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.
Zhiguo LIU
. A MULTIPLE q-EXPONENTIAL DIFFERENTIAL OPERATIONAL IDENTITY*[J]. Acta mathematica scientia, Series B, 2023
, 43(6)
: 2449
-2470
.
DOI: 10.1007/s10473-023-0608-3
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