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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)....

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Expected value of multiple iid variables conditional on number of realisations below a threshold

Consider $N$ random variables $X_1, X_2, \ldots, X_N$ that are i.i.d. distributed according to some cumulative distribution function $F$. Assume we receive a signal that says that $n$ number of the ...
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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 ...
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PMF of HGeom & Binom Distributions (Women and Men getting promoted)

Working through some book problems from "Introduction to Probability" (Blitstein) A company with n women and m men as employees is deciding which employees to promote. (a) Suppose for this ...
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Exact value of Elliptic Integrals.

I was taking currently in a elementary calculus course where i found how to find arc lengths of a smooth continuous curve. so here is how i started : $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\Rightarrow y=...
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Conjecture: $(x^\alpha+p)^\beta=\,_1F_0(\beta;;x^\alpha+p-1)$ and ideas for proof

Conjecture: For $|z|<1,\,\alpha,\beta,p\in\Bbb R$ $$(z^\alpha+p)^\beta=\,_1F_0(\beta;;z^\alpha+p-1)$$ I found this formula by noting that $$z^\alpha=\,_1F_0(\alpha;;z-1)$$ Via the simplification of ...
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Extreme-mega-ultra-crazy hypergeometric functions

I've been lusting over hypergeometric functions, and came up with some questions. Here goes. I've defined the following functions, and I want to know if there are any closed forms for them. $$f_1(2;...
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Closed form for $_2 F_2(\alpha,3\alpha;2\alpha,4\alpha;x)$

I came up with a potentially interesting hypergeometric function. I know that $$_2 F_2(\alpha,3\alpha;2\alpha,4\alpha;z)=\sum_{n=0}^{\infty}\frac{(\alpha)_n(3\alpha)_n}{(2\alpha)_n(4\alpha)_n}\frac{z^...
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Problem with a Hypergeometric function integral

I'm wondering how to solve the integral $\int_\mathbb{R}\frac{{}_2F_1\left(\Delta, 2\Delta + \frac{d - 1}{2}; 2\Delta, 1 - \frac{(x - z)^2}{(z - y)^2}\right)}{|z|^{2\Delta}|z - y|^{4\Delta - d + 1}}dz$...
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HyperGeometric distribution : Inutition for symmetry

Wikipedia page on HyperGeometric distribution says Swapping the roles of green and drawn marbles: $$ f ( k ; N , K , n ) = f ( k ; N , n , K ) $$ where in LHS, N = Total number of marbles n = ...
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Subgroup of hypergeometric equation's monodromy which gives rise to a reducible representation

I would like to find and (if possible) classify the subgroup of the Gauss hypergeometric equation's monodromy which gives rise to a reducible representation. I clarify the problem in the following. ...
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Hypergeometric, selecting together or one by one?

In hypergeometric distribution the following info is given to us. There are $N$ objects out of which $r$ objects are desirable and $N-r$ undesirable. To select $x$ objects from $r$ and $n-x$ objects ...
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Help with Mellin-Barnes Integral (product of two Hypergeometrics)

I am trying to prove that $$\int_0^1 \frac{dz}{z^2} z^{h}\cdot {}_{2}F_{1}(h,h;2h;z) \cdot {}_{2}F_{1}\left(\frac{1+2a}{2},\frac{1-2a}{2};1;\frac{z-1}{z}\right) = -\frac{\Gamma(2h)}{\Gamma{(h)}^2} \...
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Expansion of hypergeometric function for large arguments

Does anybody know of an asymptotic expansion of $${}_2 F_1 \left( {a, \ b+\lambda\atop c+\lambda}; z \right)$$ for large $\lambda$ and $z \to 1^-$? Alternative just for bounded $z$. I found a big ...
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Solutions of hypergeometric equation when $c\in\mathbb{Z}$

Let $E(a,b,c;z)$ be the hypergeometric differential equation $$ z(1-z)w''+(c-(a+b+z)z)w'-abw $$ with $w$ the unknown. It is well known that if $z\notin\mathbb{Z}$, then $E(a,b,c;z)$ has linearly ...
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Integral $\int_0^\infty dp \, \frac{p^5 \sin(p x) e^{-b p^2}}{p^4 + a^2}$: any clever ideas?

I am trying to solve the following integral, with $a>0,$ $b>0$: $I \equiv \int_0^\infty dp \, \frac{p^5 \sin(p x) e^{-b p^2}}{p^4 + a^2} $ By expanding the $\sin$, I get $I = \sum_{n=1}^\...
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Expectation of a function of a Poisson random variable

Suppose $X\sim \mathsf{Poi}(c\cdot n)$. Consider the expectation \begin{align} F(n,c) = \mathbb{E}\left[ \prod_{r = 0}^{X-1} \left ( \frac{n+r}{2n+r} \right ) \right]. \end{align} Clearly, $e^{-cn/2}=...
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Series of binomial coefficient denominators

I'm not sure how to evaluate the following : $$ \sum_{k=i}^n \frac{1}{k!(n-k)!} $$ Where $i,n \in \mathbb{N}, n > i$ are given. I don't have any working for this, I just looked it at and don't ...
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contour integration path in the complex s-plane running from $R−iW$ to $R+iW$

I would like to run the following function : "The generalized upper incomplete Fox’s H function" given by $$ \large{H}_{m,n}^{p,q}\left( z \left| \begin{array}{cc} (a_1,\alpha_1,A_1)\cdots (a_p,\...
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How to solve the given integral avoiding infinite series sum?

Question: How to solve the following integral? $$I = \int_0^\infty \dfrac{x^{N_a + N_b - 1}}{(p \Omega_1 + \Omega_2 x)^{N_a + 1}} \ln (1 + Qx) \, _2F_1\left( N_b + 1, N_b; N_b +1; \dfrac{-\Omega_3}{\...
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Numerical Inversion of an incomplete beta function expressed as gauss hypergeometric function using Mathematica

I am currently working with this hypergeometric function ${_2}F_1$, $\rho(r)=\frac{2b}{1-q}(1-(\frac{b}{r})^{1-q})^{\frac{1}{2}}{_2}F_1(\frac{1}{2},1-\frac{1}{q-1},\frac{3}{2},1-(\frac{b}{r})^{1-q})$ ...
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Computation of multiple improper integral.

In my recent work, I need to the details of the computation of the following multiple improper integral: $$\iint_{[0,1]^2}e^{-\pi x^2y^2}dxdy-\iint_{[1,\infty)^2}e^{-\pi x^2y^2}dxdy.$$ As you see, the ...
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Classification of entire solutions to hypergeometric differential equation

Question/Motivation I am trying to classify all of the entire (i.e. holomorphic) solutions to a simple-looking ODE of the form $$z\frac{\partial^2f}{\partial z^2} + (az+b)\frac{\partial f}{\partial ...
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Expressing an integral in terms of the integral representation of hypergeometric function [closed]

Can the integral, $$\int_b^x \frac{r^{15/2}}{(r^5-b^5)^{3/2}}\,dr$$ be expressed in terms of hypergeometric function? Thanks in advance.
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Two Simultaneous Cumulative Hypergeometric Distributions

Suppose we have a standard 52 card deck from which we draw five cards. What are the chances of drawing one or more Aces and one or more Kings? How do I calculate this? I know that we can calculate ...
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Integral Resulting in Hypergeometric Function

Any tips on how to prove this result? $$ \int \frac{x}{x^K + c} dx = \frac{x^2 {}_2F_1 \left(1,\frac{2}{K};\frac{K+2}{K};-\frac{x^K}{c} \right)}{2c}, $$ where $${}_2F_1 (a, b;c;z)$$ is the ...
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Continued Fraction of hypergeomteric ratio

Gautschi in his paper (Anomalous Convergence of a Continued Fraction for Ratios of Kummer Functions 1977, https://www.ams.org/journals/mcom/1977-31-140/S0025-5718-1977-0442204-3/S0025-5718-1977-...
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Hypergeometric series identity from Catalan numbers

We know that $$\sum_{n=0}^\infty C_nx^n=\frac{1-\sqrt{1-4x}}{2x}$$. But because $C_n=\frac1{n+1}\binom{2n}n=\frac{(2n)!}{n!(n+1)!}$, that sum is also $$\sum_{n=0}^\infty\frac{(2n)!}{(n+1)!}\frac{x^n}{...
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How to solve the integral $\int \frac{x^n}{\sqrt{x-x^2}}$?

I am trying to find an appropriate substitution in the following indefinite integral $\int \frac{x^n}{\sqrt{x-x^2}}$ for an arbitrary power $n$ to obtainthe intermidiate steps to the same answer as in ...
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Yet another bizarre identity involving hypergoemetric functions and gamma functions.

Let $d=4$, $T\ge d$ and $p\ge 0$ be integers. By solving Spectral densities of finite dimensional sample covariance matrices we stumbled on a following identity. \begin{eqnarray} &&\sum\...
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help with an infinite series, hypergeometric function

Hi I am doing some work in interacting particle systems. I have this sum $$ \sum_{k=0}^{\infty} r^{\frac{k}{2}(2m-1-k)} $$ where $m$ is some integer and $r>1$ is real. I don't how to work this ...
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Lerch transcendent and harmonic number simplification

I am trying to find a simplification of the following, preferably to a hypergeometric function. I have the result in Mathematica notation: ...
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Guidance on reducing this Meijer-G function

According to mathematica, for $\alpha,z>0$ we have \begin{multline} G^{3,1}_{2,3}\left(z\bigg|{0,1\atop 0,0,1-\alpha}\right)=% -\frac{z}{\alpha}\Gamma(1-\alpha){_2F_2}\left({1,1\atop 2,\alpha+1};z\...
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Mellin transform of a Gaussian Hypergeometric Function with negative x-argument

I am quite fascinated by the formula for the Mellin transform of the Gaussian Hypergeometric Function, which is given by: $$\mathcal M [_2F_1(\alpha,\beta;\gamma;-x)] = \frac {B(s,\alpha-s)B(s,\...
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Generalization of hypergeometric type differential equation

I am aware that hypergeometric type differential equations of the type: can be solved e.g. by means of Mellin transforms when σ(s) is at most a 2nd-degree polynomial and τ(s) is at most 1st-degree, ...