+1 )!}{ (2 (kj )+1 )!}p_{kj} (x )\). Then, by substituting in the first recurrence relation (22), we get
By differentiating twice, we have
Taking into account that \(p_{k}^{(2k+2)} (x )\equiv0\) and \(p''_{k+1}(x)=(2k+2)(2k+3) p_{k}(x)\), we get
Now, by putting \(p_{k} (x )\equiv y (x )\), (30) follows.
Analogously, from the second recurrence relation (27), we get
from which
With the same technique used for deriving (30), (31) follows. □
Generating function
Let \(\{p_{k} \}_{k}\) be the odd Lidstone polynomial sequence related to the numerical sequence \((\alpha_{2k} )_{k}\), \(\alpha_{0}\ne0\). In order to get the generating function, that is, a function \(G (x,t )\) such that
we consider the formal power series
The hypothesis \(\alpha_{0}\ne0\) implies that the power series (32) is invertible and, more precisely, we have the following result.
Proposition 8
The formal power series (32) is invertible and it results
where \((\beta_{2k} )\), \(k\ge0\), satisfy relation (15).
Proof
The proof follows by verifying that \(l (t )\frac{1}{l (t )}=1\). □
Corollary 2
The coefficients of the formal power series \(l (t )\) and \(\frac{1}{l (t )}\) satisfy the relations in Remark 2.
Theorem 10
(Generating function)
Let \(\{p_{k} \}_{k}\) be an element of \(\tilde{\mathcal{P}}\). Then \(\{p_{k} \}_{k}\) is an odd Lidstonetype polynomial sequence iff there exists a numerical sequence \((\alpha _{2k} )_{k}\), \(\alpha_{0}\ne0\), such that, if \(l(t)\) is defined as in (32), the following equality holds:
Proof
If \(\{p_{k} \}_{k}\in \mathit{OLS}\), then there exists the numerical sequence \((\alpha_{2k} )_{k}\), \(\alpha_{0}\ne0\), and hence we have the sequence \((\beta_{2k} )_{k}\) defined as in (15) such that, according to (21),
By multiplying the two sides of (35) by \(\frac {t^{2k+1}}{(2k+1)!}\) and adding on k, we have
Now we apply the Cauchy product for power series and the known expansion of \(\sinh tx\). Thus we have
From Proposition 8 we get (34).
On the other hand, if (34) holds, from (32) and the Cauchy product of power series, we have
and then the thesis follows. □
Corollary 3
In the hypothesis and notations of Theorem 10, the generating function of the conjugate sequences \(\{\widehat{p}_{k}\}_{k}\) is \(\frac{1}{l(t)} \sinh tx\), that is,
Relationship with Appell polynomial sequences
There is an interesting relationship between Lidstonetype and Appell polynomial sequences. In order to describe this relation, we recall some wellknown definitions and properties of Appell matrices and polynomials [15].
Definition 3
Given a numerical sequence \((\alpha_{i} )_{i}\), with \(\alpha _{0}=1\), an Appelltype matrix is an infinite lower triangular matrix \({\mathcal{A}}_{\infty}\) or, for simplicity, \({\mathcal{A}}= (a_{ij} )_{i,j\ge0}\) whose elements are
The matrix \({\mathcal{A}}_{N}:= (a_{l,k} )\), \(l=0,\ldots ,N\), \(k=0,\ldots,l\), is called Appelltype matrix of order N.
We note that \({\mathcal{A}}_{N}\) is the principal submatrix of \({\mathcal{A}}\), of order N.
Definition 4
The polynomial sequence \(\{{a}_{n} \}_{n}\) defined as
with \(a_{i,j}\) as in (36) is called an Appell polynomial sequence for the matrix \({\mathcal{A}}\) or for the sequence \((\alpha_{i} )_{i}\).
Remark 7
We note explicitly that by (36), \(\forall n\in \mathbb{N}\), \(a_{n}(x)\) can be written as
where the coefficients \(\alpha_{i}\), \(i=0,\ldots,n\), are apriori assigned, and hence independent on the degree n. Moreover, it holds
Theorem 11
([18])
A polynomial sequence \(\{{a}_{n} \}_{n}\) is an Appell polynomial sequence iff
The following theorem establishes a relationship between Appell polynomial sequences and odd Lidstone polynomial sequences.
Theorem 12
([15])
Under the previous hypothesis, the following statements hold:

1.
If \(\{{a}_{k} \}_{k}\) is an Appell polynomial sequence, there exists a unique \(\{p_{k} \}_{k}\in \mathit{OLS}\) that we said associated to \(\{{a}_{k} \}_{k}\).

2.
If \(\{p_{k} \}_{k}\in \mathit{OLS}\), there exist infinitely many Appell polynomial sequences \(\{{a}_{k} \}_{k}\) that we said associated to \(\{p_{k} \}_{k}\).

3.
If \(\{{a}_{k} \}_{k}\) is an Appell polynomial sequence and there exists \(s\in \mathbb{R}\) such that \(a_{2k+1} (\frac{s}{2} )=0\), \(\forall k \in \mathbb{N}\), then the sequence
$$p_{k} (x )=2^{2k+1}{a}_{2k+1} \biggl( \frac{x+s}{2} \biggr) $$is the associated element in OLS.
Examples
Now we consider some interesting examples of polynomial sequences belonging to OLS. The proposed sequences are associated respectively to Bernoulli and Euler polynomials, according to Theorem 12.
Example 1
(Odd Lidstone–Bernoulli polynomial sequence)
Let \(B_{k}(x)\) be the Bernoulli polynomial of degree k. Since \(B_{2k+1} (\frac{1}{2} )=0\), \(\forall k \in \mathbb{N}\) [26], from Theorem 12 the sequence \(\{ p_{k}\}_{k}\) defined by
is an odd Lidstone polynomial sequence. In fact, by known properties of Bernoulli polynomials, \(p_{k}(x)\) satisfies relations (7).
By Proposition 2, we have \(\alpha _{2k}=p'_{k}(0)=(2k+1)2^{2k}B_{2k} (\frac{1}{2} )\). Taking into account [26] that \(B_{2k} (\frac {1}{2} )=(2^{12k}1)B_{2k}\), we get
Coefficients \(\alpha_{2k}\) in (38), for \(k=1,2,\ldots\) , are connected to the coefficients of the expansion of cscht. In fact, we have
Consequently, it results
By comparing (39) with (33), we get \(\beta_{2k}=1\).
The generating function of the sequence \(\{p_{k}\}_{k}\) defined in (37) is
For the conjugate sequence \(\{\widehat{p}_{k}\}_{k}\), the generating function is
The conjugate sequence \(\{\widehat{p}_{k}\}\) satisfies conditions (7).
Moreover, for the two sequences \(\{p_{k}\}_{k}\) and \(\{\widehat{p}_{k}\}_{k}\), we get respectively \(p_{k}(1)=0\) and \(\widehat{p}_{k}'(0)=1\), so we have
and
The odd polynomial sequence \(\{p_{k}\}_{k}\), up to a constant \((2k+1)!\), coincides with the socalled Lidstone polynomial sequence [27], that is, \(\varLambda _{k}(x)=\frac{p_{k}(x)}{(2k+1)!}\), where \(\varLambda _{k}\), \(k\in \mathbb{N}\), are the Lidstone polynomials of first type.
Observe that, if \(\varOmega _{k}(x)=\frac{\widehat{p}_{k}(x)}{(2k+1)!}\), then the two sequences \(\{\varLambda _{k}\}_{k}\) and \(\{\varOmega _{k}\}_{k}\) are not conjugate sequences since \(\varLambda _{k} \circ \varOmega _{k}\neq x^{2k+1}\).
Here we do not give all the properties of classic Lidstone polynomials which can be obtained from the results of the previous sections and from a wide literature (see [5, 11, 15, 23, 27, 36, 37] and the references therein).
Example 2
(Odd Lidstone–Euler polynomial sequence)
Let \(E_{k}(x)\) be the Euler polynomial of degree k. Since \(E_{2k+1} (\frac{1}{2} )=0\), \(\forall k \in \mathbb{N}\) [26], from Theorem 12 the sequence
is an odd polynomial sequence belonging to OLS.
In fact, from the properties of Euler polynomials, \(\{p_{k}\}_{k}\) satisfies (7).
By Proposition 2, we have \(\alpha _{2k}=p'_{k}(0)=(2k+1)2^{2k}E_{2k} (\frac{1}{2} )\). Taking into account [26] that \(E_{2k}=2^{2k} E_{2k} (\frac{1}{2} )\) is the Euler number, we get
Hence we have
From (41), the coefficients \(\alpha_{2k}\) are connected to the coefficients of the expansion of secht:
Thus, according to (32), we have \(l(t)=\operatorname {sech}t\) and
By comparing (42) and (33), we get
Hence, the generating function of the sequence \(\{p_{k}\}_{k}\) defined in (37) is
The generating function of the conjugate sequence \(\{\widehat{p}_{k}\} _{k}\) is
and from (43), we get
The conjugate sequence \(\{\widehat{p}_{k}\}_{k}\) satisfies conditions (7).
Moreover, for the two sequences \(\{p_{k}\}_{k}\) and \(\{\widehat{p}_{k}\}_{k}\), we get respectively
so we have
and
Unlike the sequence in Example 1, the odd Lidstone–Euler polynomial sequence of Example 2 is not known in the literature, except in [15] and in [20, Example 3.3], hence a deeper study will be done in the future.
Even Lidstonetype sequences
Now, in analogy with the odd case, we consider the class of even Lidstonetype polynomial sequences. We only give the statements of the theorems and properties.
To this aim, let \(\widehat{\mathcal{P}} = \operatorname{span} \{x^{2i} \mid i=0,1,\ldots \}\). We denote by ELS (even Lidstone sequences) the set of polynomial sequences satisfying
From (44) it follows that

(a)
\(q_{k} (x )\), \(k=1,2,\ldots\) , are polynomials of degree 2k;

(b)
\(q_{0} (x )=\gamma_{0}\), \(\gamma_{0}\in \mathbb{R}\), \(\gamma _{0}\ne0\).
As in the case of odd Lidstone sequences, we will give a more complete characterization of the set ELS.
Proposition 9
A polynomial sequence \(\{q_{k} \}_{k}\) is an element of ELS iff there exists a numerical sequence \(\{\gamma_{2j}\}_{j}\), \(\gamma _{0}\ne0\), \(\gamma_{j}\in \mathbb{R}\) such that
Remark 8
From Proposition 9, \(\mathit{ELS}\subset\widehat {\mathcal{P}}\).
Proposition 10
If \(\{q_{k}\}_{k}\in \mathit{ELS}\), then
Matrix form
Now we introduce the even Lidstonetype matrix \(E_{\infty}\) related to a numerical sequence \((\gamma_{2k} )_{k}\), \(\gamma_{0}\ne0\). \(E_{\infty}\) is an infinite lower triangular matrix with elements \(e_{i,j}\) given by
Let \(\widehat{X}_{\infty}\) and \(Q_{\infty}\) be the infinite vectors
Then polynomials (45) can be written in a matrix form as \(Q_{\infty}=E_{\infty}\widehat{X}_{\infty}\) or
with \(Q=Q_{\infty}\), \(E=E_{\infty}\), \(\widehat{X}=\widehat{X}_{\infty}\).
If
and \(E_{n}= (e_{i,j} )_{i,j=0}^{n}\), from (47) we have \(Q_{n}=E_{n} \widehat{X}_{n}\).
Proposition 11
The infinite lower triangular matrix \(E= (E_{i,j} )_{i,j\ge 0}\) defined in (46) can be factorized as
where
and \(T_{\gamma}\) is the lower triangular Toeplitz matrix with entries \(t_{i,j}^{\gamma}=\frac{\gamma_{2(ij)}}{(2(ij))!}\).
Proposition 12
The matrix E, defined in (46), is invertible and
where D̂ is the diagonal matrix defined in (49), \(\widehat{T}_{\zeta}\) is the lower triangular Toeplitz matrix with entries \(t_{i,j}^{\zeta}=\frac{\zeta_{2(ij)}}{(2(ij))!}\), where the sequence \((\zeta_{2i} )_{i}\) is implicitly defined by
Remark 9
Given the numerical sequence \((\gamma_{2k} )_{k}\), \(\gamma _{0}\ne0\), the infinite linear system (50) allows us to determine a numerical sequence \((\zeta_{2k} )_{k}\). In fact, \(\forall i\in \mathbb{N}\), by applying Cramer’s rule, the first \(i+1\) equations in (50) give
For symmetry, the coefficients \(\gamma_{2i}\), \(i=1,\ldots,n\), have an expression similar to (51) by exchanging in (51) \(\gamma_{2i}\) with \(\zeta_{2i}\).
Conjugate even Lidstone polynomials
Let \((\gamma_{2k} )_{k}\), \(\gamma_{0}\ne0\), be a given numerical sequence and \((\zeta_{2k} )_{k}\) be the related sequence defined in (51). We consider the polynomials
From Proposition 9, \(\{\widehat{q}_{k} \}_{k}\) in (52) is an even Lidstonetype polynomial sequence. We call the two sequences \(\{q_{k} \}_{k}\) and \(\{\widehat{q}_{k} \}_{k}\) conjugate even Lidstonetype sequences.
In order to get the matrix form of the odd Lidstone sequence \(\{ \widehat{q}_{k} \}_{k}\), we set \(\widehat{Q}= [\widehat{q}_{0},\widehat{q}_{1},\ldots,\widehat {q}_{k},\ldots ]^{T}\), \(F= (f_{ij} )\), with
From (52) \(\widehat{Q}=F\widehat{X}\) and, \(\forall n\in \mathbb{N}\), \(\widehat{Q}_{n}=F_{n}\widehat{X}_{n}\).
Proposition 13
With the previous notations and hypothesis, the sequences \(\{ q_{k} \}_{k}\) and \(\{\widehat{q}_{k} \}_{k}\) are conjugate even Lidstonetype sequences iff
Corollary 4
The polynomials \(q_{k} (x )\) and \(\widehat{q}_{k} (x )\) can be written as
where \(e_{k,j}^{*}\) and \(f_{k,j}^{*}\), \(j=0,\ldots, k\), are the elements of matrices \(E^{2}\) and \(F^{2}\), respectively.
Remark 10
It is possible to endow the set ELS with an algebraic structure \(\widetilde{ELS}:= (\mathit{ELS},+,\circ )\), where + is the usual addition in ELS and ∘ is the umbral composition. This structure is a linear space which is isomorphic to the linear space of even Lidstonetype matrices.
First recurrence relation and determinant form
In order to obtain a recurrence relation, from (47) we get \(\widehat{X}=E^{1}Q=FQ\). Hence, for each \(k\in \mathbb{N}\),
where \(\widehat{X}_{k}\) and \(Q_{k}\) are defined as in (48). From (53), for \(i=0,\ldots,k\), we obtain
By the same techniques used in the case of odd Lidstone sequences, the following theorems can be proved.
Theorem 13
Let \(\{q_{k} \}_{k}\) be an element of \(\widehat{\mathcal{P}}\). \(\{q_{k} \}_{k}\in \mathit{ELS}\) iff there exists a numerical sequence \((\gamma_{2k} )_{k}\), \(\gamma_{0}\ne0\), and hence the numerical sequence \((\zeta_{2k} )_{k}\) defined as in (50) such that the following recursive relation holds:
Theorem 14
(First determinant form)
A sequence \(\{q_{k} \}_{k}\subset\widehat{\mathcal{P}}\) is an even Lidstone polynomial sequence, that is, \(\{q_{k} \}_{k}\in \mathit{ELS}\), related to the numerical sequence \((\gamma_{2k} )_{k}\), \(\gamma_{0}\ne0\), iff the following representation holds:
where \((\zeta_{2k} )_{k}\) are defined as in (50).
Remark 11
Expanding the determinant in (54) with respect to the first row, we get (45).
Theorem 15
Given a polynomial sequence \(\{q_{k}\}_{k}\), the conjugate even polynomial sequence of Lidstone type \(\{\widehat{q}_{k}\}_{k}\) related to the numerical sequence \(\{\gamma_{2k}\}_{k}\) has a determinant representation similar to (54) with \(\gamma_{2i}\) instead of \(\zeta_{2i}\), \(i=0,\ldots, k\).
Second recurrence relation and determinant form
Proposition 14
Let \(E= (e_{i,j} )_{i,j\ge0}\) be an even Lidstonetype matrix, and let \(F= (f_{i,j} )_{i,j\ge0}\) be the inverse matrix. Then the elements \(\overline{\pi}_{i,j}\), \(i\ge0\), \(j=0,\ldots , i+1\), of the production matrix Π̅ of F are given by
where \((\gamma_{2k} )_{k}\) and \((\zeta_{2k} )_{k}\) are given in Remark 9.
Theorem 16
Let \(\{q_{k} \}_{k}\in \mathit{ELS}\). Then, for all \(k\ge1\), \(q_{k}(x)\) satisfies the following recurrence relation:
where \(\overline{\pi}_{i,j}\), \(i,j\ge0\), are the elements of the production matrix Π̅ of F.
Theorem 17
If \(\{q_{k} \}_{k}\) is an even Lidstone polynomial sequence, then, for \(k\ge0\), the following determinant representation holds:
where \(\overline{\pi}_{k,j}\) are defined as in (55) and \(q_{0}(x)=\frac{1}{\zeta_{0}}\).
Generating function
In order to get the generating function for (45), we consider the formal power series
Proposition 15
The formal power series (56) is invertible and it results
where \((\zeta_{2k} )\), \(k\ge0\), satisfy relation (50).
Corollary 5
The coefficients of the series \(h (t )\) and \(\frac {1}{h (t )}\) are defined as in Remark 9.
Theorem 18
(Generating function)
Let \(\{q_{k} \}_{k}\) be an element of \(\widehat{\mathcal{P}}\). Then \(\{q_{k} \}_{k}\in \mathit{ELS}\) iff there exists a numerical sequence \((\gamma_{2k} )_{k}\), \(\gamma_{0}\ne0\), such that, if \(h(t)\) is defined as in (56), the following equality holds:
For the set ELS there holds the analogue of Theorem 12.
Theorem 19
([15])
The following statements hold:

1.
If \(\{{a}_{k} \}_{k}\) is an Appell polynomial sequence, there exists a unique \(\{q_{k} \}_{k}\in \mathit{ELS}\) that we said associated to \(\{{a}_{k} \}_{k}\).

2.
If \(\{q_{k} \}_{k}\in \mathit{ELS}\), there exist infinitely many Appell polynomial sequences \(\{{a}_{k} \}_{k}\) that we said associated to \(\{q_{k} \}_{k}\).

3.
If \(\{{a}_{k} \}_{k}\) is an Appell polynomial sequence and there exists \(s\in \mathbb{R}\) such that \(a_{2k+1} (\frac{m}{2} )=0\), \(k=0,1,\ldots\) , then the sequence \(q_{k}(x)=2^{2k}a_{2k} (\frac{x+m}{2} )\) is the associated element in ELS.
Examples
Now we present some examples of polynomial sequences in ELS. These sequences are related respectively to Bernoulli and Euler polynomials according to Theorem 19.
Example 3
(Even Lidstone–Bernoulli polynomial sequence)
Let \(\{q_{k}\}_{k}\) be the even Lidstonetype sequence defined by
where \(B_{k}(x)\) is the Bernoulli polynomial of degree k.
By known proprieties of Bernoulli polynomials, we have that the sequence \(\{q_{k} \}_{k}\) defined in (57) satisfies relations (44).
Observe that polynomials \(q_{k}(x)\) are connected to Lidstone polynomials of first type since \(q_{k}(x)=(2k)!\varLambda '_{k}(x)\) for \(k\ge0\).
Moreover, from (45) and (57) we have
Hence the generating function of sequence (57) is
The conjugate sequence of \(\{q_{k}\}_{k}\) is \(\widehat{q}_{k}(x)=\sum_{i=0}^{k}\binom{2k}{2i}\zeta_{2(ki)}x^{2i}\), \(k=0,1,\ldots\) , where \(\{\gamma_{2k}\}\) and \(\{\zeta_{2k}\}\) verify the relation
From (58), the generating function of \(\{\widehat{q}_{k}\}\) is
The even polynomial sequence (57), up to a constant \((2k)!\), coincides with the socalled Lidstone polynomial sequence of second type [19, 27], also called complementary Lidstone polynomial sequence [2, 22]. It is denoted by \(\{ v_{k}\}_{k}\), that is, \(v_{k}(x)=\frac{q_{k}(x)}{(2k)!}\).
We observe that, if \(w_{k}(x)=\frac{\widehat{q}_{k}(x)}{(2k)!}\), then the two sequences \(\{v_{k}\}_{k}\) and \(\{w_{k}\}_{k}\) are not conjugate sequences, that is, \(v_{k} \circ w_{k}\neq x^{2k}\).
Example 4
(Even Lidstone–Euler polynomial sequence)
Let \(\{q_{k}\}_{k}\) be the even polynomial sequence defined by
where \(E_{k}(x)\) is the Euler polynomial of degree k.
By known proprieties of Euler polynomials, we have that the sequence \(\{q_{k} \}_{k}\) defined in (59) satisfies relations (44). Moreover, this sequence is connected to the odd Lidstone–Euler sequence \(\{p_{k}\}_{k}\) considered in Example 2 by
The generating function of \(\{q_{k}\}_{k}\) is
In fact, let us consider the series \(h(t)=\sum_{k=0}^{\infty}\gamma _{2k}\frac{t^{2k}}{(2k)!}\) with \(\gamma_{2k}=q_{k}(0)\). Since \(q_{k}(0)=2^{2k}E_{2k} (\frac{1}{2} )=E_{2k}\), where \(E_{2k}\) are the Euler numbers, we have
The generating function of the conjugate polynomial sequence \(\{ \widehat{q}_{k}\}_{k}\) is
In fact, \(\frac{1}{h(t)}=\cosh t=\sum_{k=0}^{\infty}\frac {t^{2k}}{(2k)!}=\sum_{k=0}^{\infty}\zeta_{2k}\frac{t^{2k}}{(2k)!}\), with \(\zeta_{2k}=1\).
Remark 12
The p.s. \(\{\widehat{q}_{k}\}_{k}\), up to a constant, has been used in the context of operator approximation theory, precisely for a new generalized Szäsztype operator [13].
A new general linear interpolation problem
In order to sketch an application of these classes of polynomial sequences, we consider a linear space X of realvalued functions that are sufficiently regular; \({\mathcal {P}}_{n}\subset X\), \(\forall n\in \mathbb{N}\). Let L be a linear functional on X with \(L (x )\ne0\). We will look for the unique polynomial \(P_{n} [f ]\) of degree \(\le2n+1\) such that
and \(f(x)=P_{n} [f ](x)+R_{n}(f)\), where \(R_{n}(f)\) is the remainder.
The study of the convergence of the series \(\sum_{n=0}^{\infty}P_{n} [f ](x)\) to \(f(x)\) is interesting.
Analogously, if \(L_{1}\) is a linear functional on X with \(L_{1} (1 )\ne0\), we will look for the unique polynomial \(\tilde{P}_{n} [f ]\) of degree \(\le2n\) such that
and \(f(x)=L_{1}(f)+\tilde{P}_{n} [f ](x)+ \tilde{R}_{n}(f)\).
We will study the convergence of the series \(L_{1}(f)+\sum_{n=0}^{\infty}\tilde{P}_{n} [f ](x)\) to \(f(x)\).
We conjecture that the two problems are solvable in OLS and ELS, respectively. The solution to this problem will be investigated in detail in a subsequent paper.
Conclusions
In this paper we have introduced two new classes of polynomial sequences called respectively the odd Lidstone polynomial sequences (OLS) and the even Lidstone polynomial sequences (ELS). For each of the two classes, we studied some properties, including the matrix form, the conjugate sequences, the generating functions, some recurrence relations, and determinant forms.
Some particular cases are considered in the examples. The proposed sequences are associated respectively to Bernoulli and Euler polynomials. The sequences related to Bernoulli polynomials coincide, up to a constant, to the classic Lidstone polynomial sequences of first and second type, respectively. The other two sequences, to the knowledge of the authors, have not appeared in the literature.
In the future, applications of these polynomial classes will be considered, including general linear interpolation, operators approximation theory, solution of high order boundary value problems, numerical quadrature, and spline functions.
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Costabile, F.A., Gualtieri, M.I., Napoli, A. et al. Odd and even Lidstonetype polynomial sequences. Part 1: basic topics. Adv Differ Equ 2018, 299 (2018). https://doi.org/10.1186/s1366201817335
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MSC
 41A58
 11B83
Keywords
 Lidstone polynomials
 Polynomial sequences
 Infinite matrices