Densely generated 2D q-Appell polynomials of Bessel type and q-addition formulas

Mumtaz Riyasat

doi:10.5269/bspm.46923

Abstract

The article aims to introduce a densely generated class of $2D$ $q$-Appell polynomials of Bessel type via generating equation and to establish its $q$-determinant form. It is advantageous to consider the $2D$ $q$-Bernoulli, $2D$ $q$-Roger Szeg\"{o} and $2D$ $q$-Al-Salam Carlitz polynomials of Bessel type as their special members. The $q$-determinant forms and certain $q$-addition formulas are derived for these polynomials. The article concludes with a brief view on discrete $q$-Bessel convolution of the $2D$ $q$-Appell polynomials.

Highlights

The article aims to introduce a densely generated class of 2D q-Appell polynomials of Bessel type and to investigate their properties
The special polynomials of two variables are important from the point of view of applications in different branches of pure and applied mathematics and physics
Carlitz polynomials of Bessel type are considered as their special members and corresponding determinant forms and q-addition formulas are derived

Summary

Generating function

Motivated by the importance and relevance of the two-dimensional Bessel functions, in 2018 Riyasat and Khan introduce the two-dimensional (or 2D) q-Bessel polynomials pn,q(x, y), which are defined by means of the following generating function:. Taking y = 0 and x = 0, consecutively in above equation, the two forms of q-Bessel polynomials pn,q(x) and Pn,q(x) are deduced, which are defined by the following generating functions: eq The densely generated 2D q-Appell polynomials of Bessel type are defined by means of generating function and determinant form. Carlitz polynomials of Bessel type are considered as their special members and corresponding determinant forms and q-addition formulas are derived. We introduce a dense form of generating function for the 2D q-Appell polynomials of Bessel type.

Expanding the exponential function

Concluding remarks

Series expansions