Expectation formulas for $ q $-probability distributions: a new extension via Andrews-Askey integral

Abstract

In this paper, we utilize the $ q $-Chu-Vandermonde formula to derive a novel expectation formula for the $ q $-probability distribution $ W(x, y; q) $, extending previously known results. Several applications are presented, including a broader generalization of the Andrews-Askey integral. Although fractional $ q $-calculus is not directly employed in this work, its potential for future extensions is discussed, as non-integer order derivatives and integrals could offer deeper insights into $ q $-series and probability distributions.

Keywords

• $ q $-chu-vandermonde formula
• the probability distribution $ w(x, y;q) $
• basic hypergeometric series
• $ q $-integral