Extended \(\tau\)-hypergeometric functions and associated properties (Q2346882)

The autors introduce the extended \(\tau\)-Gauss hypergeometric function as follows \[ _2F_1^{\tau}(z):=\frac{\Gamma (c)}{\Gamma (b)}\sum_{n=0}^{\infty} \frac{(a;p)_n\Gamma (b+\tau n)}{\Gamma (c+\tau n)}\frac{z^n}{n!} \] where \[ (a;p)_n=\frac{1}{\Gamma (a)}\int_{0}^{\infty}t^{a+n-1}\exp(-t-\frac{p}{t})dt. \] is the generalized Pochhammer symbol. General properties for such hypergeometric functions are investigated, including its integral representations, derivative formulas, Mellin transforms and Riemann-Liouville fractional calculus operators.