Integral and Series Representations of q-Polynomials and Functions: Part I

2016/04/27 by Mourad E. H. Ismail, Ismail, Mourad E. H., Ruiming Zhang +1
Mathematics · #33D15 #33D45 #33D60 #42A38 #44A20 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1604.08441

openalex publication_date 2016/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By applying an integral representation for q^k² we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of q-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problems. These functions include q-Bessel functions, the Ramanujan function, Stieltjes--Wigert polynomials, q-Hermite and q^-1-Hermite polynomials, and the q-exponential functions e_q, E_q and E_q. Their representations are in turn used to derive many new identities involving q-functions and polynomials. In this work we also present contour integral representations for the above mentioned functions and polynomials.

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