On a Class of Polynomials Generated by F (xt -- R(t))

2017/03/09 by Mohammed Mesk, Mesk, Mohammed, Mohammed Brahim Zahaf +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Nonlinear Waves and Solitons #math.CA #math.CO

paper · pdf · doi:10.48550/arxiv.1703.03314

arXiv admin note: text overlap with arXiv:1605.05181

arxiv created 2017/03/09 · openalex publication_date 2017/03/09 · arxiv updated 2017/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate polynomial sets P n n≥0 with generating power series of the form F (xt -- R(t)) and satisfying, for n ≥ 0, the (d + 1)-order recursion xP_ n (x) = P_ n+1 (x) +∑_ l=0^d γ^l_n P_ n--l (x), where γ^l_ n is a complex sequence for 0 ≤ l ≤ d, P _0 (x) = 1 and P _n (x) = 0 for all negative integer n. We show that the formal power series R(t) is a polynomial of degree at most d + 1 if certain coefficients of R(t) are null or if F (t) is a generalized hypergeometric series. Moreover, for the d-symmetric case we demonstrate that R(t) is the monomial of degree d + 1 and F (t) is expressed by hypergeometric series.

Citations

• A new characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind

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