# Questions tagged [hypergeometric-function]

Hypergeometric functions often refer to a family of functions ${}_p F_q$ represented by a corresponding series, where $p,q$ are non-negative integers. The case ${}_2F_1(a,b;c;z)$ is a special case of particular importance; it is known as the Gaussian, or ordinary, hypergeometric function.

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### $\int_0^1 \sqrt{\frac{2-x^4}{1-x^4}}\, dx$ in terms of the gamma function or elliptic integrals

Question Can $$\int_0^1 \sqrt{\frac{2-x^4}{1-x^4}}\, dx$$ be expressed in closed form in terms of the gamma function at rational arguments or in closed form in terms of elliptic integrals? Thoughts ...
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### Show that $\mathcal{L}^{-1}\left\{\frac{\Gamma(n)}{s^n}e^{-\frac{2a}{s}}{}_0F_1\left(n,\frac{a^2}{s^2}\right)\right\}(1)={}_0F_1\left(n,-a\right)^2$

I am trying to show that given $n, a > 0$ $$G(s) = \frac{\Gamma(n)}{s^n} e^{- \frac{2a}{s}} {}_0F_1 \left( n, \frac{a^2}{s^2} \right)$$ the inverse Laplace transform of $G$ evaluated at $t=1$ is ...
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### Is there a closed form for $_3F_2(a, b, c; a+1, 2a+\frac{3}{2}; 1)$? [closed]

I'm trying to evaluate the hypergeometric series given by $_3F_2(a, b, c; a+1, 2a+\frac{3}{2}; 1)$. Is there a known closed-form expression for this series?
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### Problem with the transform from ${_3F_1}$ to ${_2F_2}$

I want to transform $${_3F_1}(-\frac n2,-\frac{n-1}2,\frac{n+1}2;\frac12;z^{-1})$$ into ${_2F_2}$ using equation (2.2.3.2) of the book by Lucy Joan Slater "Generalized ...
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### Problematic limit $\epsilon \to 0$ for combination of hypergeometric ${_2}F_2$ functions

In an earlier question, the integral $$I_n(c)=\int_0^\infty x^n (1+x)^n e^{-n c x^2} dx$$ was considered with particular focus on its behavior for positive integer $n$. In trying to analyze this, it ...
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### How to calculate $\int_{0}^{\infty} x^n(1+x)^n e^{-ncx^2}\text{d}x$

As the title mentioned, I want to calculate $$\int_{0}^{\infty} x^n(1+x)^n e^{-ncx^2}\text{d}x,$$ where $n$ is a positive integer, $c$ is a positive real number in the ...
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### Can I use the hypergeometric distribution to calculate draw odds for 2 variables from a single population?

I would like to program a card-drawing calculator the give the odds of there being at least x cards of set A and y cards of set B, where sets A and B are distinct populations in set C of a given ...
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### How to calculate an upper bound for $\sum_{k=0}^n \binom{n}{k} \frac{(-x)^k}{\sqrt{k+a}}$

As the title mentioned, I want to get a closed-form result of $$\sum_{k=0}^n \binom{n}{k} \frac{(-x)^k}{\sqrt{k+a}},$$ where $x\in[0,1]$ is a real number, and $a$ is a ...
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