Abstract
We present three q-Taylor formulas with q-integral remainder. The two last proofsrequire a slight rearrangement by a well-known formula. The first formula has been given in different form by Annaby and Mansour. We give concise proofs for q-analogues of Eulerian integral formulas for general q-hypergeometric functions corresponding to Erd Ìelyi, and for two of Srivastavas triple hypergeometric functions and other functions. All proofs are made in a similar style by using q-integration. We find some new formulas for fractional q-integrals including a series expansion. In the same way, the operator formulas by Srivastava and Manocha find a natural generalization.
Highlights
The aim of this paper is to continue the investigation of single and multiple q-hypergeometric series in the spirit of our book [9] and our paper [12] on the q-Lauricella functions
The fractional q-integrals and direct computations of q-integrals lead to similar results
By quoting Erdelyi and Feldheim, we have managed to save these hypergeometric formulas from oblivion; our proofs of their formulas are quite similar, these authors never wrote down their proofs
Summary
The aim of this paper is to continue the investigation of single and multiple q-hypergeometric series in the spirit of our book [9] and our paper [12] on the q-Lauricella functions. The fractional q-integrals and direct computations of q-integrals lead to similar results. We refer to previous papers with respect to convergence regions. By quoting Erdelyi and Feldheim, we have managed to save these hypergeometric formulas from oblivion; our proofs of their formulas are quite similar, these authors never wrote down their proofs. Charles Cailler in 1920 [3], [20, p. 242] and Kampe de Feriet in 1922 [19, p.
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