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 \begin{equation} 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 \begin{equation} \int^{x}_{0} \sqrt{\cosh(x)} dx. \end{equation} 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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Infinite series Sum of zeroth order Bessel Functions of first kind

I am trying to find the upper bound of $$ \sum_{n \geq 1} J_0(an) J_0(bn) \sin(cn) \sin(dn) $$ where $J_0(x)$ is the zeroth order Bessel function of first kind, and $a,b \geq 0, \textit{ and } c,d \in ...
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Lagrange inversion theorem of $x^r(x+k)$ to generalize the W Lambert function

Motivation: $2$ branches of Lambert $\text W_k(z)$ is a limit of the inverse of $x^n(x+c)$ which is expressible in terms of FoxH in Mathematica. $\text W_0(x)=\text W(x):$ $$-\lim_{a\to0}e^{\frac{(-x)^...
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Hypergeometric Variance

I have a question about finding the variance quickly which came from my textbook. Suppose there are 6 men and 4 women with 3 promotion positions available. The director decides to select 3 randomly. ...
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Under which circumstances is the incomplete beta function equal to Gauss hypergeometric function

$$B_y(x, z)= \frac{y^x}x2F1(x, 1 − z; 1 + x; y)$$ Note: 2F1 represents the Gauss hypergeometric function. I had trouble around its notation here. I have this relationship between an incomplete beta ...
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Asymptotic expansion of $U(j+a,2j+1,z)$ for $j \in \mathbb{Z}$ and $j \to \infty$

I would like to know asymptotic expansion (at least first few terms) for the following special case of Tricomi confluent hypergeometric function: $$f_j(a,z)=U(j+a,2j+1,z)$$ for $j \in \mathbb{Z}$ and $...
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Approximation of $\int_0^\pi \big[x(\pi-x)\csc (x)\big]^k\,dx \quad \forall k$

A recent post addressed the problem of the closed form of $$I(k)=\int_0^\pi \Bigg[ \frac {x(\pi-x)} {\sin(x)}\Bigg]^k \,dx$$ When $k$ is a positive integer, they seem to be known except that they ...
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How to prove the quotient of confluent hypergeometric functions of adjacent orders is convex?

Denote $F(;n;x)$ as the confluent hypergeometric function $_0F_1$, i.e. $F(;n;x)=\sum\limits_{k=0}^{\infty}\frac{x^k (n-1)!}{(n+k-1)!k!}$. How to prove $\frac{F(;n+1;x)}{F(;n;x)}$ is a convex function ...
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Show that $\binom{r}{k}(p-\frac{k}{n})^k(q-\frac{(r-k)}{n})^{r-k} < q_k < \binom{r}{k}p^kq^{r-k}(1-\frac{r}{n})^{-r}$

Show that $$\binom{r}{k}\left(p-\frac{k}{n}\right)^k\left(q-\frac{r-k}{n}\right)^{r-k} < q_k < \binom{r}{k}p^kq^{r-k}\left(1-\frac{r}{n}\right)^{-r}$$ $q_k$ is given by $$\begin{align}q_k&=\...
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Solution to this beta function integral-within-an-integral (Continued)

Continuing on from my earlier post, here, I have re-evaluated all the work-up to the integral and managed to re-formulate everything to be slightly different. Now, the aim is to solve $$ I = \int_0^\...
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Solution to this hypergeometric function-beta function product integral

I am trying to calculate the energy of a system, and have reduced part of it down to an integral; $$ \int^\infty_0 \frac{ t^{\frac{5}{\alpha}-7}}{(1+t)^8}\space_2F_1(1,\frac{3}{\alpha}-6;\frac{3}{\...
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Hypergeometric Function and it's relation to Gamma function

I've been trying to solve a Quantum mechanics problem, where I have to do the integration below: $$ \int_{-\infty}^{+\infty} \frac1{(x^2+b^2)^{2n}}dx $$ with $b$ and $n$ not specified (the problem ...
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Proving Jacobi's Triple Product Identity using Heine's Summation formula

I am taking a course in Special Functions in Number theory and was given Basic Hypergeometric Series by Gasper and Rahman as the text book. In that, Jacobi's triple product is proved using Heine's ...
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A Hypergeometric probability distribution and Zeilberger

On page 107 of book 'The Concrete Tetrahedron' by Manuel Kauers . Peter Paule A Hypergeometric probability distribution is given : consider an urn containing N balls, m green ones and $N - m$ blue ...
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Evaluating $\mathbb{E}[(X/(X+a))^2]$ for $X$ a Binomial random variable

Suppose $X$ is Binomial with parameters $n, p$. Is it possible to compute $$ \mathbb{E}\Big[\Big(\frac{X}{X + a}\Big)^2\Big] $$ in closed form when $a > 0$? I tried writing it as $(1- a/(X+a))^2$, ...
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How to solve the following set of integrals by using Hypergeometric functions?

Any recommendations to help me to solve this integral $$ \int_{-\sqrt{x}+y}^{1-y} \left(1-(t+y)^2\right)^{r} \left(1-x(t-y)^2 \right)^{r} dt $$ where $\{r,x,y\} \in \mathbb{R} , \, |y|<1+x, ...
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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: \begin{equation} 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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Finding a simple differential equation to define an inverse of $\,_2\text F_1(a,b;c;z)$ with respect to $z$ with the Gauss Hypergeometric function.

An “Inverse Gauss Hypergeometric function” with respect to $z$ in terms of a differential equation would define many special case inverse functions. Define: $$\,_2\text F_1(a,b;c;z)=\sum_{n=0}^\infty \...
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3 votes
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Laplace transforms of products of modified Bessel Functions

I am dealing with integrals of the form $$\int_0^\infty e^{-t}I_0(xt/a)^a\ \mathrm{dt}$$ where $I_0(x)$ is the modified Bessel function of the first kind. Clearly this is just a Laplace Transform $\...
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2 votes
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Methods to prove closed-form solution to series over hypergeometric functions?

I'm trying to solve a problem in graph theory, and I've stumbled upon the sum: \begin{equation} T_{nlm}=\frac{1}{(\alpha+m+n)_l}\sum_{k=0}^\infty\frac{\Gamma(\alpha+m+l)}{\Gamma(\alpha+m+k)}\frac{(-n)...
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Is Maple's simplification assuming integer rigorous?

I have a rather long and complicated expression that should be zero that I cannot work out by hand. But using Maple and assuming the variable is an integer, it simplifies to zero, as expected. How ...
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Integrating the second modified Bessel function squared

I would like to compute the integral of the second modified Bessel function which has the following form $$ \int_0^\infty dz \;K^2_{\nu}(z)\;z^\alpha $$ where $K_\nu(z)$ is the second modified Bessel ...
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3 answers
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Integral of second modified Bessel function [closed]

I would like to compute the integral of the second modified Bessel function which has the following form $$ \int^{\infty}_{\epsilon}{dz \;z^a K_{\nu}(z)} $$ where I have some power of $z$ multiplied ...
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Expectation under Cauchy distribution [duplicate]

I recently discovered this relationship but I cannot understand why it holds. $$\int_0^\infty \frac{x^\alpha}{1+x^2} \, dx = \frac{\sin(\pi\alpha/2)}{\sin(\pi\alpha)}$$ when $0<\alpha<1$. As we ...
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How to find a closed form formula for such integral using special functions

I am trying to find a closed form formula for this integral $$\int_{}^{} \left(1-x^2\right)^n \frac{\mathrm{d^i} }{\mathrm{d} x^i} \left(1-x^2\right)^n dx $$ where $i=0,1,2,...\in \mathbb{N}$ and $n\...
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2 votes
2 answers
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Trigonometric Integrals and Hypergeometric function

I'm dealing with these two integrals: $$I_1=\int_{-\pi}^{\pi} \frac{\cos(x)\cos(nx)}{(1+e \cos(x))^3} \mathrm{d}x, \quad I_2=\int_{-\pi}^{\pi} \frac{\sin(x)\sin(nx)}{(1+e \cos(x))^3}\mathrm{d}x$$ is ...
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What is this exponential related series?

Does the following series have a solution in terms of standard-ish funcitons (e.g. hypergeometric)? If not, does it at least have a name, or papers discussing it? (It seems so simple that it ought to ...
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Evaluation of $\displaystyle \int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt[3]{\frac{1}{3} + \sin^2 x}} $ [duplicate]

I want to evaluate the following integral in a closed form involving elementary functions: $$I = \displaystyle \int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt[3]{\frac{1}{3} + \sin^2 x}} = \frac{\sqrt[3]{3}\...
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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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1 vote
2 answers
72 views

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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1 answer
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Help me to find good references about this equation. [closed]

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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1 vote
1 answer
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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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14 votes
3 answers
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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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$\sum _{k=0}^{\infty } \sum _{v=0}^{\infty } \frac{ (a+b x)^k (a-b x)^v}{(k!)^2 v!}U(v+1,k+v+2,x)$ with $U$ being confluent hypergeometric function

I am wondering if I can convert the expression $$f(x)=\sum _{k=0}^{\infty } \sum _{v=0}^{\infty } \frac{ (a+b x)^k (a-b x)^v}{(k!)^2 v!}U(v+1,k+v+2,x)\\{\rm with}\,\,x,a\in\mathbb{R}_{>0},b\in\...
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4 votes
1 answer
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Proving $\int_{0}^{1} \frac{K(x)K\left ( \sqrt{2} \sqrt{(1-x^2)/(2-x^2)} \right ) }{2-x^2}\text{d}x =\frac{\pi^3}{8\sqrt{2} } {}_6F_5(...)$

I encountered an integral identity: $$\int_{0}^{1} \frac{K(x)K\left ( \sqrt{2} \sqrt{\frac{1-x^2}{2-x^2} } \right ) }{2-x^2}\,dx =\frac{\pi^3}{8\sqrt{2}} {}_6F_5\left ( \frac{1}{4},\frac{1}{4},\...
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5 votes
2 answers
412 views

$\int_{-\infty}^{\infty}\frac{1}{3 t^2+1} {\rm exp}\left(\frac{at^2+ibt}{3 t^2+1}+itx\right){\rm d}t$

How to solve the integral? $$ f(x)=\int_{-\infty}^{\infty}\frac{1}{3 t^2+1} {\rm exp}\left(\frac{a t^2+i b t}{3 t^2+1}+itx\right){\rm d}t\tag{1} \\{\rm with}\,\, x,b\in \mathbb{R},a\in\mathbb{R}_{<...
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1 vote
1 answer
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Indefinite integral with geometric series

I was trying to calculate the following integral $$\int \frac{\text{d}x}{(1 + x^n)^n}$$ for $n > 0$. I tried this road: $$\int \frac{\text{d}x}{\left(x^n\left(1 + \frac{1}{x^n}\right)\right)^n} = \...
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1 answer
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Solve this integral by hypergeometric function

I was trying to find the normalization constant of the following distribution:$$p(x)=(1-(1-q)x^2)^{\frac{1}{1 - q}}$$ where $1<q<3$, which done by integration over $p(x)$ from $-\infty$ to $\...
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1 vote
1 answer
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Swapping limits when simplifying a hypergeometric function

I have been trying to find a simplified expression for ${}_3F_2\left(\begin{smallmatrix}1,1,\frac{3}{2}\\2,3 \end{smallmatrix}; x\right)$. After manipulating the Pochhammers in the definition, I ...
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2 votes
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Differential Equation involving hypergeometrics

My earlier question did not get answered so I'm presenting a related question with the hope that the answer to this will help me figure out the other one myself. I have come across the following ...
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Writing the expectation of a function of gamma random variable in terms of special functions

I have to calculate expectations of the form $\int_0^\infty f(x)\frac{x^{k-1}e^{-\frac{x}{\theta}}}{\theta^k\Gamma(k)}dx$, basically $\mathbb{E}[f(x)]$ where $x\sim\text{Gamma}(k,\theta)$. ...
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first inverse moment of hypergeometric distribution

I am interested in the Petersen Estimator for the Mark and Recapture method. By standardarguments one can construct the estimator via Maximum-Likelihood as follows: $$\hat N(k)=\frac{R m}{k}$$ Where $\...
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3 votes
0 answers
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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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4 votes
2 answers
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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 $\text P_a^b(z)$ ...
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