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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$\int_0^1 \sqrt{\frac{2-x^4}{1-x^4}}\, dx$ in terms of the gamma function or elliptic integrals
Question
Can
$$\int_0^1 \sqrt{\frac{2-x^4}{1-x^4}}\, dx$$
be expressed in closed form in terms of the gamma function at rational arguments or in closed form in terms of elliptic integrals?
Thoughts
...
- 609
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0
answers
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Show that $\mathcal{L}^{-1}\left\{\frac{\Gamma(n)}{s^n}e^{-\frac{2a}{s}}{}_0F_1\left(n,\frac{a^2}{s^2}\right)\right\}(1)={}_0F_1\left(n,-a\right)^2$
I am trying to show that given $n, a > 0$
$$ G(s) = \frac{\Gamma(n)}{s^n} e^{- \frac{2a}{s}} {}_0F_1 \left( n, \frac{a^2}{s^2} \right) $$
the inverse Laplace transform of $G$ evaluated at $t=1$ is
...
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0
answers
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Is there a closed form for $_3F_2(a, b, c; a+1, 2a+\frac{3}{2}; 1)$? [closed]
I'm trying to evaluate the hypergeometric series given by $_3F_2(a, b, c; a+1, 2a+\frac{3}{2}; 1)$.
Is there a known closed-form expression for this series?
1
vote
2
answers
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Limit of confluent Hypergeometric function [closed]
I have the following confluent Hypergeometric function :
$$
M(A,n+1,nx^2)
$$
Here $0\leq x \leq 1 $. $n$ is an integer and I wish to calculate the asymptotic behavior as $n\rightarrow \infty$. Also $A ...
- 13
2
votes
0
answers
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What is the relationship between the zeroes of contiguous confluent hypergeometric functions?
Confluent hypergeometric functions differing from $F=_1F_1(a,b,z)$ by $\pm1$ in either parameter $a$ or $b$ are called contiguous to $F$. For rational $a, b$, assume I know $z_0$ is a zero of $F$. ...
1
vote
2
answers
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evaluating ${}_1F_1(a;b;x)$ and $\Gamma(x)$ at negative arguments
I am trying to evaluate the following integral:
\begin{equation*}
\int_{\sqrt{\varepsilon}}^{\infty } u^{m+1} \left(u^2-\varepsilon\right)^{j+\frac{k}{2}-1} \exp \left(-\frac{u^2}{2 \varsigma}\right) ...
- 329
0
votes
0
answers
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hypergeometric function on Wikipedia
I am not a mathematician, but I recently encountered the hypergeometric function (for the first time) in a paper about gravitational collapse in General Relativity.
According to Wikipedia here, the ...
- 141
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votes
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answer
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Upper bound on absolute value of Confluent Hypergeometric Function with real argument and parameters.
A hypergeometic E-function is
$$
{}_pF_q\left(\left.\begin{array}{ll}
a_{1}, \ldots, & a_{p} \\
b_{1}, \ldots, & b_{q}
\end{array} \right\rvert\, \lambda z^{q-p}\right)=\sum_{n=0}^{\infty} \...
0
votes
1
answer
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Eigenfunctions of the Fokker-Planck operator for the Rayleigh process
I am studying the eigenfunction method to solve Fokker-Planck equations from Gardiner's book on stochastic processes, section 5.2.5. In particular, I am trying to solve one of the examples proposed in ...
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0
votes
1
answer
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How to tell whether a function is a hypergeometric function knowing its first series coefficients?
Suppose one knows the first coefficients $\alpha_i$ of the series expansion of a function $f(t)$ at point $t=0$
\begin{equation}
f(t) \approx \sum_{i=0}^{n} \alpha_i \cdot t^i
\end{equation}
Question: ...
- 213
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votes
1
answer
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How to solve the integral (antiderivative) of $\sqrt{1+x^3}$?
The antiderivative:
$$
\int{\sqrt{1+x^3}dx}
$$
(the real part, with $x>-1$)
is not an elementary function, using Mathematica it gives me some Hypergeometric function.
How to tackle such indefinite ...
- 61
1
vote
0
answers
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Asymptotics for $f(n+1) = (a n + b) f(n) + c \sum_{i=0}^{n} f(i) f(n-i)$?
I was wondering about
$$f(0) = 1$$
$$f(n+1) = (a n + b) f(n) + c \sum_{i=0}^{n} f(i) f(n-i)$$
for given positive integers $a,b,c$.
How fast does this grow ?
Such equations occur often and are ...
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1
vote
1
answer
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How can I prove $(\star)$? - I nice property of $_2F_1\left(\frac{5}{6},1;\frac{11}{6};-t^6 \right)$
Conjecture (Goal)
Prove or disprove the following:
$$_2F_1\left(\frac{5}{6},1;\frac{11}{6};-t^6 \right) = \frac{5}{t^5}\left(\frac{\pi}{3}- \int_{t}^{\infty} \frac{x^4}{1+x^6} dx \right) \label{conj} ...
- 891
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Calculating a definite improper integral - part 2
This question is regarding my previous question on the site. I want to calculate the following integral
$$\int_0^{\infty} \frac{x^p(1+x)^{-p-1}}{r+sx}dx, p,r,s\in \mathbb{R}^+$$
according to the ...
- 3,271
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votes
0
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Calculating a definite improper integral
I want to calculate an integral of the form
$$\int_0^{\infty} \frac{x^p(1+x)^{-p-1}}{r+sx}dx$$
The answer can be found in terms of Gauss hypergeometric function. However, there this formula (equation ...
- 3,271
2
votes
0
answers
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A recurrence for $f(n) = \sum_{k = 0}^{\infty} \frac{1}{(2k + 1)^{n}\binom{k - 1/2}{k}}$
For an integer $n \geq 2$ consider the absolutely convergent series:
$$f(n) = \sum_{k = 0}^{\infty} \frac{1}{(2k + 1)^{n}\binom{k - 1/2}{k}} = \sum_{k = 0}^{\infty} \frac{k!^2}{(2k)!(2k + 1)^{n}}4^k$$
...
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votes
2
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how to find closed form for $\int_0^{\frac{\pi}{2}} \ln(1+\sqrt{\sin x}) dx$?
I tried to solve the integral
$$I=\int_0^{\frac{\pi}{2}} \ln(1+\sqrt{\sin x}) dx$$
and by series and beta function I got that
$$ I=2\sqrt{\pi}\sum_{k=0}^\infty \frac{(-1)^k}{k+1} \frac{\Gamma\left(\...
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2
votes
2
answers
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On a derivative of Appell's $F_1$ function with respect to a parameter
I'm trying to compute
$$\left.\frac{\partial}{\partial\alpha}F_1(2,\alpha,\alpha;3;x,y)\right|_{\alpha=0},$$
where $F_1$ is Appell's hypergeometric function in two variables. What I tried so far is ...
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votes
1
answer
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Is there a closed form for $\sum _{n=0} ^{\infty} \frac{1}{(2n+1)^3 \binom{n-1/2}{n}}$
I was trying to find the closed form for the series
$$
S = \sum_{n = 0}^{\infty} \frac{1}{\left(2n + 1\right)^{3}\binom{n - 1/2}{n}} \approx 1.12269
$$
Although it is a bit messy, we can represent ...
- 431
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votes
1
answer
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Find coefficients of the inverse of a matrix holding sum of binomial coefficients
For a given $n$, we define the coefficient $\beta_{i}(k)$ with the following (some context here):
\begin{align*}
\beta_{i}(k) = \sum_{u=0}^{k} (-1)^u \cdot {n \choose u} \cdot {n-i \choose k-u}
\end{...
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votes
1
answer
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Series of confluent hypergeometric functions with one extra Pochhamer symbol
I'm trying to solve analytically the integral
$$Z = \int_0^a \exp(-\alpha x^4 - \beta x^2 - \gamma) \mathrm{d}{x}.$$
To do this, I wrote it as
$$Z = e^{-\gamma} \sum_{n=0}^{+\infty} \frac{(-1)^n \beta^...
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votes
0
answers
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Ways of finding upper bounds of hypergeometric functions
I realized that I don't really know any good ways of finding bounds of hypergeometric functions after ${}_2 F_1$. For example, numerical evaluation convinced me that the generalized hypergeometric ...
- 431
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0
answers
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how to find closed form for ${}_3F_2(-1)$ function identity?
I tried to prove this identity I got it from wolfram functions site
$$\Omega_a={}_3F_2(1,1,a;2,3-a;-1)=\frac{2-a}{2(1-a)} H_{1-a} $$
firstly from its series I got that
$$ \Omega_a=\frac{\Gamma(3-a)}{\...
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Converting a 2nd order homogeneous equation into a known form of an Special function.
I got the following 2nd order fuchsian equation.
$z^2 u''(z)+z u'(z)+\left(\frac{A \beta }{(z-1)^2}+\frac{A (\alpha +\beta )}{z-1}-A
E\right)u(z)=0$
I am pretty impressed by how close it looks to ...
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1
answer
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Asymptotics of hyperrgeometric 2F1 for large integer parameters
I have stumbled upon $$\frac{\Gamma(n+\frac{1}{2})\Gamma(n-\frac{1}{2})}{\Gamma(2n+1)} \ {}_2F_1\left(n+\frac{1}{2}, \ n-\frac{1}{2}; \ 2n+1, \ z\right)$$
Where the integer $n \to \infty$. I need to ...
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vote
1
answer
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Card Game Opening Hand Calculator with subgroups
So I'm trying to create a Yugioh Card Game opening hand calculator in a google spreadsheet. This can also be seen as drawing balls from an urn without replacement, but every ball has multiple ...
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votes
0
answers
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Looking for an identity to simplify an infinite sum involving hypergeometric 2F1's
From a physics context I have landed into
$$\chi(h,N) = \sum_{q=0}^{\infty} \ (\frac{4h}{(1+h)^2})^{qN} \cdot \frac{ \Gamma(qN+\frac{1}{2})\Gamma(qN-\frac{1}{2})}{2 \pi\Gamma(2qN+1)} \cdot {}...
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vote
0
answers
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Finding the general convolution of probability function with hypergeometric PDFs.
I am trying to find the generalized convolution of this PDF distribution.
$$f(n, \sigma; v)= \frac{2^{\frac{3}{2}-\frac{n}{2}} \sigma ^{-n-1} v^\frac{n - 1}{2} K_{\frac{ n - 1}{2}}\left(\frac{v}{\...
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vote
0
answers
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Regularized pFq to 2F1 for specific values of parameters
I have encountered a regularized generalized hypergeometric function as
$${}_3 \tilde{F}_2[ \{-\frac{1}{2}, \frac{1}{2}, 1 \}, \{ 1 - a, 1 + a \}, z] = \frac{1}{\Gamma(1-a) \Gamma(1+a)} \ {}_3{F}_2[ \{...
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Question of proving Pfaff-Saalschütz Theorem
I may be asking a stupid question. In this website, the proof of this theorem is posted. But in the last step, it uses that
\begin{equation*}
_3F_2
\left[
\begin{array}{cc}
-x,-y,-z \\
n+1,-x-y-z-n
\...
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vote
0
answers
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Is ${}_3 F_2(a,b,c;d,e;x)$ irrational for $ a,c,d,e,x \in \mathbb{Q} / \{0 \} $
My understanding of G-functions is simply nonexistent but I do know that they can assume algebraic values at nonzero rational arguments. But could those assumed values be rational?
Specifically I was ...
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1
vote
2
answers
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Closed form for the recurrence $S_n = 1 + S_{n-1} + \frac{2}{n} S_{n-2}$, where $S_1=1$ and $S_2=2$?
How would you go about getting an expression for $S_n$ where $S_1=1$, $S_2=2$, and $S_n = 1 + S_{n-1} + \frac{2}{n} S_{n-2}$?
I'm using this to try and solve a separate problem which involves the ...
- 75
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votes
1
answer
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Explanation of Mathematica's result for $\frac1\pi\int_0^\omega d\Omega\ \sin(\Omega(1+n\eta))\cot\left(\eta\frac{\Omega}2\right)\eta$ [closed]
I would like to understand the motivation behind the equality (given $n \in \mathbb{Z}$, $\eta > 0$)
$$\frac{1}{\pi} \int_{0}^{\omega} d \Omega \ \sin \left( \Omega (1 + n \eta) \right) \cot \left( ...
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vote
0
answers
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Integral involving a rational function inside a hypergeometric ${}_2F_1$
I want to know if the following integral can be expressed in terms Hypergeometric (or any other rational) functions
\begin{equation}
\int_{0}^1 \int_0^1 dr\, dt\,r^{-1-i p} t^{-1-i p} (1-r)^{i p+2 q-1}...
- 759
1
vote
0
answers
50
views
An integral identity involving a generalized hypergeometric function.
Let $\theta \ge 0$, $S\ge 0$, $\zeta \ge 0$ and $x \ge 0$ be real numbers and let $q \ge 1$ be an integer. Then the following identity below holds true:
\begin{equation}
G_\zeta(x) := \int\limits_0^\...
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0
votes
0
answers
26
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Simplifying a multivariable hypergeometric function
The confluent multivariable Luaricella's hypergeometric function is defined as
$$\Phi^{(n)}_2\left(b_1,\cdots,b_n;c;x_1,\cdots,x_n\right) = \sum_{m_1=0,\cdots,m_n=0}^{\infty}\frac{\left(b_1\right)_{...
- 3,271
1
vote
0
answers
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Lower bound of Hypergeometric function ${}_2F_1(d,1,d+1, z)$ [closed]
I am looking for a (non-trivial) lower bound of the Gauss hypergeometric function ${}_2F_1(d,1,d+1, z)$ where $d\in\mathbb{N}$ and $0\leq z <1$. Ideally, the bound would be valid $\forall d\in\...
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1
vote
0
answers
35
views
Systems of partial differential equations and multiple hypergeometric functions
The following differential equation system show up in my research (it is related to the group $PSL(2,\mathbb{R})$):
$$x_1(1-x_1)\partial_1^2f+x_2(1-x_2)\partial_2^2f+(x_1+x_2+x_3-2 x_1 x_2 -1)\...
- 111
1
vote
0
answers
40
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Mellin transform of confluent Lauricella hypergeometric function
The $F_D^{(n)} $ Lauricella's hypergeometric function can be defined as follow
$$F_D\left(a,b_1,\cdots,b_n;c;x_1,\cdots,x_n\right) = \sum_{m_1=0,\cdots,m_n=0}^{\infty}\frac{\left(a\right)_{m_1+\cdots+...
- 3,271
8
votes
1
answer
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Generating function for the products of pairs of Narayana numbers
The Narayana numbers OEIS sequence A001263 are given by:
$$\operatorname{N}(n, k) = \frac{1}{n} {n \choose k} {n \choose k-1},$$
and have the generating function:
$$G(z,t) = \sum_{n=1}^\infty \sum_{k=...
0
votes
0
answers
62
views
Solving a pair of PDEs by Kampé de Fériet function and other functions
I have a pair of PDEs as follow:
$$\left(1+2x\frac{\partial}{\partial x}+6y\frac{\partial}{\partial y}\right)\left(2+2x\frac{\partial}{\partial x}+6y\frac{\partial}{\partial y}\right)w=\left(2+x\frac{\...
- 107
1
vote
1
answer
94
views
Problem with the transform from ${_3F_1}$ to ${_2F_2}$
I want to transform
\begin{equation}
{_3F_1}(-\frac n2,-\frac{n-1}2,\frac{n+1}2;\frac12;z^{-1})
\end{equation}
into ${_2F_2}$ using equation (2.2.3.2) of the book by Lucy Joan Slater "Generalized ...
- 243
6
votes
2
answers
268
views
Problematic limit $\epsilon \to 0 $ for combination of hypergeometric ${_2}F_2$ functions
In an earlier question, the integral $$I_n(c)=\int_0^\infty x^n (1+x)^n e^{-n c x^2} dx$$ was considered with particular focus on its behavior for positive integer $n$. In trying to analyze this, it ...
- 17.6k
8
votes
3
answers
238
views
How to calculate $\int_{0}^{\infty} x^n(1+x)^n e^{-ncx^2}\text{d}x$
As the title mentioned, I want to calculate
\begin{equation}
\int_{0}^{\infty} x^n(1+x)^n e^{-ncx^2}\text{d}x,
\end{equation}
where $n$ is a positive integer, $c$ is a positive real number in the ...
- 243
0
votes
1
answer
319
views
Evaluating: $ \int_0^{\infty}\left[1+\frac{\alpha}{\kappa}\left(\sqrt{p^2+1}-1\right)\right]^{-\kappa-1} p^2 \,dp $
I try to evaluate the following integral :
$$
\operatorname{I}\left(\kappa,\alpha\right) =
\int_{0}^{\infty}
\left[1 + \frac{\alpha}{\kappa}\left(\sqrt{\,{p^{2} + 1}\,} - 1\right)\right]^{-\kappa - 1}\...
- 243
0
votes
1
answer
91
views
PGF of hypergeometric distribution
The hypergeometric distribution is given by the law: $$P(x)= \frac{\binom{M}{x} \binom{\theta - M}{n - x}}{\binom{\theta}{n}}, \theta, M, n\in\mathbb{N}\cup\{0\}, M\leq\theta, n\leq\theta, x\in\{\max(...
3
votes
1
answer
53
views
Can I use the hypergeometric distribution to calculate draw odds for 2 variables from a single population?
I would like to program a card-drawing calculator the give the odds of there being at least x cards of set A and y cards of set B, where sets A and B are distinct populations in set C of a given ...
- 33
2
votes
0
answers
117
views
How to calculate an upper bound for $\sum_{k=0}^n \binom{n}{k} \frac{(-x)^k}{\sqrt{k+a}}$
As the title mentioned, I want to get a closed-form result of
\begin{equation}
\sum_{k=0}^n \binom{n}{k} \frac{(-x)^k}{\sqrt{k+a}},
\end{equation}
where $x\in[0,1]$ is a real number, and $a$ is a ...
- 243
4
votes
2
answers
156
views
The sum of $\sum_{k=0}^{n} \binom{n}{k} \Gamma\left(\frac{k}{2}+a\right)$
As the title mentioned, I want to have the value of the sum
\begin{equation}
\sum_{k=0}^{n} \binom{n}{k} \Gamma\left(\frac{k}{2}+a\right),
\end{equation}
where $a$ is a positive number, and $\binom{n}{...
- 243
0
votes
1
answer
197
views
Asymptotic of the integral $\int_0^1 x^{k-1}(1-x)^{l-1}(1-ax)^{\frac{m}{2}}\, dx$
Consider the integral
$$
I(n,a) = \int_0^1 x^{k-1}(1-x)^{l-1}(1-ax)^{\frac{m}{2}}\, dx,
$$
where $k,l\in\mathbb{N}$,
$$
k+l=n\in\mathbb{N},\quad 0<a<1,\quad m\in\mathbb{N}\quad\mbox{and}\quad m\...
- 83