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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
61 views

${_3F_3}\left({2,a,a\atop 1,1+a,1+a};z\right)=e^z{_3F_3}\left({?,?,?\atop ?,?,?};-z\right)$?

I'm interested in a transformation of the form $$ {_3F_3}\left({2,a,a\atop 1,1+a,1+a};z\right)=e^z{_3F_3}\left({?,?,?\atop ?,?,?};-z\right). $$ Many times, hypergeometric functions of the form ${_nF_n}...
0 votes
0 answers
54 views

A Fourier series inverting $\frac{\tan(y)}y$?

$\def\k{\operatorname k}\def\W{\operatorname W}$ Using a Dirac $\delta(x)$ Fourier series and this post $$\begin{align} y\cot(y)=\frac1x\implies\frac1{\sec^2(y)-x}=\int_0^\frac\pi2\delta(\tan(t)-xt)dt-...
  • 7,279
1 vote
0 answers
133 views

Sum with gamma value.

Context: Reading this paper: https://arxiv.org/abs/1309.1140. (On proving some of Ramanujan's formulas for $1/\pi$ with an elementary method). We find on page 26 that the author claims: $$\sum_{n=0}^{\...
0 votes
0 answers
23 views

Question about the proof of Dougall's 7F6 formula by L.J.Slater

I am reading L.J.Slater"Generalized Hypergeometric Functions". I have a question about the proof of Dougall's 7F6(1) Formula in this book. Equation(2.3.4.3) is\begin{eqnarray}&&(1+a-...
1 vote
1 answer
180 views

Inverse Fourier transform of $\frac{1}{ib|\omega|^a - \omega}$

I would like to know if there might be a connection between the following inverse Fourier transform (with some physics conventions): $$f(t) = \int_{-\infty}^\infty d\omega e^{i\omega t}\frac{1}{ib|\...
0 votes
0 answers
12 views

Hypergeometric Fox H-function representation with Mellin-Barnes-type contour integrals problem.

I need help with the double complex integral (Mellin-Barnes type).  I am solving the following integral: \begin{equation}  \int_0^{\infty}x^{b-1}\mbox{}_1{\rm{F}}_1(a,b, -c  x)e^{-s x^\delta}e^{-\...
0 votes
0 answers
25 views

Relation between HyperGeometric function and Gamma function

It seems numerically that this equality holds for $n_1,n_2 \in \mathbb{N}$ : $\frac{\Gamma (n_1+1) \Gamma (n_2+1)}{\Gamma (n_1+n_2+2)} = \frac{1}{2^{n_1+n_2+2}} \left(\Gamma (n_1+1) \, _2F_1^{(reg)}\...
  • 1
3 votes
2 answers
112 views

How to integrate $\int^1_{-1}\frac{\pi}2 e^{ix}\operatorname{sech}\left(\frac{\pi x}{2}\right)\text{ d}x$?

I have an integral I need to integrate, as follows $$\int^1_{-1}\frac{\pi}2 e^{ix}\operatorname{sech}\left(\frac{\pi x}{2}\right)\text{ d}x$$ This is quite troublesome to do, since I cannot use ...
  • 2,733
10 votes
1 answer
235 views

Evaluate $\int_{0}^{1}\frac{K(x)\ln\left(1-x^2\right)}{\sqrt{1-x^2}}\text{d}x,\int_{0}^{1}\frac{xK(x)^2\ln\left(1-x^2\right)}{\sqrt{1-x^2}}\text{d}x$

I am recently interested in integrals containing an elliptic integral $K(x)$, which is defined by $\int_{0}^{1} \frac{1}{\sqrt{1-t^2}\sqrt{1-x^2t^2} }\text{d}t$ for $|x|<1$ and $x$ is the elliptic ...
1 vote
1 answer
48 views

Limit of hypergeometric function when first three parameters are all large

I have encountered an interesting limit involving a particular parameterisation of the hypergeometric function. The function of interest to me uses the parameters $1 \leqslant k \leqslant n \leqslant ...
  • 3,930
-1 votes
0 answers
52 views

Why can't I use the binomial distribution to model the probability of choosing k of n winning lottery numbers

I'm currently working through John Rice's Mathematical Statistics and Data Analysis. In Ch. 2 of his book, Rice uses the the hypergeometric distribution to model the probability of selecting $k$ of 6 ...
4 votes
1 answer
284 views

Evaluate a strange integral involving generalized hypergeometric function ${}_1F_2$

$$ \small I=\int_{0}^{\infty}x^2e^{-4x} \left (x^{-2/3}\Gamma\left ( \frac13 \right )^2\,_1F_2 \left ( \frac{1}{3};\frac{2}{3},\frac43;x^2 \right )^2 -\frac{9}{4}x^{2/3}\Gamma\left ( \frac{2}{3} \...
0 votes
1 answer
41 views

Integration using hypergeometric function

I would like to calculate the definite integral $\int_0^\infty \frac{dx}{x(x^a + x^{-b})}$. From Wolfram Alpha, the indefinite integral is $x^b\ _2F_1(1;b/(a+b);b/(a+b)+1;x^{a+b})/b$ Where $_2F_1$ is ...
  • 33
0 votes
0 answers
47 views

Function $f$ subordinate to function $g$

I'm trying to study a research paper, during that i come across this word Subordinate, i searched this a lot on google but didn't get any satisfactory answer. I understand the definition they gave in ...
  • 1,052
3 votes
0 answers
67 views

Integral of $\arccos$ function combined with exponontial function

Is it possible to calculate this integral? $$\int_0^1 x^{j}\ e^{ax^2}\arccos(x)\ dx,\qquad a\in\mathbb{R}_+,\ j\in\mathbb{N}. $$ Although the integral looks neat and fairly simple in form. I tried ...
  • 69
0 votes
0 answers
30 views

Solving hypergeometric like differential equations using Integral transforms.

It is known that for 2nd order differential equations with 3 singularities that are regular, can be converted into the hypergeometric differential equation. As an example of this, we can see the first ...
  • 53
0 votes
0 answers
43 views

Using Inverse Laplace Transform to solve a differential equation involving F(s), F(s+a),F(s-a) and others .

I'm trying to get the inverse Laplace transform of the following expression: \begin{equation} \begin{split} 2 F(s)=\frac{-\alpha ^2 A}{\left(-B -\alpha ^2 A -s^{2}\right)}F(s+2 \alpha )+\frac{-\...
  • 53
1 vote
0 answers
25 views

Hypergeometric differential equation with fractional polynomial coefficient

I have a differential equation of the form $$ y''(x)+P(x)y'(x)+Q(x)y(x) =0 $$ where $P$ and $Q$ are rational functions $$ P(x)=\frac{c-2ia-(c+2b-2ia)x}{cx(1-x)}\\ Q(x)=\frac{(-1+(1-x)^{2/c})a^{2}+x(-b^...
8 votes
2 answers
443 views

Integral: $\int_{1}^{\infty} \operatorname{arctanh} \left(k\,\sqrt{\frac{x^2-1}{x^2}} \right)\, e^{-\alpha \, x \,} dx$

Is there any chance to find a closed form for these integrals? $$I_1(k,\alpha)=\int_{1}^{\infty} \operatorname{arctanh} \left(k\,\sqrt{\frac{x^2-1}{x^2}} \right)\, e^{-\alpha \, x \,} dx$$ $$I_2(k,\...
  • 189
2 votes
2 answers
64 views

Solve $\int_0^1\frac{x(x+1)^b}{\alpha-x}dx$ using hypergeometric functions

How can we solve this integral? $$\int_0^1\frac{x(x+1)^b}{\alpha-x}dx$$ I used geometric series $$=\int_0^1(x+1)^b\sum_{n=0}^\infty\left(\frac{x}{\alpha}\right)^{n+1}dx=\sum_{n=0}^\infty\int_0^1\left(\...
  • 1,628
2 votes
1 answer
39 views

How to merge odd series and even series of hypergeometric function of Legendre polynomials into one hypergeometric function?

On the Wolfram MathWorld page of Legendre Differential Equation, Legendre polynomials are represented as $$ P_l(x) = c_n \begin{cases}\begin{align*} &_2F_1\left(-\frac{1}{2}(l), \frac{1}{2}(l + 1);...
  • 282
2 votes
0 answers
34 views

Integral representaion of confluent hyper-geometric function

How can I prove these two integrals are equivalent ($\alpha,\beta, z>0$): \begin{equation} \frac{1}{\Gamma(\beta)}\int_0^\infty t^{\beta-1}(1+t)^{-\alpha}e^{-zt} dt, \end{equation} and \begin{...
  • 73
2 votes
0 answers
115 views

Asymptotics of ratio of q-Pochhammer symbols

In (9.23) of https://arxiv.org/abs/1908.08875 , they claim that it is "easy" to show the following limits $$\lim_{y\to 0}\frac{(yq^\frac{2-r+m}{2};q)_\infty}{(y^{-1}q^\frac{r+m}{2};q)_\infty}...
2 votes
1 answer
77 views

Converting a differential equation to a hypergeometric equation

I would like to transform the following differential equation into a hypergeometric equation: ${z^2}\frac{{{d^2}W}}{{d{z^2}}} + z\frac{{{d^{}}W}}{{d{z^{}}}} + \left( {{A^2} - {B^2}\left( {1 - {C_{}}z ...
2 votes
0 answers
19 views

Which numerical quadrature is the most suitable for an integral that integrates an exponentially increasing function?

I have an integral function of the form $$ F_n(x) = \int_0^x du \left( K_n(u)L_n(x) - K_n(x)L_n(u) \right)u^p K_n(u) $$ where $$ K_n(x) = x^{(m+1)/2}e^{-x/2}M(a_n,m+1,x), \\ L_n(x) = x^{(m+1)/2}e^{-x/...
  • 357
1 vote
1 answer
73 views

Integration of Gauss ${}_{2}F_{1}$ hypergeometric function

The indefinite integral representation of Gauss hypergeometric function is $$\int {{}_2{F_1}\left( {a,b;c;z} \right)dz = \frac{{c - 1}}{{\left( {a - 1} \right)\left( {b - 1} \right)}}} {}_2{F_1}\left( ...
1 vote
1 answer
83 views

The integral of $(1+x^2)^{-a} \ln (1+x^2)$

Consider the integral $$ \int_0^{\infty} (1+x^2)^{-a} \ln (1+x^2) dx$$ where $a > 0$ is big enough. Is it expressible in terms of some known special functions (gamma function, hypergeometric ...
  • 1,053
1 vote
0 answers
86 views

Reduction of $_3\text F_2(a,a,1-b;a+1,a+1;x)$ with the hypergeometric function

A derivative of the incomplete beta function $\text B_x(a,b)$ uses hypergeometric $_3\text F_2$ $$\frac{d\text B_x(a,b)}{da}=\ln(x)\text B_x(a,b)-\frac{x^a}{a^2}\,_3\text F_2(a,a,1-b;a+1,a+1;x)$$ Now ...
  • 7,279
1 vote
2 answers
124 views

How to evaluate $\int\frac{x^ndx}{\sqrt{ax^2+bx+c}}$ for natural $n$

How do we evaluate the following integral? $$\int\frac{x^ndx}{\sqrt{ax^2+bx+c}}$$ I am strictly looking for a solution for natural $n$ I wanted to try a trig substitution $$\int\frac{x^ndx}{\sqrt{ax^2+...
  • 1,628
0 votes
0 answers
28 views

What values of $m$ does the elliptic integral of the first kind $\text F(x,m)$ or the Jacobi amplitude relate to a single hypergeometric series?

Similarly to: Does the Incomplete Beta function have forms of Elliptic E besides $\frac14 \text B_{\sin^2(2x)}\left(\frac12,\frac34\right)=\text E(x,2)$? we have simpler transformations for the ...
  • 7,279
4 votes
1 answer
138 views

Asymptotic of $_3F_2(1, \frac{1}{2}+d+n, 1+c+n; 1+2c, 2+2n;1)$ as $n\to \infty$

Let $c,d$ be in a small neighborhood of $0$; I think the limit $$\begin{aligned}&\quad \lim_{n\to \infty} 4^{-n} n^{3c-d} {_3F_2}(1, \frac{1}{2}+d+n, 1+c+n; 1+2c, 2+2n;1) \\ &= \lim_{n\to \...
  • 17.7k
1 vote
0 answers
17 views

Solving second order ODE - Hermite polynomial and Kummer confluent hypergeometric function

I have the following system of two second order ODEs: \begin{align} (L -D)V+Q^{-1}g(x)=0 \end{align} where \begin{align} L = \frac{1}{2} \sigma_{x}^{2} I \partial_{x}^{2}+(A - a_1 x) I \partial_{x} \...
  • 183
0 votes
1 answer
49 views

$\sum_{i=1}^{n} \frac {x^{2i-1}}{\sqrt{2i}}$ as polylogarithm

$$\sum_{i=1}^{n} \frac {x^{2i-1}}{\sqrt{2i}}$$ It is very clear for me that it has to be polylogarithm function but as it is partial sum I tried to split the series as $$\sum_{i=1}^{\infty} \frac {x^{...
1 vote
1 answer
86 views

How to find a closed form expression for the integral of an exponential divided by a polynomial over an infinite range?

I am trying to find a general solution of integrals with the following form $$I= \int_{-\infty}^{\infty}\frac{e^{-(x-\mu)^2/\sigma^2}}{(1+a x^2)^b} dx$$ where $\{a,b,\mu, \sigma\}\in\mathbb{R}$, $a,b,\...
  • 107
1 vote
1 answer
64 views

How to simplify RMM limit of integral?

This was posted a while back as an analysis problem from Romanian Mathematical Magazine. It is prove that $\lim _{n\to \infty }\frac{4^n}{\sqrt{n}}\int _0^1\frac{\sqrt{x^2+\frac{1}{e}}}{\left(x^{\frac{...
0 votes
0 answers
30 views

Proving that a Fuchsian differential equation has solutions in terms of the hypergeometric function.

I am trying to prove that some kind of fuchsian differential equations have solutions on the form of trygonometric functions. What we know From the book,"Linear Differential Equations and Group ...
  • 53
3 votes
0 answers
154 views

$\sum\limits_{n=1}^\infty\frac{(-1)^\frac{(5+\sqrt3i)n}6x^n}{n!}(\frac ns )^{(n-1)}\,_2\text F_1\left(1-n,-sn;2-sn;s\right)$ closed form or integral.

From: Completing the explicit $\lim\limits_{c\to0}\text I^{-1}_{cx}(r,c)$/ inverse $\int_0^x\frac{t^{r-1}}{1-t}dt,r\in\Bbb Q$ series expansion here is a Lagrange inversion expansion with $r=\frac23$ ...
  • 7,279
1 vote
0 answers
30 views

Solutions of 2nd order Fuchsian differential equations as Heun or Hypergeometric functions.

I have been trying to find a solution on any of these Fuchsian differential equations... $v''(z)+\frac{A_1+B_1z }{z (z-1) }v'(z)+\frac{C_1+ D_1 z}{z^2 (z-1)^2 }v(z)=0$ $u''(z)+\frac{A_0+B_0 z^2 }{z(z-...
  • 53
3 votes
1 answer
93 views

Laplace transforms of Appell functions F1, F2, F3, F4 relative to one of the two variables

Appell's two-variable functions $F_1, F_2, F_3$ and $F_4$ are known to have numerous uses in applied mathematics, notably mathematical Physics. I am looking for generalized Laplace transforms (if they ...
2 votes
0 answers
44 views

Generalization of Hermite polynomials for negative, fractional degree.

After solving a differential equation I've gotten a solution given as a linear combination of Hermite polynomials and confluent hypergeometric functions. The caveat is that the Hermite polynomials ...
  • 31
0 votes
0 answers
45 views

Does this second order linear ODE admits a solution?

I am considering the following ODE in $x$: $$x^2v''(x) + \alpha x v'(x) + \beta v(x)=0, $$ where $\alpha,\beta \in\mathbb{R}$ and one of the boundary conditions says $v(0)=\gamma>0$. If I consider ...
1 vote
0 answers
35 views

Can one express hypergeometric function with 1 as the first and -1 as the last argument by some factorials or gamma functions [closed]

As a result of integration in Mathematica I got: \begin{equation} _{2}F_{1}(1,\frac{1}{2}(2+n-r),\frac{1}{2}(2-n-r),-1)\quad n,r\in\mathbf{N} \end{equation} I try to express it as some factorials (as ...
1 vote
0 answers
38 views

Integral representation of ${}_3 F_2$ allowing negative integers

There exist many integral representations of the generalised hypergeometric function ${}_3 F_2$ but these are all assuming that the entries are positive integers. I have a problem involving, $${}_3F_2 ...
  • 1,676
2 votes
1 answer
101 views

Quotient of generalised hypergeometric series ${}_3 F_2$

I have a function, defined for $x>0$ as $$f(x) = (1+x)^3 \cdot \frac{{}_3F_2\left(\left\{1,\frac{1}{2}+\frac{1}{2x},\frac{1}{2x}\right\},\left\{1+\frac{1}{2x},\frac{3}{2}+\frac{1}{x}\right\},1\...
  • 82
1 vote
1 answer
60 views

Evaluate $\sum^{\infty}_{n=0} (-1)^n \frac{z^{4n+1}}{(4n+1)!}$

I want to evaluate $\sum^{\infty}_{n=0} (-1)^n \frac{z^{4n+1}}{(4n+1)!}$ but while doing my research, I noticed that \begin{align*} &\sin(z) = \sum^{\infty}_{n=1} (-1)^n \frac{z^{2n+1}}{(2n+1)!} \\...
  • 1,596
1 vote
2 answers
69 views

Ideas/Methods for solving an non linear DE with non constant coefficients

I have to solve the following equation. $2 A x^2 u''(x)-2 A x u''(x)+2 A x u'(x)-A u'(x)+B u(x)+\frac{C u(x)}{x}$ I have been thinking in a way of solving this since it is pretty complicated. I tried ...
  • 53
7 votes
1 answer
167 views

Transformation Identities of the $_2F_1$ function

From Wolfram Functions we have the following identities for the hypergeometric function $_2F_1$: $$\begin{align} _2F_1\left(a,c-b;c;\tfrac{z}{z-1}\right)&=(1-z)^a\,_2F_1(a,b;c;z)\tag1\\ _2F_1\left(...
  • 15.9k
1 vote
1 answer
50 views

Solution of a hypergeometric equation

I am thinking the differential equation $(1+e^x)\frac{d^2 y}{dx^2}+(2+e^x-\delta)\frac{dy}{dx}+y=0$. Here $\delta$ is a parameter I want it to approaches to zero. I think this can be transformed into ...
3 votes
1 answer
86 views

A method for approximating a hypergeometric function to an elementary function

I found the following hypergeometric function as the antiderivative of some function in the interval where x is between 0 and 1. ${}_2F_1(-\frac{1}{k}, \frac{1}{k}; 1 + \frac{1}{k}; x^k)x$ Or by ...
3 votes
1 answer
45 views

Closed form for integer moments of Poisson random variable?

I'm interested in a closed-form (special functions included) for the integer moments of the Poisson distribution. Let $X\sim\operatorname{Poisson}(\lambda)$. Then we have for the moment generating ...

1
2 3 4 5
26