New posts in hypergeometric-function

About the confluent versions of Appell Hypergeometric Function and Lauricella Functions

integration definite-integrals hypergeometric-function

Evaluating $\lim_{n\to\infty}\frac{1}{n+1}(\omega+\nu)^{(n+1)}z^\nu{_2F_1}(1,\omega+\nu+1;n+2;1-z)$

sequences-and-series gamma-function hypergeometric-function zeta-functions stirling-numbers

Integral $\int_0^\infty x^2\,e^{-x^2}\operatorname{erf}(x)\,\log(x)\,dx$

integration definite-integrals improper-integrals closed-form hypergeometric-function

Crazy $\int_0^\infty{_3F_2}\left(\begin{array}c\tfrac58,\tfrac58,\tfrac98\\\tfrac12,\tfrac{13}8\end{array}\middle|\ {-x}\right)^2\frac{dx}{\sqrt x}$

calculus integration definite-integrals closed-form hypergeometric-function

How to show an infinite number of algebraic numbers $\alpha$ and $\beta$ for $_2F_1\left(\frac13,\frac13;\frac56;-\alpha\right)=\beta\,$?

calculus radicals modular-forms hypergeometric-function conjectures

Expressing the integral $\int_{0}^{1}\frac{\mathrm{d}x}{\sqrt{\left(1-x^3\right)\left(1-a^6x^3\right)}}$ in terms of elliptic integrals

integration definite-integrals closed-form hypergeometric-function elliptic-integrals

Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

calculus sequences-and-series special-functions closed-form hypergeometric-function

Prove ${\large\int}_{-1}^1\frac{dx}{\sqrt[3]{9+4\sqrt5\,x}\ \left(1-x^2\right)^{2/3}}=\frac{3^{3/2}}{2^{4/3}5^{5/6}\pi }\Gamma^3\left(\frac13\right)$

calculus definite-integrals special-functions closed-form hypergeometric-function

How to prove $\int_0^\infty J_\nu(x)^3dx\stackrel?=\frac{\Gamma(1/6)\ \Gamma(1/6+\nu/2)}{2^{5/3}\ 3^{1/2}\ \pi^{3/2}\ \Gamma(5/6+\nu/2)}$?

special-functions gamma-function closed-form conjectures hypergeometric-function

Integral $\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$

calculus integration special-functions closed-form hypergeometric-function

Evaluating the sums $\sum\limits_{n=1}^\infty\frac{1}{n \binom{kn}{n}}$ with $k$ a positive integer

sequences-and-series summation binomial-coefficients improper-integrals hypergeometric-function

Integral $\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$

calculus integration logarithms closed-form hypergeometric-function

Closed form for $\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$

special-functions definite-integrals gamma-function closed-form hypergeometric-function

Closed form for $ S(m) = \sum_{n=1}^\infty \frac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$?

taylor-expansion riemann-zeta hypergeometric-function pattern-recognition catalan-numbers

Derivation of mean and variance of Hypergeometric Distribution

probability-distributions hypergeometric-function means

List of integrals or series for Gieseking's constant $\rm{Cl}_2\big(\tfrac{\pi}3\big)$?

sequences-and-series definite-integrals big-list hypergeometric-function constants

Evaluating $\sum_{n=0}^\infty\frac{(1/2)_n}{n!}(H_n-H_{n-1/2})$

sequences-and-series special-functions hypergeometric-function harmonic-numbers pochhammer-symbol

Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$

calculus definite-integrals special-functions closed-form hypergeometric-function

Prove ${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$

calculus sequences-and-series definite-integrals improper-integrals hypergeometric-function

Are elementary and generalized hypergeometric functions sufficient to express all algebraic numbers?

calculus algebraic-number-theory special-functions hypergeometric-function radicals