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# 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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### Analytic continuation of ${_0F_{\beta-1}}(-;\mathbf 1;H^\beta)$ in $\beta$

I am working with series of the form $$\sum_{k=0}^\infty\left(\frac{H^k}{k!}\right)^\beta,$$ where $\beta\in(0,1)$. By the ratio test this series converges for all $\beta>0$. If $\beta\in\Bbb N$ ...
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### What is the surface area of the finite part of the Cayley nodal cubic surface?

The Cayley nodal surface is defined by the equation $x^2+y^2+z^2-2xyz=1$. The finite part of the surface is the tetrahedral part bounded by the 4 nodes $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, $(-1,-1,1)$....
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### How to calculate this limit (if exist)?

I have just asked the calculation of the following summation see here $$S(a,b,m,n_1,n_2)=\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k},$$ which is motivated by the calculation of the ...
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### Closed form for ₄F₃(n,n,n,2n;1+n,1+n,1+n;−1)

For positive integer $n$ the following value of a hypergeometric function $$_4F_3(n,n,n,2n,1+n,1+n,1+n,-1)$$ based on the first few terms looks like $$R_1(n) + R_2(n) \pi^2$$ where $R_{1,2}(n)$ are ...
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### Fractional partial derivatives and integrals of De Bruijn's $H_t(z)$-function. Does a simpler form exist for the $z$ derivative/integral?

$\newcommand\KummerU{\text{KummerU}}$ $\newcommand\Hypergeom{\text{Hypergeom}}$ During 2018/2019, the polymath 15 project managed to successfully reduce the upper bound of the De Bruijn-Newman ...
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### Classification of Hypergeometric Functions: How Fundamental is the Order?

Horn classified the hypergeometric functions of two independent variables into 34 different convergent functions, where 14 are complete and 20 are confluent. However, Carlson notes that linear ...
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### Why does a given image value of a function $x H^3(x)$ where $H(x)$ contains hypergeometric functions have preimage value?

Prof. Edward B. Bender is a famous combinatorial mathematician. In his paper of A Survey of Asymptotic Behaviour of Maps published in Journal of Combinatorial Theory, there is a part of analyzing the ...
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### Any name for this special function?

We know $$\sum_{m=0}^\infty \frac{x^m}{(a-m)!m!} = \frac{1}{a!}(1+x)^m$$ where we understand the factorial as Gamma function $\Gamma(x)$ such that it is divergent if the argument is negative integer....
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### Generating function of the product of Legendre polynomials

The generating function of the product of Legendre polynomials for the same $n$ is given by \begin{aligned} \sum_{n=0}^{\infty} z^{n} \mathrm{P}_{n}(\cos \alpha) \mathrm{P}_{n}(\cos \beta)&=\frac{\...
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### Functional inverse of $z=1+w+\cdots+w^{n-1}$

Migrated from the MSE. I am interested in the functional inverse of $$z=1+w+\cdots+w^{n-1},\quad w\geq0,\ n>1.$$ This function is strictly increasing on $w\geq0$ and thus admits an inverse. By ...
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### Estimates for the absolute value of the hypergeometric function ${}_2F_1(2-n,n+2,2;x)$ on $[0,1]$

I would like to know whether the estimate $$|{}_2F_1(2-n,n+2,2;x)| \le \frac{n}{2}, \quad x \in [0,1]$$ holds and in that case where to find a reference.
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### $\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} z^{2k}$ is an elementary function

I try to calculate the following series \begin{align*} S_{n,m}(z)=\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} \, z^{2k}, \end{...
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### Hyper geometric series reference

Can someone point out a reference for the proof of this identity? Thanks in advance. https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric4F3/03/03/01/0002/
### Is there a special function for $\sum_{m=0}^{\infty} x^m/(m!)^s$?
Is there a special function for the following series? $$\sum_{m=0}^{\infty} {x^m \over (m!)^s}$$ Here, $s$ is a positive real number. When, $s$ is an integer, $s=n \in \mathbb{Z}$, this series can be ...