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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Closed form of ${_2}{F}_1\left(\frac{1}{2},s,\frac{3}{2};-\frac{1}{a^2}\right)$

While solving an integral, I came acorss the term $$ \tilde{I}(a,s)= {_2}{F}_1\left(\frac{1}{2},s,\frac{3}{2};-\frac{1}{a^2}\right).$$ To be precise, it came in the following calculations \begin{align*...
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How to simplify this integral

$$\int_{0}^{\infty} e^{i 2m x - i\Omega \sin^{-1}(\tanh(x))} \,dx$$, whose solution is given to be $$4i\pi m e^{-i\pi(\frac{2m + i\omega}{2})}cosech(\pi\omega) {}_2F_1(1-2m, 1-i\omega, 2; 2) $$, can i ...
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Asymptotics of Hypergeometrical function ${}_2F_1$ when integrality is not satisfied

I am looking for an expression of the asymptotic behaviour for $|z|\rightarrow\infty$ of the Hypergeometric function ${}_2F_1(a,b,c,z)$ when the condition $a-b \not\in {\mathbb Z}$ is not satisfied, ...
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How to estimate or get an upper bound on the entropy of multivariate hypergeometric distribution?

When we allocate $N:=k\cdot Z$ balls, where $\frac{N}{2}$ are red balls and $\frac{N}{2}$ are black balls, into $k$ groups, the number of red balls in each group follows a multivariate hypergeomtric ...
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Help plotting a function that depends of the confluent hypergeometric function of the second kind

From wolfram mathematica I got this result. $\sum\limits_{i=0}^{k}\frac{k!x^i}{i!(k-i)!^2}$ This is $\frac{e^{i\pi k}x^k U(-k,1,-1/x)}{k!}$ Where U(a,b,z) is the confluent hypergeometric function of ...
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Proving Rodrigues' formula from Murphy's formula. Legendre polynomials

I'm following the book "Special functions of mathematical physics and chemistry" by Ian N. Sneddon where he derives Rodrigues formula for the Legendre polynomials $$ P_n(\mu) = \frac{1}{2^n ...
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Is there an even function $f(k)$ such that the coefficients in its Fourier cosine series are the $3F2$ hypergeometric functions of the type below? [closed]

I have $f_{z}(k) = \frac{1}{2} a_0(z) + \sum\limits_{n=1}^{\infty} a_n(z) \cos n k$. where $a_n(z) = {}_3F_2(\{1/2, 1, 1\}, \{ 1-n, 1+n \}, z)$. Is it possible to find $f(k)$ explicitly?
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Constructing hyperbolic triangles with specific angles

I’m trying to investigate hyperbolic tessellations and how to construct them. After doing lots of reading I’ve found that given a hyperbolic triangle with angles $\frac{\pi}{m}, \frac{\pi}{n}, \frac{\...
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Any simplification of $\sum_{k=0}^{n}\binom{n}{k} \frac{(-1)^{k}}{(N-k)^2}$

I am trying to find a compact expression for $\sum_{k=0}^{n}\binom{n}{k} \frac{(-1)^{k}}{(N-k)^2}$, where $N > n \geq 0$. Maple simplifies it using hypergeometric functions, which is not very ...
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Integral of hypergeometric function [closed]

Good morning, I need help with the following integral $$ \int_0 ^1 \left(\log (y) \frac{y^{a -1} {}_2F_1(a ,a ;2 a ;y)}{y-x} - \log (x) \frac{x^2 y^{a -3} {}_2F_1(a ,a ;2 a;y)}{y-x}\right) dy$$ ...
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Closed form for a finite sum

Define $$f_n(x_1,x_2,x_3,x_4)=\sum_{s_1=0}^n a_{n,s_1,s_1,s_1,s_1} x_1^{s_1}x_2^{s_1}x_3^{s_1}x_4^{s_1}+\sum_{s_i\neq s_j, \forall i\neq j,s_i\in[0,n]} a_{n,s_1,s_2,s_3,s_4} x_1^{s_1}x_2^{s_2}x_3^{s_3}...
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I'm stuck in calculating $_2F_1(1,1;\tfrac12;\tfrac12)$

By using $(71)$ and $(70)$ identities of Mathworld's hypergeometric function page, $${_2}F{_1}(1,1;\tfrac12;\tfrac12)\stackrel{(71)}{=}2\,{_2}F{_1}(1,-\tfrac12;\tfrac12;-1)\stackrel{(70)}{=}2\left(\...
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How to write Tricomi's confluent hypergeometric function in terms of Meijer-G function

I am calculating a closed form expectation and I encountered the Tricomi's confluent hypergeometric function (aka confluent hypergeometric function of the second kind) given by integral $U\left( a,b,z ...
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Exact bounds for the Median of the Hypergeometric distribution

I am wondering if anybody knows good exact bounds for the Median of a Hypergeometric distribution: $$ X \sim \text{Hypergeometric}(N,K,n) $$ One exact bound could be given by exploiting the inequality ...
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Asymptotic Behaviors of Hypergeometric function for two large parameters

I am studying the asymptotic behaviors of the hypergeometric function ${}_2F_1(a,b;c;z)$ by reading this paper. However, the paper does not mention the case when $b$ and $c$ are large. I tried looking ...
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Relation between Gaussian integral and Kummer's hypergeometric function

Short version: I need to find out what known identity is used to convert the following integral into the formula with Kummer's confluent hypergeometric function, ${}_1F_1(a;b;z)$: $$ (2\pi\sigma^2)^{-...
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Approximation formula for the Kummer function ${}_1F_1(a, b, x)$ for large $x$?

The question here: Asymptotics of Kummer Hypergeometric Function in first argument addresses asymptotics for the first argument of a Kummer function. However, it is a very special case, and does not ...
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Approximate inverse Mellin transform

I need to evaluate the following complex integral (which is essentially an inverse Mellin transform): $$\int_{-c-i\infty}^{-c+i\infty} \Gamma^2 (-s) \Gamma (s+1) \Gamma (a-s) \mbox{}_1F_1(s;1;-y) x^s{...
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Value of $ {_3F_2}{\left(1,n+\frac32,1-r;\frac32,n+2;1\right)} $

This sum can be calculated using a CAS as a product of Gamma and Generalized Hypergeometric functions $$\displaystyle \sum_{k=0}^n \frac {\dbinom{n}k \dbinom{n+r}k}{\dbinom{2n}{2k}} = \dfrac{(2n+1) \...
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Solve Convoluted Integral with Gauss Hypergeometric function

I'm working on an operator problem that requires solving the following complicated integral, involving the Gauss Hypergeometric function \begin{equation} \int_{-\infty}^{\infty} \int_{-\infty}^0 \...
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How to differ between Binomial and Hypergeometric Distributions while solving problems?

I'm listing 3 questions: 1.Suppose that a batch of 100 items contains 6 that are defective and 94 that are not defective. If X is the number of defective items in a randomly drawn sample of 10 items ...
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Compute this numerical series

i would like to find a formula that is true for $n \geq 1$ for that : $\displaystyle \sum _{k=1}^n \frac{n^k}{k! \times k}$ I already tried many things like telescoping the sum or symetrising it but i ...
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Integral of product of zeroth-order bessel functions times cosine $\int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x$

I am new to Bessel functions and need to solve the following integral \begin{equation} \int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x \end{equation} with $J_{0}$ ...
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Is there a asymtopic bound to a confluent hypergeometric function of ${_1F_1}(n,1.5n,nz)$?

As the title mentioned, is there a good way to approximate $${_1F_1}(n,1.5n,nz)$$ where $n \in \mathbb{N}$, and $z$ is a real positive number. If the direct bound is difficult, an asymptopic bound (as ...
jobs adam's user avatar
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$\displaystyle \int\frac{x^m}{(a^n\pm x^n)^p}\mathrm{d}x$ in terms of elementary functions

Context For curiosity I wanted to calculate this integral: $$\displaystyle \int\frac{x^m}{(a^n\pm x^n)^p}\mathrm{d}x\qquad m\in\mathbb{N},\ n,p\in\mathbb{N}^{+}, \ a\in\mathbb{R}$$ For brevity: $m\...
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Transformation from Hypergeometric to Whittaker.

I'm working on an eigenvalue problem involving 2 operators, which we will call $\hat{A}$ and $\hat{B}$. They are related for very small values on a coordinate $x \approx 0$, in which $\hat{A}$ ...
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For all $z\in\mathbb{C}$ and an arbitrary $b\in\mathbb{R}$, ${}_2F_1(a,b,b,z)=(1-z)^{-a}$

The Gaussian hypergeometric function is given by: $${}_2F_1[a,b,c,z]=\sum_{n=0}^{\infty}\dfrac{(a)_n(b)_n}{(c)_n n!}z^n,$$ where $(a)_n=a(a+1)\dots(a+n-1)$ for $n\in\mathbb{N}$ and $(a)_0=1$, is the ...
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Equivalent of a special Generalized hypergeometric function

Is there any equivalent or tight upper bound with an "elementary function" of following generalized hypergeometric function: ${}_k F_{k-1}(2,\dots,2,1-m;1,\dots,1;-1)$ when especially $m$ ...
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An integral involving Gaussian hypergeometric function and exponential function [closed]

The integral is $\int_{0}^{1}x^a (1+x)^b e^{-cx^2}{_2F_1}(-\frac{n}{2},n-1,n-0.5,x)dx$, where $a$, $b$ and $n$ are positive integers, $c$ is a positive real number. Is there a closed-form result of ...
jobs adam's user avatar
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Riemann surface of the hypergeometric function

The hypergeometric function $_2F_1(a,b,c,z)$ has a branch cut extending from $z=1$ to $z=\infty$. Does this define an infinite-sheeted Riemann surface (like that for $\log{z}$) or one with a finite ...
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Is the Gaussian hypergeometric function ${}_2F_1$ bigger than 1?

I want to know if the Gaussian hypergeometric function ${}_2F_1$ satisfies $$ {}_2F_1(a,b;c;x) \geq 1 $$ for all $a,b\leq 0$, $c > 0$, $0 < x < 1$. (If not, how would we restrict $a,b$ ...
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Closed form for the following sum of Binomial coefficients [duplicate]

I am trying to find a closed form for the following for any $p$ $$ \sum_{n = 0}^{m - 1} p^{n} \binom{2m}{n} $$ I typed this into Wolfram Alpha and got an answer in terms of the hypergeometric function,...
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Sequential testing of enriched categories

In order to improve an existing method to test enrichment of some terms (biology), I developed a new approach using special permutations while the standard approaches are using hyper-geometric test. ...
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In Pascal's triangle without the $1$s, what is the sum of $n$th powers of the reciprocals?

Consider Pascal's triangle without the $1$s. Let $S(n)$ be the sum of $n$th powers of the reciprocals, where $n$ is an integer greater than $1$. ($S(1)$ diverges because the harmonic series diverges.)...
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Closed form expression of the countour integral of Gamma functions combined with a Gaussian hypergeometric function?

Is there any way of writing the following contour integral as a closed form expression ? $$ \int_{-i \infty}^{+i \infty} dp \int_{-i \infty}^{+i \infty} ds \Gamma(d/2 + s + p -1) \Gamma(-\Delta - s - ...
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Limit case of formula with hypergeometric functions ratio

I need help with evaluating $f = [x(1-x)]^{\frac{1}{2}}\left|\frac{ab}{c}\frac{F(a+1,b+1;c+1;x)}{F(a,b;c;x)}\right|$ when $x \rightarrow 1$. The result should be (according to this paper: M.S.Silver ...
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Hypergeometric Functions (2F1) positive everywhere

Five years ago, there was a question about the positivity of the hypergeometric functions $_{1}F_{2}$ on a short interval $(0,5)$. I would like to ask a very similar question but for the ...
QA Ngô's user avatar
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A probability may related to the Gaussian Hypergeometric function and its combination with other special functions.

I hope that I should describe the original question rather than the forms after some simplifications. Given two independent $N$-dimensional Gaussian vectors $n_{1}\sim\mathcal{N}(0,\sigma^{2}_{1}\...
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Asymptotic Expansions for the Hypergeometric function for large parameters and bounded z?

I have stuck on an asymptotic expansion question for the Gaussian Hypergeometric function for large $a$ and bounded $z$ as the following form with $\lambda$ a constant. \begin{equation} {}_2F_{1}(a,a+\...
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Proof of transformation of Hypergeometric to Whittaker.

I am working on this paper, regarding the spectrum of a certain operator in the hyperbolic plane, and at a certain point are presented with an hypergeometric function \begin{equation} \text{}_2 F_1\...
MultipleSearchingUnity's user avatar
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Asymptotic of $\sum_{k=0}^\infty\frac{\Gamma (k+n+1) \Gamma (3 k+n+1)}{\Gamma (k+2) \Gamma (3 k+2 n+2)}$ as $n\to \infty$

Let $a,b,c,d$ be fixed and in a small neighborhood of $0$, so the series $$f(n) = \sum_{k=0}^\infty\frac{\Gamma (a+k+n+1) \Gamma (b+3 k+n+1)}{\Gamma (c+k+2) \Gamma (d+3 k+2 n+2)}$$ converges, the ...
pisco's user avatar
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How to prove that $\big(\operatorname{sinc}(x) + \operatorname{sinc}(x-1) \big)^{*r} = \frac{\Gamma{(r+1)}}{\Gamma{(x+1)}\Gamma{(r-x+1)}}$

First thing to note is that $*r$ on the upper index is defined to be the convolutional power. This means that the affected function is convoluted with itself $r$ times. The idea of the question is, ...
A. Kristian Winstén's user avatar
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Proving $_4F_3\left(\frac{1}{2},\frac{3}{4},\frac{3}{4},1;\frac{5}{4},\frac{5}{4}, \frac{3}{2};1\right)=\frac{\Gamma^8(1/4)}{768\pi^3}$

I have been trying to Prove the following Sum. $$\frac{\Gamma^2(5/4)}{\Gamma^2(3/4)}\sum_{r=0}^{\infty}\left(\frac{1}{2r+1}\right)\frac{\Gamma^2(r+3/4)}{\Gamma^2(r+5/4)}=\frac{\Gamma^8(1/4)}{768\pi^3}$...
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Asymptotic limit of Hypergeometric function into Whittaker function.

I started with the differential equation \begin{equation} -\sinh^2(x) \left( \frac{d^2}{dx^2} - \left( k - \frac{b}{\tanh(x)} \right)^2\right) \phi_1(x) = \lambda \phi_1(x) \end{equation} and found a ...
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Spectrum of Klein-Gordon operator in Hyperbolic space.

I'm working on obtaining the spectrum of the Klein-Gordon operator in Hyperbolic space $H_2$ in a given coordinate system, but I'm having trouble solving for the continuous part of the spectrum. The ...
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Analytic continuation of incomplete beta function

The incomplete beta function $B_x(a,b)$ is defined for $x\in [0,1]$ by the integral $$B_x(a,b)=\int_0^x dt t^{a-1}(1-t)^{b-1}.\tag{1}$$ I'm interested in two aspects associated to that. First is the ...
Gold's user avatar
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How and why does $\frac{(k+1)(c+k)}{(a+k)(b+k)}=1+\frac{1+c-a-b}{k}+O\left(\frac{1}{k^2}\right)$ hold for integer $k$ and complex $a,b,c$?

The question is rather simple: How and why does this identity hold? $$\frac{(k+1)(c+k)}{(a+k)(b+k)}=1+\frac{1+c-a-b}{k}+O\left(\frac{1}{k^2}\right)$$ where $k$ is an integer and $c,a,b$ are complex. ...
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Prove an inequality with confluent hypergeometric functions

In some biological modeling work I am doing, I end up with an expression involving Kummer's confluent hypergeometric function ${}_1F_1(a;b;z)$. The expression is as follows: $f(x,n) = \frac{1}{(n+2)} {...
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Tricky "Divergent" Integral: Correction to Groundstate

I am trying to rederive the results presented in the paper, in particular equation (30). That is, I am trying to compute the correction to the ground-state energy of a dipolar condensate due to beyond-...
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Hypergeometric functions with constant modulus over the positive real line

I am interested in finding examples (and more optimistically, a full classification) of hypergeometric functions $_2F_1(a,b;c;x)$, that evaluate to a phase (i.e. they have constant modulus) for x real ...
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