Sequences of Non-Gegenbauer-Humbert Polynomials Meet the Generalized Gegenbauer-Humbert Polynomials

Tian-Xiao He,Peter J.-S Shiue

doi:10.5402/2011/268096

Tian-Xiao He, Peter J.-S Shiue

Open Access

https://doi.org/10.5402/2011/268096

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Abstract

Here, we present a connection between a sequence of polynomials generated by a linear recurrence relation of order 2 and sequences of the generalized Gegenbauer-Humbert polynomials. Many new and known transfer formulas between non-Gegenbauer-Humbert polynomials and generalized Gegenbauer-Humbert polynomials are given. The applications of the relationship to the construction of identities of polynomial sequences defined by linear recurrence relations are also discussed.

Highlights

Many number and polynomial sequences can be defined, characterized, evaluated, and classified by linear recurrence relations with certain orders
In [6], Aharonov, Beardon, and Driver have proved that the solution of any sequence of numbers that satisfies a recurrence relation of order 2 with constant coefficients and initial conditions a0 = 0 and a1 = 1, called the primary solution, can be expressed in terms of Chebyshev polynomial values
Similar to the case of the Jacobsthal polynomial sequence shown in Example 4, we have the inverse formulas: Un(x) = Sn(±2x), Pn+1(x) = (∓i)n Sn (±2xi), Fn+1(x) = (∓i)n Sn (±xi), Bn(x) = (±1)nSn(±(x + 2))

Summary

Introduction

Many number and polynomial sequences can be defined, characterized, evaluated, and classified by linear recurrence relations with certain orders. In [6], Aharonov, Beardon, and Driver have proved that the solution of any sequence of numbers that satisfies a recurrence relation of order 2 with constant coefficients and initial conditions a0 = 0 and a1 = 1, called the primary solution, can be expressed in terms of Chebyshev polynomial values. In [5], the authors presented a new method to construct an explicit formula of {an(x)} generated by (1) Inspired with those results, in [9], the authors and Weng established a relationship between the number sequences defined by recurrence relation (1) and the generalized Gegenbauer-Humbert polynomial value sequences.

Main results

Examples and applications