On the generalized Euler integrals (Q2768814)

This article deals with properties of the generalized (according to Wright) confluent hypergeometric function \(_1\Phi_1^{r}(a;c;z)={\Gamma(c)\over\Gamma(a)}\sum_{n=0}^{\infty}{\Gamma(a+\tau n)\over\Gamma(c+\tau n)}{z^{n}\over n!}\), where \(a\) and \(c\) can be complex, \(c>a>0\), \(\tau\in R,\;\tau>0\), \(a+\tau n\neq 0,-1,-2,\ldots\) as \(n=0,1,2,\ldots\); \(a\) and \(c\) are such that \(\Gamma(a+\tau n)\), \(\Gamma(c+\tau n)\) are finite for \(n=0,1,2,\ldots\) In particular, the author proves that if \(\text{ Re} c>\text{Re} a>0\), \(\tau\in R,\;\tau>0\), then the following integral representation is valid: NEWLINE\[NEWLINE_1\Phi_1^{r}(a;c;z)={\Gamma(c)\over\Gamma(a)\Gamma(c-a)}\int\limits_{0}^{1} t^{a-1}(1-t)^{c-a-1}e^{zt^{r}}dt.NEWLINE\]NEWLINE Some applications of the function \(_1\Phi_1^{r}(a;c;z)\) to new generalizations of gamma and beta functions are presented.