Some Convolution Formulae Related to the Second-Order Linear Recurrence Sequence

Abstract

The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is f ( t ) = 1 1 + a t + b t 2 , we can give the exact coefficient expression of the power series expansion of f x ( t ) for x ∈ R with elementary methods and symmetry properties. On the other hand, if we take some special values for a and b, not only can we obtain the convolution formula of some important polynomials, but also we can establish the relationship between polynomials and themselves. For example, we can find relationship between the Chebyshev polynomials and Legendre polynomials.

Some New Identities of Second Order Linear Recurrence Sequences

“…is a fourth-order recurrence sequence, and it satisfy the recurrence Formula (12). This completes the proof of Theorem 2.…”

“…Many other papers related to Fibonacci numbers, Fibonacci polynomials, Chebyshov polynomials and second-order linear recurrence sequences can also be found in references [5][6][7][8][9][10][11][12][13][14][15][16][17][18], here we will no longer list them one by one.…”

General Linear Recurrence Sequences and Their Convolution Formulas

“…Starting from the results by Chen Zhuoyu and Qi Lan [9], we have shown convolution formulas and linear recurrence relations satisfied by a generating function containing several parameters. This can be used for number sequences (assuming x = 1) or polynomial sequences, depending on several parameters.…”

“…In a recent article Chen Zhuoyu and Qi Lan [9] introduced convolution formulas for second order linear recurrence sequences related to the generating function [1] of the type…”

Tribonacci Numbers and Some Related Interesting Identities

“…As an interesting corollary of [3], Chen Zhuoyu and Qi Lan proved that, for any positive integer k, one has the identity:…”