# 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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### Congruences between hypergeometric functions coming from modular forms

Consider the formal power series $${}_2F_1(1/12,5/12;1;z)^{24}-{}_2F_1(1/12,7/12;1;z)^{24}=-z/3-z^2/2-(320293/559872)z^3-\cdots$$ It follows from a theorem on modular forms that for $p\ge5$ the ...
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### Factorial series $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$ and hypergeometric functions

For positive integer $D$, define $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$. For $D \le 6$, sage finds closed form in terms of hypergeometric functions at algrebraic arguments and fails to find closed ...
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### When can GKZ setup encompass HMS?

Are there any instances when the Landau-Ginzburg superpotential describing the mirror of a smooth projective Fano variety $X_\Sigma$ is encompassed by a GKZ hypergeometric system? In some sense I am ...
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### Why don't Zeilberger and Gosper's algorithms contradict Richardson's theorem?

Richardson's theorem proves that whether an expression A is equal to zero is undecidable. A is in this case an expression, constructed from $x,e^x,\sin(x)$ and the constant function $\pi$ and $\ln(2)$ ...
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### A hypergeometric series for $\sqrt3\pi$ with converging rate $1/9$

Recently, I found a (conjectural) new series for $\sqrt3\pi$: $$\sum_{k=1}^\infty\frac{(8k-3)\binom{4k}{2k}}{k(4k-1)9^k\binom{2k}k^2}=\frac{\sqrt3\pi}{18}.\label{1}\tag{1}$$ The series converges fast ...
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### How to calculate the Integral with confluent hypergeometric function

How to prove this.Thank you in advance Let $\delta,\beta>0$ How to prove this \begin{align} & \int^1_0 \frac{w^{1-\beta}}{(1-w)^{1+\delta}} (-t.s w)^{\frac{-\delta}{2}} e^{-\frac{w}{1-w}(s+t)}...
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### Two particular combinations of Gauss hypergeometric functions

Browsing this site and the web I could not find a reference on the following combinations of Gauss hypergeometric functions $F={}_2F_1$, for which I have reason to believe that they can be simplified ...
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### Simplification of hypergeometric Function

First of all I am not at all a math expert, but I have some working knowledge. That said, please excuse "dumb" questions. I am looking at the following process: Assume you are on the 2-...
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### Is the hypergeometric function ${}_1F_2(1;a,a+\frac12;-x^2)$ an elementary function? How about its positivity, monotonicity, and convexity in $x$?

Is the generalized hypergeometric function ${}_1F_2\bigl(1;a,a+\frac12;-x^2\bigr)$ for $a>-1$ and $x>0$ an elementary function? How about the positivity, monotonicity, and convexity of the ...
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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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Question: How to calculate this summation $S=\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k}$? Where $m<n_1,m<n_2$ Remark1: When $a=b$, I know the above summation $S=a^m\sum_{k=0}^m {... • 57 1 vote 0 answers 157 views ### polynomial approximation of hypergeometric function 2F1 I have the following function$T(k_1,k_2)$resulting from multiphoton transition matrix elements calculations:$T(k_1,k_2)=\gamma^{-k_2}\sum_{j=0}^{k_1}(j+2)_{l+1}\binom{k_1}{j}(k_1+1)_3(\gamma-1)^{j}{...
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 ...