On a generalized homogeneous Hahn polynomial

We investigate a generalized homogeneous Hahn polynomial in some detail. This polynomial includes as special cases the homogeneous Hahn polynomial and the homogeneous Rogers-Szegő polynomial. A generating function, which contains a known generating function as a special case, is given. We also give a finite series generating function. Some results on the asymptotic expansion for this polynomial are derived. Certain results on zeros are also obtained. We deduce several results on zeros of certain entire functions involving this generalized Hahn polynomial. As results, one of Zhang (2017)’s results as well as others is obtained. Finally, we derive several general results on q-congruences of the generalized q-Apéry polynomials, from which two q-congruences involving the generalized homogeneous Hahn polynomial are deduced.