New posts in hypergeometric-function

Integral involving hypergeometric function $\int_0^1[{}_2F_1(\frac13,\frac23;1;x^3)]^2dx$

integration definite-integrals special-functions hypergeometric-function modular-forms

$I_k=\int_0^1 \frac{1}{\mathbf{B}(\alpha , \beta )} \cos^k (\pi \theta) \theta^{\alpha -1} (1-\theta)^{\beta -1}d\theta $

integration sequences-and-series definite-integrals hypergeometric-function

Arc length of $x^n$ found using Hypergeometric function and series. Alternate representations and solution verification needed.

sequences-and-series solution-verification gamma-function hypergeometric-function arc-length

Closed form for $f(x)=\ _3F_2\left(\tfrac12,\tfrac12,\tfrac12;\tfrac32,\tfrac32;x\right)$

integration definite-integrals special-functions hypergeometric-function

On the integral $\int_{(0,1)^n}\frac{\prod\sin\theta_k}{\sum\sin\theta_k}d\mu$

definite-integrals special-functions bessel-functions hypergeometric-function

About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions

definite-integrals closed-form modular-forms hypergeometric-function elliptic-functions

Closed-form expression for an iterated integral

integration multivariable-calculus definite-integrals hypergeometric-function elliptic-integrals

The integral of an elliptic integral: $\int_{0}^{1}\frac{x\mathbf{K}^2\left ( x \right )}{\sqrt{1-x^{2}}}\mathrm{d}x$

calculus integration definite-integrals hypergeometric-function elliptic-integrals

Incomplete hypergeometric function

special-functions hypergeometric-function beta-function

Prove ${_2F_1}\left({{\tfrac16,\tfrac23}\atop{\tfrac56}}\middle|\,\frac{80}{81}\right)=\frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$

calculus special-functions closed-form hypergeometric-function

Prove $\int_0^1 \frac{4\cos^{-1}x}{\sqrt{2x-x^2}}\,dx=\frac{8}{9\sqrt{\pi}}\left(9\Gamma(3/4)^2{}_4F_3(\cdots)+\Gamma(5/4)^2{}_4F_3(\cdots)\right)$

definite-integrals closed-form gamma-function hypergeometric-function trigonometric-integrals

How to prove ${}_{2}F_{1}\left(\frac{1}{3},\frac{2}{3};\frac{3}{2}; \frac{27}{4}z^2(1-z)\right) = \frac{1}{z}$

integration sequences-and-series number-theory summation hypergeometric-function

The closed form of $\lim_{x\to\frac{4}{3}}\frac{\partial}{\partial x}\left[\,_2{\rm{F}}_1\left(\frac{1}{3},1;x;-1\right)\right]$

calculus real-analysis limits derivatives hypergeometric-function

Inequality between binomial sums

combinatorics inequality binomial-coefficients hypergeometric-function

Infinite product of sinc functions

infinite-product hypergeometric-function

Tied chess matches and the monotonicity of $\sum_{k=0}^n \binom{2n}{k,k,2n-2k} (pq)^k (1-p-q)^{2n-2k}$

probability sequences-and-series hypergeometric-function multinomial-coefficients multinomial-distribution

Prove $_2F_1\!\left(\frac76,\frac12;\,\frac13;\,-\phi^2\right)=0$

calculus special-functions hypergeometric-function golden-ratio

On the integral $\int_{0}^{1/2}\frac{\text{Li}_3(1-z)}{\sqrt{z(1-z)}}\,dz$

integration reference-request special-functions hypergeometric-function polylogarithm

How to show an infinite number of algebraic numbers $\alpha$ and $\beta$ for $_2F_1\left(\frac14,\frac14;\frac34;-\alpha\right)=\beta\,$?

calculus definite-integrals radicals hypergeometric-function conjectures

Yet another difficult logarithmic integral

integration sequences-and-series special-functions hypergeometric-function polylogarithm