Extended hypergeometric and confluent hypergeometric functions (Q702678)

The functions under consideration are the extended Gaussian hypergeometric function \[ F_p(a,b;c,z)= {1\over B(b,c- b)} \int^1_0 t^{b-1}(1- t)^{c-b-1}(1- zt)^{-a}\exp\Biggl[-{p\over t(1- t)}\Biggr]\,dt \] and its confluent counterpart \(\Phi_p(b;c;z)\) with \(\exp(zt)\) in place of \((1- zt)^{-a}\). The authors discuss differentiation with respect to \(z\), Mellin transforms, linear transformations, recurrence relations, and asymptotic behaviour. Many results are distinguished from the classical ones only by the presence of \(p\). For instance, the analogue of Kummer's first transformation reads, \[ \Phi_p(b;c;z)= \exp(z)\Phi_p(c- b; c;- z). \] On the other hand, the analogue of Gauss's summation theorem reads, \[ F_p(a,b;c;1)= {B(b,c- a-b;p)\over B(b,c- b)}, \] where the numerator is the extended beta function, obtained by inserting the factor \(\exp[-p/(t(1- t))]\).