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.