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 2h comment Solution of Integral $\int_0^\infty \frac{x^{r/\beta}e^{-\alpha x}}{\left(1-(1-\beta)e^{-x}\right)^{\alpha+1}} \, dx$ @SAAN Thanks for you great trial. But isn't the first approach in fact suitable for $|1-\beta|<1$ , i.e. $0<\beta<2$ , and the second approach in fact suitable for $|1-\beta|>1$ , i.e. $\beta>2$ ? 2h comment Some properties about the Kampé de Fériet function But how to prove $\sum\limits_{s_1,\ldots,s_n=0}\dfrac{\prod\limits_{j=1}^p(a_j)_{s_1+\ldots s_n}}{\prod\limits_{l=1}^q(b_l)_{s_1+\ldots s_n}}\dfrac{x_1^{s_1}}{s_1!}\ldots\dfrac{x_n^{s_n}}{s_n!}=\sum\limits_{k=0}^\inf‌​ty\dfrac{\prod\limits_{j=1}^p(a_j)_{k}}{\prod\limits_{l=1}^q(b_l)_{k}} \dfrac{(x_1+...+x_n)^k}{k!}$ ? Natural is not the reasonable reason in maths! 2h revised Can anyone solve this integral function? edited tags; edited title 3h answered Infinite series of Hypergeometric function 3h comment Solution of Integral $\int_0^\infty \frac{x^{r/\beta}e^{-\alpha x}}{\left(1-(1-\beta)e^{-x}\right)^{\alpha+1}} \, dx$ @user64494 But at least she gives the expressable series form, don't you know that the integral is more complicated than Lerch zeta function? 22h asked Some properties about the Kampé de Fériet function 22h revised Evaluting $\int_{1}^{2} \frac{\tan^{-1} x}{\tan^{-1} \frac {1}{x^2-3x+3}} \operatorname dx$ edited tags 1d comment Closed form for an infinite sum over Gamma functions? @Jack D'Aurizio Converting series back to the integral form is not the expected way of simplification. 1d comment Expressing ${}_2F_1(a, b; c; z)^2$ as a single series @glebovg at least it relates to the Legendre function in this way: functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/… 1d revised Closed form for an infinite sum over Gamma functions? added 349 characters in body 1d revised Closed form for an infinite sum over Gamma functions? deleted 778 characters in body 2d revised Closed form for an infinite sum over Gamma functions? added 566 characters in body 2d answered Closed form for an infinite sum over Gamma functions? 2d answered solving the integral of $e^{x^2}$ 2d revised How to prove and evaluate an Improper Integral edited tags Apr21 revised How to solve over-determined linear system of equations? edited tags Apr21 revised Limit of solution equal to solution at limit edited tags Apr21 revised Two variable equation added 5 characters in body; edited tags Apr21 revised Find a line parallel to a known line that intersects a known circle at one point. edited tags Apr21 revised Question regarding arbitrary parameters edited tags