{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T11:15:20Z","timestamp":1760181320715,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,2]],"date-time":"2020-11-02T00:00:00Z","timestamp":1604275200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Our investigation is motivated essentially by the demonstrated applications of the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, in many diverse areas. Here, in this paper, we use two q-operators T(a,b,c,d,e,yDx) and E(a,b,c,d,e,y\u03b8x) to derive two potentially useful generalizations of the q-binomial theorem, a set of two extensions of the q-Chu-Vandermonde summation formula and two new generalizations of the Andrews-Askey integral by means of the q-difference equations. We also briefly describe relevant connections of various special cases and consequences of our main results with a number of known results.<\/jats:p>","DOI":"10.3390\/sym12111816","type":"journal-article","created":{"date-parts":[[2020,11,2]],"date-time":"2020-11-02T09:04:46Z","timestamp":1604307886000},"page":"1816","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["A Note on Generalized q-Difference Equations and Their Applications Involving q-Hypergeometric Functions"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari M.","family":"Srivastava","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7173-0591","authenticated-orcid":false,"given":"Jian","family":"Cao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hangzhou Normal University, Hangzhou City 311121, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6699-3525","authenticated-orcid":false,"given":"Sama","family":"Arjika","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Informatics, University of Agadez, Post Office Box 199, Agadez 8000, Niger"},{"name":"International Chair of Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Post Office Box 072, Cotonou 50, Benin"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,2]]},"reference":[{"unstructured":"Gasper, G., and Rahman, M. 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