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

Qiuxia Hu; Bilal Khan; Serkan Araci; Qiuxia Hu; Bilal Khan; Serkan Araci

doi:10.3934/math.2025067

AIMS Mathematics

2025, Volume 10, Issue 1: 1448-1462. doi: 10.3934/math.2025067

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

1.
Department of Mathematics, Luoyang Normal University, Luoyang 471934, China
2.
Institute of Mathematics, Henan Academy of Sciences, Zhengzhou 450046, China
3.
Department of Computer Engineering, Faculty of Engineering, Hasan Kalyoncu University, TR-27010, Gaziantep, Türkiye

Received: 02 December 2024 Revised: 07 January 2025 Accepted: 15 January 2025 Published: 22 January 2025
MSC : Primary 05A19, Secondary 33D15

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.
- $ q $-Chu-Vandermonde formula,
- the probability distribution $ W(x, y;q) $,
- basic hypergeometric series,
- $ q $-integral
Citation: Qiuxia Hu, Bilal Khan, Serkan Araci. Expectation formulas for $ q $-probability distributions: a new extension via Andrews-Askey integral[J]. AIMS Mathematics, 2025, 10(1): 1448-1462. doi: 10.3934/math.2025067

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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.

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