Hypergeometric functions are the analytic functions defined by Taylor expansions of the shape $\sum_{n \geq 0} a_n x^n$, where $a_{n+1}/a_n$ is a rational function of $n$. This general family of functions encompasses many classical functions. The hypergeometric functions play an important role in many parts of mathematics.
Questions tagged [hypergeometric-functions]
269 questions
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Is there a way to express hypergeometric identities in terms of D-modules?
A hypergeometric function is a solution to
(*) w'' + p(z) w' + q(z) w = 0
where q has at most simple poles and q has at most double poles at 0,1,infty.
That differential equation is equivalent to the data of a flat connection D = D(p,q) on the…
ya-tayr
- 824
5
votes
1 answer
Series solution of the trinomial equation
The roots of trinomial equations $x^p+x-q=0$ ($p\in\mathbb{N}$) can be expressed in terms of the hypergeometric functions. I am wondering if at least one real root, for instance given by the following iterative solution $x_n=\frac{q - (1 - p)…
yarchik
- 482
5
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1 answer
Can you get the following asymptotic expression for an hypergeometric function?
I conjecture that the following statement holds for large values of $N$
$$
{}_3F_1\left(-N+1,1,1;2;-\frac{1}{N}\right)\to\frac{1}{2}\bigg({}_2F_1(1,1;2;1-\frac{1}{N})+\log 2+\gamma\bigg)
$$
where $\gamma$ is the Euler-Mascheroni constant. PLugging…
PhoenixPerson
- 247
4
votes
2 answers
Product of Hypergeometric Functions
I am looking for the product of Gaussian hypergeometric functions of the form
$_2F_1(a,b;c;\lambda z) _2F_1(d,e;f;\phi z)$. I would like, but do not expect, an answer of the form $_pF_q(\mathbf{a},\mathbf{b};(\lambda*\phi) z)$, where $\mathbf{a}$…
Joe Lucke
- 51
4
votes
0 answers
Upper bounds on hypergeometric function $_3F_2$ for stipulated parameters
Are there any existing bounds on the hypergeometric function $_3F_2(-m,-n,1/2;1,1;4)$ where $m,n \geq 1$? Or any thoughts how one can be obtained?
Most of the inequalities I've seen apply to the cases whose acceptable parameter ranges are…
Derek
- 41
4
votes
1 answer
proof that the schwarz map defined as ratios of gauss hypergeometric functions is univalent
The ratio of two linearly independent solutions of the Guass hypergeometric differential equation defines a map from the upper half plane to a Schwarz triangle. Everything I read tells me that this map is injective, but I cannot find a proof. Is…
Dan
- 41
3
votes
2 answers
Logarithm of a hypergeometric series
I am sorry if the answer to my question is well-known. I am quite new in this topic, so it will be also nice to have a reference, if it exists.
I was wondered if there exists a nice closed formula for a logarithm of an arbitrary hypergeometric…
user79456
- 401
3
votes
0 answers
New $_2F_1$ identity?
The following identity arises in (a new) derivation of the distribution of the estimate $r$ of the binormal correlation coefficient $\rho$. Here formulated in terms of $x = r\rho$.
For $-1 < x < 1$, we have:
$\qquad{}_2\mathrm{F}_1\big(a,b ;…
japalmer
- 141
3
votes
1 answer
Proving Clausen hypergeometric identity
How do I show the Clausen identity
$$
{}_2F_1\left(a, b; a+b+\frac{1}{2}; z\right)^2 = {}_3F_2\left(2a, 2b, a+b; a+b+\frac{1}{2}, 2a+2b, z\right)?
$$
I saw this on MathWorld but am unsure how to progress. I tried the Cauchy product but that was no…
user161698
3
votes
1 answer
Asymptotic formula for Gauss hypergeometric function
My problem refers to the asymptotic formula for Gauss hypergeometric function $F(n, b; 2n; z)$, where $n$ is a fixed positive integer, $z$ is a fixed positive real number less than unity, and the large complex parameter is $b$, $|b|\gg1$.…
Velk
- 33
2
votes
1 answer
hypergeometric closed form for z=1/4,-1/3
There exist the linear identities for the 2f1 hypergeometric function where z is either -1, 1, or 1/2
using the quadratic transdormations it is easy to derive new identities in terms of gamma functions for z = -1/8,8/9 and 1/9
I have seen some…
mathers
- 21
2
votes
0 answers
A minimizing problem involving Gauss hypergeometric functions
Recently I am considering a geometric question, which is reduced to the following problem.
Given $L<0$, let $a\in [L/2,0]$ and $b=L-a$. For any $c>0$, let $p,1-p$ solve
$$x^2-x+c^2=0,$$
and $q,1-q$ solve
$$x^2-x-c^2=0.$$
Using Gauss hypergeometric…
Changwei Xiong
- 87
- 6
2
votes
1 answer
How to calculate the Lauricella function of type A by using matlab?
Does anyone know how to calculate the Lauricella hypergeometric function of type A with multiple variables by using Matlab?
I saw in a paper that it's a function that can be computed directly by using a software supplied by Exton (2007). But I…
2
votes
1 answer
Partial sum of Hypergeometric2F1 function
Can we get a closed form for the following series:
$$\sum\limits_{x=1}^c \dfrac{(x+c-1)!}{x!} {}_2F_1(x+c,x,x+1,z)$$
where $c$ is a positive integer and $z$ is a real number less than -1?
Any suggestions or hints are appreciated.
user43753
2
votes
2 answers
Hypergeometric sum 3F2 at 1
Is there a closed-form sum for the hypergeometric series $_3F_2(a+c, c+d, 1; c+1, a+b+c+d \mid 1)$ where $a, b, c, d$ are all positive and not necessarily integers?
Update: The motivation for this question comes from equation 4 of this tech report…
John D. Cook
- 5,147