Geodesic
A note on $q$-partial difference equations and some applications to generating functions and $q$-integrals
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 3, pp. 671-694

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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.
DOI : 10.21136/CMJ.2018.0470-17
Classification : 05A30, 11B65, 33D15, 33D45, 33D50, 35C11
Keywords: $q$-partial difference equation; homogeneous generalized Al-Salam-Carlitz polynomial; generating function; Andrews-Askey integral; Ramanujan $q$-beta integral 
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Niu, Da-Wei; Cao, Jian. A note on $q$-partial difference equations and some applications to generating functions and $q$-integrals. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 3, pp. 671-694. doi: 10.21136/CMJ.2018.0470-17

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