Gottlieb Polynomials and Their <i>q</i>-Extensions

Abstract

Since Gottlieb introduced and investigated the so-called Gottlieb polynomials in 1938, which are discrete orthogonal polynomials, many researchers have investigated these polynomials from diverse angles. In this paper, we aimed to investigate the q-extensions of these polynomials to provide certain q-generating functions for three sequences associated with a finite power series whose coefficients are products of the known q-extended multivariable and multiparameter Gottlieb polynomials and another non-vanishing multivariable function. Furthermore, numerous possible particular cases of our main identities are considered. Finally, we return to Khan and Asif’s q-Gottlieb polynomials to highlight certain connections with several other known q-polynomials, and provide its q-integral representation. Furthermore, we conclude this paper by disclosing our future investigation plan.

Keywords

• Gottlieb polynomials in several variables
• <i>q</i>-Gottlieb polynomials in several variables
• generating functions
• generalized and generalized basic (or -<i>q</i>) hypergeometric function
• Lauricella’s multiple hypergeometric series in several variables
• <i>q</i>-binomial theorem