Questions tagged [hypergeometric-functions]
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.
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Sturm Liouville differential equation and hypergeometric functions
I'm trying to understand how to solve this differential equation:
$ [z^2(1-z)\dfrac{d^2}{dz} - z^2 \dfrac{d}{dz} - \lambda] f(z) = 0 $
I know the solution is related to the hypergeometric function ...
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Yet another real-rooted polynomial
In this entry I asked for the real-rootedness of a polynomial, and two very interesting answers were given: one using Malo's theorem and the other a clever rewriting of the expression using Jacobi ...
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Gegenbauer's addition theorem for Jacobi polynomials
I have the following identity,
$$\int_{-1}^{1} \! dz \, j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) \, P_n(z)
= 2 \, j_n(x) \, j_n(y) \;,$$
where $x, y > 0$, $P_n$ is a Legendre polynomial, and $...
user114668
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Can the integral $ \int_0^R\quad J_{m-n}(a r)J_m(b r) dr$ be explicitly represented in a closed form?
Doe the following definite integral have an explicit representation in terms of a Bessel functions or a generalized hypergeometric function ${}_pF_q$?
$$
\int_0^R\quad J_{m-n}(a r)J_m(b r) dr, \quad \...
user54423
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answer
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An integral of $|\sin(x)\cos(nx)^{-2/n}|$ from $-\pi$ to $\pi$
For an integer $n \geq 3$, define
$$A_n = \int\limits_{-\pi}^\pi\frac{|\sin(x)|}{|\cos(nx)|^{2/n}}dx.$$
It is a fact that $A_n$ is finite for all such $n$. I am interested in the behaviour of a ...
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Asymptotic approximations or upper bounds for ${}_{2}F_{1}(x+1,x+1,1,z)$ when $x \gg 1$?
I have recently encountered the hypergeometric function
$$
{}_{2}F_{1}(x+1,x+1,1,z),
$$
where $x$ is an integer and $z$ is a real number with $x \ge 1$ and $0<z<1/2$. This is the first time I ...
user54423
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Estimation of Hypergeometric function ${_3F_2}$ [closed]
Is there any way to estimate the following function, which is a result of sum of ratios of Gamma functions?
$$
{_3F_2}\begingroup
\renewcommand*{\arraystretch}
% your pmatrix expression
\left[
\begin{...
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Reflection formula for the ${}_2\!F_1$ hypergeometric function of a matrix argument
According to my implementation of the hypergeometric function of a matrix argument, the so-called "Reflection formula" for ${}_2\!F_1$ given on DMLF (formula 35.7.8) is not true.
On the Wikipedia ...
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Why do we define the hypergeometric function of a matrix argument for symmetric matrices only?
The hypergeometric function of a matrix argument has form
$$
{}_pF_q^{(\alpha)}(a_1, \ldots, a_p; b_1, \ldots, b_q;X) =
\sum_{k=0}^\infty\sum_{\kappa\vdash k} c^{(\alpha)}_{a,b} J^{(\alpha)}_\kappa(X)...
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Does the hypergeometric function of a matrix argument depend on $\alpha$ for a $1\times 1$ matrix?
I already posted this question on maths.SE but got no answer.
The hypergeometric function of a matrix argument has form
$$
{}_pF_q^{(\alpha)}(a_1, \ldots, a_p; b_1, \ldots, b_q;X) =
\sum_{k=0}^\...
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Evaluation of hypergeometric type continued fraction
Is there a (possibly hypergeometric-type) explicit evaluation of the
continued fraction
$$a-\dfrac{1.(c+d)}{2a-\dfrac{2.(2c+d)}{3a-\dfrac{3.(3c+d)}{4a-\ddots}}}$$
Even the special case $d=0$, $a=1$ ...
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Evaluate Gaussian Hypergeometric Function $_2F_1(1;1;c;z)$?
I need to evaluate the Gaussian hypergeometric function $_2F_1(a;b;c;z)$ for the inputs $a=1,b=1,c\in \left\{\frac{n}{2} : n \in \mathbb{N} \setminus \{0,1\}\right\}$, and $z \in [0,1)\subset \mathbb{...
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A hypergeometric identity related to Bessel functions
The identity in my recent answer can be stated in a particularly neat form:
$${}_2F_0\left({-n, n+1\atop{}};\frac{x}{2}\right) ~\cdot~ {}_2F_0\left({-n, n+1\atop{}};-\frac{x}{2}\right) ~=~ {}_3F_0\...
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Hypergeometric representation of Eisenstein series
It is well known (Fricke ?) that $E_4^{1/4}$ and $E_6^{1/6}$ can be represented as Gauss hypergeometric functions of $1728/j$ and $1728/(1728-j)$ respectively.
The same result is true in levels $2$, $...
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Hypergeometric equation in a particular case
I have a question to make in relation to the solution of the hypergeometric differential equation. Let us consider the aforesaid equation,
\begin{equation}
y(1-y)h'' + [c-(1+a+b)y]h' -abh=0,
\end{...
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question about equality series containing hypergeometric term and a simple term
I need a help about the following:
Maple gave that the following equality is true for n =1,2,3,4,5, $$ \sum_{h=0}^{\infty}\binom{n+h}{n}{_3}F_2\left( \substack{-h,n+1,n+1\\ 1,1}; x\right)= \frac{1}{x^{...
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A special solution to the Hermite Differential Equation
I know that the general form solution to the Hermite differential equation
$$ y''-2xy'+2\lambda y=0$$
is
$$y(x)=a_1 M(-\frac{\lambda}{2},\frac{1}{2},x^2)+a_2 H(\lambda,x),$$
where $M(\cdot,\cdot,\cdot)...
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"Find a representation [using Mellin transform] of 𝑓(𝜇,𝛽) as Gauss hypergeometric function in variable 𝜇"
This is a follow-up to the first comment (by Nemo) to the posting Compute the two-fold partial integral, where the three-fold full integral is known . (I also just asked this as a comment to that ...
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Asymptotic expansion of hypergeometric function near $z=1$
Given the hypergeometric function $_2F_1[a,b,c,z]$
in the interval $z\in(1,\infty)$. What is the proper asymptotic expansion of the aforesaid function near $z=1$, when one is approaching from $z>1$....
- 21
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Perform a univariate integral, involving a Gauss hypergeometric function
This is a follow-up question to the one posed in Compute the two-fold partial integral, where the three-fold full integral is known . (I hope that doing so is viewed as a legitimate step. If not so, I ...
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votes
4
answers
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Compute the two-fold partial integral, where the three-fold full integral is known
I have the following trivariate ($\rho_{11}, \rho_{22}, \mu$) function
\begin{equation}
4 \mu ^{3 \beta +1} \rho_{11}^{3 \beta +1} \left(-\rho_{11}-\rho_{22}+1\right){}^{3 \beta +1}
\rho_{22}^{3 \...
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What partial sum formulae exist for this basic hypergeometric series?
I've run into:
$$\sum_{x=1}^{\infty} {x^a\over 1-q^{x}}, \ s.t.\ q\in \mathbb N>1 \ or \ q\in (0, 1),\ a \in \mathbb N$$
I am interested mostly in the cases where $a = 1$ or $ a = 2$
Things I'...
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Correction terms in the asymptotic expansion of hypergeometric function
I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of $\rho$ below),
$$\rho=\frac{2b}{1-q}\left(1-\left(\frac br\right)^{1-q}\right)^{1/2}\left(_2F_1\left(\frac{1}...
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On the convergence of MIller's Algorithm for special function evaluation (hypergeometric 1F1)
This is going to a longish question, so the short version first: Is there a way to sanity-check which solution to the 3-term recurrence relation an application of Miller's algorithm has converged on?
...
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Asymptotic expansion of an integral involving hypergeometric function
I need to consider
$$
\int_0^\infty d\tau \ \ {}_2F_1\left(\Delta, \Delta, 2\,\Delta, -A \cosh^2\left(\frac{\tau}{2}\right)
\right),\qquad A>0,\ \Delta>0
$$
and I am interested in the asymptotic ...
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Euclidean volumes under the matrix norm of one-parameter subsets of unit balls of $2 \times 2$ matrices
Prove that the Hilbert-Schmidt volume (normalized to equal 1 at $\varepsilon=1$) of the subset of the unit ball in operator norm (http://mathworld.wolfram.com/OperatorNorm.html) of the $2\times 2$ ...
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Conjectural nonvanishing of some combinatorial sums (6j symbols)
From various considerations and with the help of J. Van der Jeugt, I was led to conjecture the following property of a class of Wigner 6j-symbols:
for any integers $k,m$ with $m\ge k\ge 2$,
$$
\left\{...
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best-possible inequalities for hypergeometric functions
In what follows, let $n$ be a positive integer and $0<a<1/2$. I am interested in the Gauss hypergeometric functions, $_{2}F_{1}( -n, -n-a; 1-a; z)$. Notice that these are polynomials, if that is ...
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Integration of hypergeometric product for legendre polynomials
I'm looking for a general solution to the integral:
$\int_{0}^1 {_2}F_1(m-n,m+n+1,m+1;z){_2}F_1(m-k+1,m+k+2,m+2;z) dz$
where $m,n,k\in \mathbb{N}$ and $m\leqslant n$ and $m+1 \leqslant k$.
To give ...
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lower bound for absolute value of a hypergeometric function
I am working with a certain Gauss hypergeometric function, $_{2}F_{1}(a,a-b;2a;1-z)$, where $a, b \in {\mathbb R}$ with $a, a-b > 0$ and $0<b<1$.
It appears that $\left| _{2}F_{1}(a,a-b;2a;1-...
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Uniform Asymptotic Approximation of the Whittaker function
I would like to know if there exist a uniform asymptotic approximation of the Whittaker function $W_{\kappa,i\mu}(x)$ for $\kappa<0$, $x >0$, and with $\mu \to +\infty$. The case of $\kappa \ge ...
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Limit of a hypergeometric function(1F2)
I don't have experience with hypergeoemtric functions, but wish to compute the following limit:
$\lim_{x→\infty}{F([1],[a,b];-\frac{x^2}{4})}$, where $a,b$ are non-integer real parameters.
I tried ...
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Understanding the branch cut and discontinuity of the hypergeometric function
DISCLAIMER: This question comes from math.stackexchange (where it has an active bounty). The link is here.
UPDATE: the question has been answered on math.stackexchange at the previous link, and the ...
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1
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Value of the hypergeometric function
Let $n$, $m$ and $k$ be some (positive) integers such that $(k+3/2)-(n+m/2)<0$. Can the hypergeometric function
$$F\left (n+\frac{m}{2},n+\frac{m+1}{2};k+\frac{3}{2};-\tan^2{\phi}\right) \tag{1}$$
...
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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,...
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How to use this generalised 'generating function' for the Gegenbauer polynomials
Cohl (2011) gives a generalisation of the standard generating function for the Gegenbauer polynomials $C_n^\mu(x)$:
$(1 -2tx + t^2)^{-\nu} = A_{\mu,\nu} \frac{(1-t^2)^{-\nu+\mu+1/2}}{t^{\mu+1/2}} \...
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Compute Confluent Hypergeometric Function 1F1
I am attempting to compute the (Kummer's) confluent hypergeometric function (see also here)
\begin{align}
M\left(\frac{n}{2}, n +\frac{3}{2}, -z\right) = {}_1F_1\left(\frac{n}{2};n +\frac{3}{2};-z\...
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Polya urn: Mean number of draws to get a specific sequence of colors?
Polya urn model: At time $0$ an urn initially contains $b$ $\tt{B}$lue balls and $r$ $\tt{R}$ed balls. At time $1$, a ball is drawn uniformly at random (removing it) from the urn, and two balls of the ...
- 240
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integral involving hypergeometric function of matrix argument
This conjecture comes from an observation on simulations of the matrix variate noncentral Beta distribution (similar to this observation, but I open a new question because yet I'm not sure it is ...
- 2,433
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1
answer
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Basis for solutions hypergeometric differential equation
In the book "Theorie der gewöhnlichen Differentialgleichungen" by Bieberbach, page 240, there is a solution to the hypergeometric differential equation
$z(z-1)w^{\prime \prime}+(2z-1)w^{\prime}+\frac{...
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0
answers
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Does the Riemann characterization of the hypergeometric function have a q-analog?
This question is inspired by another recent one here, Characterization of the hypergeometric function. The latter is about the classical result of Riemann characterizing the hypergeometric functions ...
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0
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Is there a name for the hypergeometric function with more parameters than Lauricella but more variables than Kampé de Fériet?
Some background:
The Appell functions generalise the hypergeometric function ${}_2F_1$ to two variables.
The Lauricella functions generalise this to even more variables.
The Kampé de Fériet ...
- 301
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votes
0
answers
222
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Generalized hypergeometric function at $z=1$
I wonder if there is a closed formula for the following generalized hypergeometric function at $z=1$:
$${}_4F_3\left(a,a,a,a;a+1,a+1,2a;1\right)$$
Specifically, I would like to have a formula in ...
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votes
1
answer
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Efficient numeric routines for computing $_2F_0$
As part of a project related to kinematic fluid dynamics, the following integral appeared in the moment expansion
$$
\int_0^{\alpha} t^m\,_2F_0\left(\begin{matrix}-\ell-\frac{1}{2},m+2\\-\end{matrix};\...
- 249
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1
answer
193
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What is the name of this special function?
I have asked this question on a different forum. I am asking it here as well in order to increase the number of different people who see it.
Consider a special function defined as:
$$f(a_1,a_2,a_3;...
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votes
0
answers
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Characterization of the hypergeometric function
One of the definition of the hypergeometric function $_2 F_1$ rely only on its global properties around the singularities (and not on a differential equation or a serie expansion)
In modern language (...
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Multivalued functions with three independent branches
Let $n$ be a positive integer and $f: \mathbb{C} \rightarrow \mathbb{C}$ be a multivalued function, analytic everywhere except for branch points at $0$, $1$ and $\infty$. Around one of those ...
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Perform an integration involving the product of two hypergeometric functions
I've encountered the following product,
\begin{equation}
\, _2F_1\left(3 d+2,3 d+2;6 d+4;1-\frac{1}{t^2}\right) \,
_3F_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;t^2\right)
\...
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0
answers
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Generalized Lauricella functions?
The Lauricella function $F_D$ in $n$ variables can be essentially written as
$$\int...\int(1-\sum_{r=1}^n x_r)^\delta(1-\sum_{j=1}^nc_jx_j)^{\gamma}\prod_{i=1}^n x_i^{a_i}dx_i$$
up to some gamma ...
- 321
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votes
1
answer
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Perform an integration over the unit interval of a two-parameter expression involving a Gauss hypergeometric function
In a quantum-information-theoretic context, I've encountered the problem
of integrating over $r \in [0,1]$, the function
\begin{equation}
r^{2 d-1} \, _2F_1\left(-\frac{d}{2},\frac{d}{2};\frac{d+2}{2}...
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