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Keywords:
$q$-partial difference equation; homogeneous generalized Al-Salam-Carlitz polynomial; generating function; Andrews-Askey integral; Ramanujan $q$-beta integral
Summary:
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.
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