Czechoslovak Mathematical Journal, first online, pp. 1-24


A note on $q$-partial difference equations and some applications to generating functions and $q$-integrals

Da-Wei Niu, Jian Cao

Received October 10, 2017.   Published online December 27, 2018.

Abstract:  We study the condition on expanding an analytic several variables function in terms of products of the homogeneous generalized Al-Salam-Carlitz polynomials. As applications, we deduce bilinear generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials. We also gain multilinear generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials. Moreover, we obtain generalizations of Andrews-Askey integrals and Ramanujan $q$-beta integrals. At last, we derive $U(n+1)$ type generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials.
Keywords:  $q$-partial difference equation; homogeneous generalized Al-Salam-Carlitz polynomial; generating function; Andrews-Askey integral; Ramanujan $q$-beta integral
Classification MSC:  05A30, 11B65, 33D15, 33D45, 33D50, 35C11
DOI:  10.21136/CMJ.2018.0470-17

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Affiliations:   Da-Wei Niu (corresponding author), School of Mathematical Science, East China Normal University, 500 Dongchuan Road, Shanghai 200241, P. R. China, e-mail: nnddww@163.com, nnddww@gmail.com; Jian Cao, Department of Mathematics, Hangzhou Normal University, Hangzhou City, Zhejiang Province, 310036, P. R. China, e-mail: 21caojian@gmail.com


 
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