Closed-form solution of a general three-term recurrence relation
© Gonoskov; licensee Springer. 2014
Received: 21 March 2014
Accepted: 25 June 2014
Published: 23 July 2014
We present a closed-form solution for n th term of a general three-term recurrence relation with arbitrary given n-dependent coefficients. The derivation and corresponding proof are based on two approaches, which we develop and describe in detail. First, the recursive-sum theory, which gives the exact solution in a compact finite form using a recursive indexing. Second, the discrete dimensional-convolution procedure, which transforms the solution to the non-recursive expression of n, including a finite number of elementary operations and functions.
where (), is unknown function of n, and are arbitrary given functions of n, and , are initial conditions (we assume not to have the trivial case when and ).
This well-known relation has a large number of applications and plays an important role in many areas of mathematics and physics. We recall here a few of them to underline the importance of the proposed statement.
First, the recurrence relation Eq. (1) corresponds to the finite difference equation for the general second order differential equation with unknown function and arbitrary given , namely: . Therefore, it is widely used for the analytical and numerical analysis (and approximations) in corresponding physical and mathematical applications; see for example .
Second, three-term recurrence relations appear naturally when one uses the Frobenius method for solving some linear differential equations and studying some special functions; see .
Third, Eq. (1) corresponds to a continued fraction with arbitrary given coefficients. Basically, the three-term recurrence relation corresponds to the expressions for the numerators and denominators of this continued fraction (see Chapter IV in ), which were derived by Euler.
In this manuscript we obtain a closed-form solution of a canonical three-term recurrence relation, which is equivalent to Eq. (1) in the case of (, ∀n). Our goal is to obtain the solution consisting of a finite number of terms, rather than a variety of methods with infinite series . In that way we develop two approaches for the expressions with a finite number of terms. We start with the recursive-sum theory, which is presented in Section 3. It allows one to obtain the exact solution of the three-term recurrence relation in a compact form using recursive indexing. Next, we develop the discrete dimensional-convolution procedure, which allows one to eliminate recursive indexing and to represent the solution as a closed-form expression. It means that the final closed-form expression depends on arbitrary given coefficients and includes a finite number of elementary operations, such as: ‘+’, ‘−’, ‘×’, ‘÷’; and elementary functions, such as the Heaviside step function (or unit step function) and the floor function for integer division.
where includes all terms of the power p. In the next sections we demonstrate that each is equal to the corresponding recursive-sum, and we rigorously prove all the propositions.
3 Recursive-sum theory
In this section we start from the definition of a general recursive-sum (R-sum). Next, we present and prove some of its properties, which determine the R-sum algebra. It could be useful for solving different recurrence relations, but, particularly for the solution of the canonical three-term recurrence relation it is sufficient to use a particular case of the R-sum, namely the reduced R-sum, which is described in the next section.
Next, we present some of the elementary properties of the R-sum, P.(1)-P.(3), which follow directly from the definition.
which proves the key lemma. □
4 Exact solution of a canonical three-term recurrence relation via finite R-sum expansion
In this section we start with a definition of a reduced R-sum, a particular case of the general R-sum. Then we construct the exact solution of the canonical three-term recurrence relation, by using the R-sum key lemma.
Now, we will rigorously prove a theorem about the exact solution of a canonical three-term recurrence relation Eq. (4).
To prove it for all n, we consider below two cases. First case: n is an arbitrary even number, , . Second case: n is an arbitrary odd number, , .
which proves the first case of the theorem.
Second case, .
which proves the second case and Theorem 1. □
5 Discrete dimensional-convolution procedure
In this section we develop a procedure which could be used for the transformation of the exact solution Eq. (21) to the expression without recursive indexing.
where (), , is an arbitrary given natural number, is an array of arbitrary given natural numbers, and is an arbitrary given function of .
Now we construct one-to-one mapping between the global index q and the index numbers . For that we need to solve Eq. (36), i.e. to express any certain index number as a function of q (not depending on any other index numbers). We do that separately for , , and all others, (, ).
Summarizing, the proposed procedure allows one to calculate a reduced R-sum without recursive indexing.
6 Closed-form solution of the canonical three-term recurrence relation
where , (, ), and is the unit step function (see Eq. (33)).
Remark The integer-valued function determines the index number of arbitrary given in the canonical three-term recurrence relation; see Eq. (4). The first term in the d-products in Eq. (48) is according to Eq. (46) and the corresponding condition , so for the brevity in our notation we imply: .
7 Fibonacci numbers
The Fibonacci number () corresponds to the total number of terms (d-products) in the solution of the canonical three-term recurrence relation since the exact solution consists of a sum of unique combinations of products; see Eqs. (4)-(5) and Eqs. (46)-(48).
This expression gives the exact solution of the Fibonacci sequence without involving irrational numbers; the solution includes only certain combinations of natural numbers.
In summary, we obtain the closed-form solution of a canonical three-term recurrence relation Eq. (4) with an arbitrary given n-dependent coefficient . The final non-recursive expression Eq. (48) includes a finite number of elementary operations and functions.
Possible applications of the developed approaches, namely the R-sum theory and the discrete dimensional-convolution procedure, are not limited by the considered statement. Due to its universality, they could be used for solving other recursive problems, in particular many-term recurrence relations.
An interesting and open question for the author is how the solution Eq. (48) could be efficiently used for approximations and solving differential equations.
We use the following notations: , .
We define double factorial for an arbitrary number as follows: .
IG would like to thank his wife Alla for her help and understanding during the preparation of the manuscript.
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