# Questions tagged [hypergeometric-functions]

Hypergeometric functions are the analytic functions defined by Taylor expansions of the shape $\sum_{n \geq 0} a_n x^n$, where $a_{n+1}/a_n$ is a rational function of $n$. This general family of functions encompasses many classical functions. The hypergeometric functions play an important role in many parts of mathematics.

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### How can I show that this product is equal to a product of Gamma functions? [migrated]

$\prod_{n=0}^{x-1}\left( 1+\frac{a}{an+b}\right) = \frac{\Gamma\left(\frac{a}{b}\right)\Gamma\left(x+\frac{a+b}{b}\right)}{\Gamma\left(\frac{a+b}{b}\right)\Gamma\left(x+\frac{a}{b}\right)}$ I found ...
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### A recurrence formula for the Legendre function $P_\mu^\nu(x)$

Im looking for a recurrence formula of type: $$(\mu-\nu) x P_\mu^\nu(x) + P_{\mu-1}^\nu(x)=?, \quad \mu,\nu\in \mathbb R$$ where $P_\mu^\nu(x)$ is the Legendre function of the first kind (solution to ...
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### An integral of $|\sin(x)\cos(nx)^{-2/n}|$ from $-\pi$ to $\pi$

For an integer $n \geq 3$, define $$A_n = \int\limits_{-\pi}^\pi\frac{|\sin(x)|}{|\cos(nx)|^{2/n}}dx.$$ It is a fact that $A_n$ is finite for all such $n$. I am interested in the behaviour of a ...
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### Asymptotic approximations or upper bounds for ${}_{2}F_{1}(x+1,x+1,1,z)$ when $x \gg 1$?

I have recently encountered the hypergeometric function $${}_{2}F_{1}(x+1,x+1,1,z),$$ where $x$ is an integer and $z$ is a real number with $x \ge 1$ and $0<z<1/2$. This is the first time I ...
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