The mission of this talk is to find two general q-operational equations together with the expansion issue of the bivariate q-Laguerre polynomials from the perspective of q-partial differential equations. We also give some applications including some q-Hardy--Hille type formulas and the generalized Askey--Wilson polynomials. In addition, we present the Rogers-type formulas and the U(n+1)-type generating functions for the bivariate q-Laguerre polynomials by the technique based upon q-operational equations. Moreover, we derive a new generalized Andrews--Askey integral and a new transformation identity involving the bivariate q-Laguerre polynomials by applying q-operational equations. Finally, we find general q-operational equations and q-partial differential equations for the generalized Askey--Wilson polynomials and give some applications.

曹健，杭州师范大学教授，硕士生导师，从事组合数学与特殊函数领域的研究，主持国家及浙江省基金等多项，已在本领域(Trans. London Math. Soc.、Stud. Appl. Math.、Adv. Appl. Math.等)重要学术刊物上以独立或通讯作者发表40余篇SCI论文，入选杭州市属高校中青年学术带头人、杭州市“131”人才等，多次在全国组合数学与图论、海峡两岸图论与组合数学、英国肯特大学及伦敦大学学院等国内外学术会议上作报告。