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Questions tagged [hypergeometric-function]

In mathematics, the Gaussian or ordinary hypergeometric function ${}_2F_1(a,b;c;z)$ is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.

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An identity on $\small{}_pF_q\left(\left.\begin{array}{c} a_1+1,a_2+1,\dots ,a_p+1\\ b_1+1,b_2+1,\dots ,b_q+1\end{array}\right| z\right)$

I stumbled upon this relation while trying to answer this post. I was trying to find a relation between the two generalized hypergeometric functions, $$A=\,_3F_2\left(\color{blue}{\tfrac12,\tfrac12},\...
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On $\sum_{n=0}^{\infty}{2n+3\choose n+1} \left(\frac{1}{2^n}\cdot\frac{3}{2n+1}\right)^4$ and Gieseking's constant

I. Intro While trying to solve this post about the function, $$F(k)=\sum_{n=0}^{\infty}{2n+3\choose n+1} \left(\frac{1}{2^n}\cdot\frac{3}{2n+1}\right)^k$$ for $k=3$, I found out Mathematica can ...
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Rewriting Appell's Hypergeometric Function $F_1$ in terms of Gauss' Hypergeometric Function $_2F_1$

While going through David H. answer on What is $\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx$? I have encountered a step in between I do not really understand. Within the second ...
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Correction terms in the asymptotics of hypergeometric function

I am interested in obtaining the asymptotic expansion of $r(\rho)$ $($which is the inverse of $\rho$ below$)$, $$\rho=\frac{2b}{1-q}\left(1-\left(\frac br\right)^{1-q}\right)^{1/2}\left(_2F_1\left(\...
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What hypergeometric transformation rules might I apply to try to simplify a certain expression?

I have (https://mathematica.stackexchange.com/questions/189538/sum-a-certain-hypergeometric-function-based-expression-pertaining-to-an-integrat) a Mathematica expression involving the following six (...
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An identity for generalized hypergeometric function

I think the following identity is true, $$ \frac{4 (4 s+9)}{3 \Gamma \left(s+\frac{5}{2}\right) \Gamma \left(s+\frac{7}{2}\right)}-\frac{16 (s+2)}{3 \Gamma (s+3)^2}=\frac{\, _3F_2\left(2,s+\frac{5}{2}...
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Hypergeometric identity

I was trying to solve this integral problem and I noticed something that may be true $$ \int((1-x^r)^{1/r}-x)^{2 n} \mathrm dx = \frac{1}{2 n+1}\sum _{j=1}^{2 n+1} (-1)^{j+1} x^j \binom{2 n+1}{j} \, ...
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Integral involving a Gaussian hypergeometric function and a rational function

Let $x_0,x \in (0,1/10)$ and define: \begin{equation} g(x):= F_{2,1}\left[\frac{1}{13},\frac{1}{17},\frac{1}{5}; 100 x^2 \right] \end{equation} Then the following identity holds true: \begin{eqnarray} ...
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Integral involving hypergeometric function

I've worked out the projection of a spherically symmetric power law volume density profile $\rho(r)=br^a$, i.e. its surface density $\sigma(R)$, and am now trying to integrate this in a series of ...
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Identities involving the Gaussian hypergeometric function

By applying the algorithm from Solving linear ordinary, 2nd order differential equations via global integral bases. to the five parameter family of ODEs defined in my first answer to Gauge ...
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How to find integration of function, in form of hypergeometric function, given below?

I would like to prove the left side to right hand side which is in form of hypergeometeric function. Looking for your hints, suggestions and solultions. $$ \alpha_{1} \int_{0}^{1} (1-z)^{\alpha_{1}+\...
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which one is worse in terms of probability

There are 2N white balls and N red balls (all balls are same except for the color), to put into K different boxes, such that every box contains 3N/k balls. We say event A happens, if any box has more ...
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Functions whose input is the same as the output?

Given the Dedekind eta function $\eta(\tau)$ and complex number $\tau$. I came across these family of functions, $${f_2(\tau)= \frac{i}{\sqrt{2}}\frac{\,_2F_1\left(\tfrac14,\tfrac34,1,\,1-\alpha_2\...
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Convergent or divergent series? Asymptotic of the series?

\begin{align} A&=\sum_{n\ge 0} \frac{(-1)^n}{n!} \frac{b^n}{(n+1)^3}\\ &= \sum_{n=0}^{\infty}\frac{(-1)^{n}b^{n}}{n!}\frac{1}{(1+n)^{3}} = 1\,{}_{3}F_{3}(1,1,1;2,2,2;-b) \end{align} The series ...
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Hypegeometric differential equation

0 I was wondering if someone, a smart person, can tell me how to transform the following equations (see below) into the well-known Hypergeometric equations. I know that this can be done provided The ...
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Solving an integral equation with inverse Laplace transform

Let $\alpha,\beta,\mu>0$. I am looking for a solution, i.e. a function $g(x)$, that satisfies $$ \frac{\beta^{\alpha}}{\Gamma(\alpha)}\int_0^\infty g(x)x^{\alpha-1}e^{-\beta x}\,\mathrm dx=\left(\...
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bounds for hypergeometric 2F2 function

Looking for lower- and upper- bounds for the following case of the hypergeometric ${}_pF_q$ function: $$ {}_2F_{2}(1,k;\;k+1/2+b,k+1/2-b;\;z), $$ where $k\in\mathbb{N}$, $b\in[0,1/2)$, and $z\ge0$. ...
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Confused on deciding whether its Negative Binomial or Binomial and Negative Hypergeometric or Geometric

I am currently struggling on deciding whether a question is a Negative binomial or a Binomial, and Negative Hypergeometric or a Hypergeometric SRV (Special Random Variable), As I seem to always ...
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Is the hypergeometric function $_2F_1\left[\frac{1}{2},n+\frac{1}{2};n+1;z\right]$ expressible in terms of more elementary functions?

Is this special case of hypergeometric function expressible in terms of more elementary functions : $$_2F_1\left[\frac{1}{2},n+\frac{1}{2};n+1;z\right]$$ It will also be helpful for me to know, if ...
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Integral involving power, exponential and Confluent Hypergeometric function from 1 to Infinity

I am trying to integrate the following $\int_{1}^\infty e^{-a\,y}\,y^{b-2}\,W_{\kappa,\nu}(c\,y)\,dy$ where $a,b,\nu\,\in\Bbb R$, with $a,b,\nu>0$ and $c,\kappa\in\Bbb C$, with $Re(c)=0$, $Im(c)&...
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Sum of hypergeometric function

I am trying to evaluate the following sum $$\sum_{n=1}^N {}_2F_1(-n,n-N,1,x) y^n $$ I notice that according to wolfram alpha, $$\sum_{n=1}^\infty {}_2F_1(-n,b,c,x) y^n = \frac{_2F_1(1,b,c,\frac{...
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$a_n=(n-1)a_{n-1}+a_{n-2}$

In an answer to a recent question of mine, I was introduced to the recurrence relation $$a_n=(n-1)a_{n-1}+a_{n-2}$$ Where $a_1=0,a_2=1$. I have computational evidence that $a_n=a_{-n}$, as I have ...
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Laplace transform of generalized hypergeometric distribution

What is please the Laplace transform (moment generating function $M(t)$) of a generalised hypergeometric distribution shown below $$p_X(x)=K\cdot\frac{(a_1)_x\dots(a_p)_x}{(b_1)_x\dots(b_q)_x}\cdot\...
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The accuracy of relations related to hypergeometric function

Kummer differential equation is: $$xy^{''}+(b-x)y^{'}-ay=0$$ It is written in the handbook of mathematical functions(Abramowitz and Stegun 1972) that, this equation has 8 answers, the most general of ...
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Inverse of the asymptotic expansion of Gauss Hypergeometric function

I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of $\rho$ below). Basically I want to series expand $\rho$ for large $r$ (i.e. as $r\to \infty$) and then ...
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Evaluation of $\int_{e}^{\pi} \frac{dx}{4(x^2-1)- (x^2-1)^{\frac14}2^{-\frac14 }}$

I tried many ways to evaluate the below integral but i didn't succeed , I have used Hypergeometric functions but it fails , now My question is the below integral has known closed form ? and how i can ...
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What is this generalization of the Chebyshev polynomials?

For $\varepsilon>0$ consider the tridiagonal matrix $$L_{\varepsilon}=\begin{bmatrix} 0 & 1 & \ & \ & \ & \ & \ & \ \\ 1 & \varepsilon & 1 & \ & \ &...
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Is the hypergeometric function a simple resurgent function

In the study of Airy functions you obtain, by Borel transforming components of a transseries, a hypergeometric function $_2F_1(a,b,c|z)$ where the constant are positive. Now I would like to know if ...
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Tricomi confluent function with the negative argument

What is the relation between the Tricomi hypergeometric function with the negative argument and the Tricomi hypergeometric function with the positive argument or Kummer hypergeometric function? That ...
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hypergeometric function with special Pochhammer symbol

What is the relationship between these two hypergeometric functions? Can the following function be written as another function of some hypergeometric functions ? $$1F1(a+b,2a,x)$$ and $$1F1(a+b,a,x)$$ ...
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Solve the equation similar to the Kummer equation

In my calculations, I arrived at the differential equation below. What should I do to solve this equation, which is similar to the Kummer equation? Can the Kummer equation be obtained by changing the ...
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Integral of squared Hypergeometric Function

I am trying to integrate the following $\int_{0}^{1} {_2{F}_1}\big(-n,1+2m+n,1+m,1-z\big)^2 dz$, where $m\in\Bbb Z$ and $n\in\Bbb Z$ with $m>0$, $n\geq 0$. (Basically I want to normalise the ...
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an approximation to the generalized hypergeometric function

Relating to the article An approximation to the generalized hypergeometric function, I would like to calculated example of the Poisson distribution $Po(10)$. If you have the possibility, please see ...
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Tricomi confluent hypergeometric function

Let $$U(\alpha,\delta,z)=\frac{\Gamma(1-\delta)}{\Gamma(\alpha-\delta+1} {}_1\!F_1(\alpha,\delta,z) + z^{1-\delta}\frac{\Gamma(\delta-1)}{\Gamma(\delta)}\!F_1(\alpha-\delta+1, 2-\delta,z)$$ be the ...
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The convergence and reality of Gauss Hypergeometric Function

The Gauss Hypergeometric function is defined via a power series: $$F(a,b,c;z)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}\frac{z^n}{n!}$$ In ...
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Closed form for $c_m = \sum_{n=|m|}^{\infty} \left(\dfrac{1}{2}\right)^{2n} (-1)^{m+n}{2n \choose m+n}$, $m$ integer

What is a closed form for $c_m = \sum_{n=|m|}^{\infty} \left(\dfrac{1}{2}\right)^{2n} (-1)^{m+n}{2n \choose m+n}$ with $m$ an integer? (N.B. the starting index is an absolute value.) I already ...
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How to get a formula for the definite integral mentioned below

I came upon the problem to get a formula depending on $k\ge 1$ for the definite integral $$\int_0^1 \frac{t^{2k}}{(1-t^2)^k} dt$$ Please note that 1 is a singularity of the integrand. I tried on ...
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A connection between the confluent hypergeometric function and Bessel functions.

Let $b$ and $w$ be real parameters subject to $b\neq 1$. Let $x \in {\mathbb R}$. Define: \begin{equation} {\mathfrak N}(w,b):= \frac{2^{\frac{1}{2 (1-b)}+1} \left(\frac{1}{b-1}\right)^{\frac{1}{2 (b-...
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Do we have a formula to express $_3F_2(L+r-1+2q,1,r+2;L+r+2,r+3;1)$ in closed form?

Note: This is not a homework. I am interested to calculate $_3F_2(L+r-1+2q,1,r+2;L+r+2,r+3;1)$ where $L,r\geq 1$ are integers and $q\in [0,1]$ is a real number. In particular, Question: Do we ...
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How to calculate the Kampé de Fériet function?

This is a continuation of this post. The following is my original question in that post. Question: Is it possible to express $$\sum_{l=0}^\infty \sum_{r=0}^\infty\frac{\Gamma(L+r-2q)}{\Gamma(L+...
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Simplifying specific $\,_4F_3$ hypergeometric function?

Consider the hypergeometric function $$\,_4F_3\left({{a,b+n,c-n,d}\atop{a+1,b,c}};1\right)$$ with $a,b,c,d\in\mathbb{C}$ and $n\in\mathbb{N}$. Is there any way to simplify it to bits and pieces that ...
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multiplying a hypergeometric series

We are able to calculate the value of the sum $\sum_{k=0}^\infty \frac{(a_1)_k(a_2)_k\dots(a_p)_k}{(b_1)_k(b_2)_k\dots(b_{p-1})_k}\cdot\frac{x^k}{k!}$, which equals the generalised hypergeometric ...
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Simplify $\sum_{l=0}^\infty \sum_{r=0}^\infty\frac{\Gamma(L+r-2q)}{\Gamma(L+r-1+2q)} \frac{\Gamma(L+r+l-1+2q)}{\Gamma(L+r+l+2)}\frac{r+1}{r+l+2}$

This question is a continuation of this post. Let $r,l,L\geq 1$ be integers. Assume that $q\in [0,1]$ is a real number. The authors obtained the following equation $36$ in their paper (I express in ...
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If $a,b,c$ are positive integers with $c\leq a+b,$ can I conclude that $_2F_1(a,b;c;1)$ diverges?

Recall that the hypegeometric series is defined by $$_2F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}$$ where $z\in \mathbb{C}$ with $|z|<1$ and $(a)_n = a(a+1)...(a+n-1)...
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Probability Function Hypergeometric distribution

The question: A shipment of 2500 car headlights contains 200 defective. You choose from this shipment without replacement until you have 18 which are not defective. Let X be the number of defective ...
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Understanding Mathematica's formula for $ \int_0^{\infty } x^a \exp \left(-\frac{c x^2+f x}{b}\right) \, dx $

My goal is to integrate the following function: $$ \int_0^{\infty } x^a \exp \left(-\frac{c x^2+f x}{b}\right) \, dx $$ where, $a, b, c > 0$ and $a, b, c, f \in \mathbb{R}$. Mathematica gives me ...
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Is a hypergeometric sum the minimum of a “potential” function?

I'm wondering if values of a generalized hypergeometric function can be written as solutions to an optimization problem, like this: $$_q F_p (a_1, \dots, a_p;b_1,\dots,b_q;x)=\min_{t}\psi(a_1, \dots, ...
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My attempt at $f''(x)+\sin(f'(x))=0$ does it work?

Here's an attempt at a fairly innocent-looking separable differential equation. $$f''(x)+\sin f'(x)=0$$ $$\int\frac{f''(x)}{\sin f'(x)}dx=-(x+c)$$ let $y=f'(x)$ then $dy=f''(x)dx$ $$\int\csc y\,dy=-(x+...
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Computation of a sum involving gamma functions

Let $l$ denote a positive integer and $m$ be an integer $-l \leq m \leq l$. I would like to prove the following identity: $$\sum_{0 \leq j \leq \left\lfloor\frac{l - m}{2}\right\rfloor}\sum_{0 \leq k ...
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hypergeometric function - express in terms of hypergeometric function of another variable

I am very new at this, so please be happy to correct my mistakes. Suppose I have a Hypergeometric function $ {}_2F_1(a,b;c;z) $, where $z$ is the form $ z = Ax^2 + Bx, \quad where \space A,B > 0 ...