# 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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### How to find the enveloping curve of this family of polynomials?

I was studying the Rule 90 cellular automaton and came across a family of polynomials defined by D_n(x)=\begin{cases} \displaystyle\sum_{k=0}^{m}(-1)^{m+k}\binom{m+k}{m-k}x^{2k}\ , &...
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### Integral of $\sqrt{\cosh(x)}$ with respect to x

I am trying to obtain a solution for the integral $$\int^{x}_{0} \sqrt{\cosh(x)} dx.$$ A CAS system yields an answer depending on an elliptic integral of the second kind ...
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### From incomplete beta function sum $\frac1{(\text B(a,b)c)^2}\sum_{k=0}^\infty\frac{\text B_y(2a+r+k,b)(1-b)_k}{(a+k+r)k!}$ to hypergeometric function.

The goal is to integrate Inverse Beta Regularized $\text I^{-1}_{z}(a,b)$ to a constant power with respect to $z$ twice for a future identity. Notice the Incomplete Beta function $\text B_z(a,b)$ and ...
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### Double sum over Gauss hypergeometric function.

I've been dealing with sums and integrals over hypergeometric functions quite a bit lately, and the latest problem is the following double sum: F(x,y;\alpha,t)=\sum_{n,m=0}^\infty\...
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### Hypergeometric ordinary differential equation

$$r^2 u′′+r(a+bn r^s)u′+(c+d r^s)u=0$$ How does one convert this equation into the Whittaker equation?
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### Numerical Approximation of Hypergeometric For Maximum Likelhood Estimation Overflows

I am trying to improve my implementation of the maximum likelihood (ML) estimator for the multiple squared correlation (https://onlinelibrary.wiley.com/doi/10.1111/j.1467-842X.1985.tb00559.x). The ML ...
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### Prove $\lim_{k\to\infty}\sum_{n=0}^\infty \frac{1}{(n!)^k}=2$

How to prove that $$\lim_{k\to\infty}\sum_{n=0}^\infty \frac{1}{(n!)^k}=2 \,\,?$$ A plot shows that the values seem to quickly converge to $2$. Cannot exclude a duplicate but couldn't find it in the ...
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What are some recommended references disucssing Gauss's Hypergeometric Equation? Specifically, I would like references discussing: the origin of the equation, how to obtain it, the solution by the ...
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### Asymptotic approximation of $_{2}F_{1}(\{1/2- n/2, -n/2\},\{3/2 - n\};z\}$ for $-1/z\rightarrow0$

I'm looking at hypergeometric functions at the moment in relation to the $n$'th term of the Taylor series of $\sqrt{1-a x^2}$. From this consideration a $_{2}F_{1}$ arises that I'd like to approximate ...
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### Functional inverse of $z=1+w+\cdots+w^{n-1}$

Migrated to MO. 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. My attempt:...
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### Evaluation of a summation involving hypergeometric functions

I need help in evaluating the following tricky summation mainly involving a product of two Kummer's confluent hypergeometric function, ${}_1 F_1(a;b;z)$. Is there some identity of ${}_1 F_1(a;b;z)$ ...
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### An integral involving hypergeometric functions

As part of a research project, I have arrived at the following integral which I need to evaluate:  I_n(r) = \frac{\gamma_+(r)}{\sqrt{5}} \int dr \frac{r \gamma_-(r)}{(2+r^2)^2} P_n(r) - \frac{\...
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### The special case of Pochhammer Symbol at Zero?

I am interested in a property of Pochhammer Symbol. So I need an information about it. Let $a^{\bar{n}}$ Pochhammer symbol or rising factorial. As you know in the literature $a^{\bar{0}}=1.$ I ...
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### Evaluating $\lim_{n\to\infty}\frac{1}{n+1}(\omega+\nu)^{(n+1)}z^\nu{_2F_1}(1,\omega+\nu+1;n+2;1-z)$

I recently found a proof for the following sum \begin{align*} S_n & =\sum_{k=0}^n\mathcal S_n^{(k)}(\Phi(z,-k,\omega)-z^\nu\Phi(z,-k,\omega+\nu))\\ & =\frac{1}{n+1}(\omega+\nu)^{(n+1)}z^\nu{...
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### Solve $\sum_{n=0}^\infty \text P_{-n}^{-n}(z)$ and $\sum_{n=0}^\infty \mathsf P_{-n}^{-n}(z$) with Associated Legendre P functions of type $1$ and $3$
Here is a simple looking sum which should have an alternate form since it is just a double hypergeometric series with the Associated Legendre P function of the First (aka Second) Type $\text P_a^b(z)$ ...