Acta mathematica scientia,Series B ›› 2023, Vol. 43 ›› Issue (6): 2449-2470.doi: 10.1007/s10473-023-0608-3

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A MULTIPLE q-EXPONENTIAL DIFFERENTIAL OPERATIONAL IDENTITY*

Zhiguo LIU   

  1. School of Mathematical Sciences, Key Laboratory of MEA Ministry of Education & Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China
  • Received:2022-06-06 Revised:2023-05-29 Published:2023-12-08
  • About author:Zhiguo LIU, E-mail: zgliu@math.ecnu.edu.cn; liuzg@hotmail.com
  • Supported by:
    This work was supported by the National Natural Science Foundation of China (11971173) and the Science and Technology Commission of Shanghai Municipality (22DZ2229014).

Abstract: Using Hartogs' fundamental theorem for analytic functions in several complex variables and $q$-partial differential equations, we establish a multiple $q$-exponential differential formula for analytic functions in several variables. With this identity, we give new proofs of a variety of important classical formulas including Bailey's $_6\psi_6$ series summation formula and the Atakishiyev integral. A new transformation formula for a double $q$-series with several interesting special cases is given. A new transformation formula for a $_3\psi_3$ series is proved.

Key words: $q$-hypergeometric series, $q$-exponential differential operator, Bailey's $_6\psi_6$ summation, double $q$-hypergeometric series, $q$-partial differential equation

CLC Number: 

  • 05A30
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