6 - q-Extensions of Some Special Functions and Polynomials

Abstract

The hypergeometric functions can be generalized along the lines of basic (or q-) number, resulting in the formation of q-extensions (or q-analogues). This chapter provides an overview of the q-Extensions of some special functions and polynomials. These extensions are potentially useful in variety of fields such as theory of partitions, number theory, combinatorial analysis, finite vector spaces, Lie theory, particle physics, nonlinear electric circuit theory, mechanical engineering, theory of heat conduction, quantum mechanics, cosmology, and statistics. The chapter illustrates qdevelopments around the zeta and related functions which include Jackson's q-integral, the q-Gamma function, and the q-Beta function, multiple q-Gamma functions, q-Bernoulli numbers, and q-Bernoulli polynomials, q-Euler numbers and polynomials, q-zeta functions, and multiple q-zeta functions.