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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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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 $z>1$....
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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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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(\...
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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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A two argument sepcial function related to Legendre polynomial and Meixner polynomial

This problem raised when I was trying to evaluate a complicated integral. A polynomial with 2 arguments emerged and I could not recognize it. Let's call it $F_n(k,x)$, what I know is that $F_n(0,x)=...
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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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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 ...
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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 ...
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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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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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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 ...
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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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Asymptotic behavior

What is the asymtotic behavior of the hypergeometric function :$_2F_{1}(a,b,c,z)$ near $|z|=1$. Thanks a lot in advance.
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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};\...
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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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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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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 ...
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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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Linear relation between $_3F_2$ at $z=1$

A colleague of mine found the following linear relation between values at 1 of the $_3F_2$ hypergeometric function using a very indirect argument. I am aware that there are myriads of such relations, ...
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Integral involving Laguerre, Gaussian and modified Bessel function

I am trying to prove that the integral \begin{align} \int_{0}^{\infty } e^{-\frac{r^2}{2B}} r^{l-n} L_n^{l-n}\left(\frac{r^2}{C}\right) I_{l-n}\left(\rho r \right) r dr \end{align} has ...
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Relation between elliptic curves and hypergeometric functions

If we are given an elliptic curve of the form $y^2 = (x-e_1)(x-e_2)(x-e_3) $, where $ e_1 > e_2 > e_3 $ and all $ e_i $ are real, then we can evaluate the period and the dual period, which will ...
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Double Series involving Gamma function

Does anyone have any ideas on howto verify $$\sum_{n,m=0}^\infty \frac{\Gamma(n+m+3x)}{\Gamma(n+1+x)\Gamma(m+1+x)}\cdot \frac{1}{3^{n+m+3x-1}} = \Gamma(x)$$ for $x>0$? I posted this question also ...
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Proving a particular “Abel type” identiy

I have reduced solving this question to proving the following identity, for $n, \ell \ge 0$: $$ (n-2\ell+1)^{n-1} \binom{n}{\ell-1} = \\ \frac{1}{2} \sum_{n_1+n_2=n-1}\left[ (n_1+1)^{n_1-1} (n_2-2\...
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Identity with binomial coefficients and k^k

In process of doing some computations on Hilbert schemes, I stumbled across the following identity, for $k \ge 2$: $$ k^{k-3} = \frac{1}{2} \sum_{i=1}^{k-1} \binom{k-2}{i-1} i^{i-2} (k-i)^{k-i-2} $$ ...
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Integral with Bessel function and hypergeometric function ${}_2F_2$: explicit expression for these polynomials?

This question follows this one, where the general problem has apparently no simpler form than the integral one. I focus now on the limit case: \begin{align} \int_0^T e^{-x}\frac{nI_n(x)}{x}dx=\int_0^T ...
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225 views

Legendre functions as Hypergeometric functions

Is there some approximation or regularization that goes in tacitly in the following equality: $Q_{\lambda }(z) = \frac{\sqrt{\pi } \Gamma (\lambda +1)}{2^{\lambda +1} \Gamma \left(\lambda +\frac{3}{2}...
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Closed form for $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$

EDIT: Some additional details and corrections, I would appreciate any information about the highlighted expression. I try to solve $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$ where $I_n(x)$ is the ...
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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 ...
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Solving a partial difference equation with variable coefficients explicitly

I'm trying to solve the following partial difference equation with variable coefficients: $$a_{i,j,k}=-\frac{j+1}{i}a_{i-1,j+1,k-1}- \frac{k+1}{i}a_{i-1,j-1,k+1}, $$ defined on the grid $(i,j,k) \in \...
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On the Integration and Manipulation of Expressions Involving Hypergeometric Functions

I would like to ask the following two: For the integral: \begin{equation} \int_{0}^{t}\left( 1-s^p \right)^{\frac{1-p}{p}}ds \end{equation} I know that it is reduced to the following product ...
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287 views

On modified Bessel solutions to complex ODE's using Kummer's series

I am trying to reduce the following ODE to Bessel's ODE form and hence solve it: $$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .$$ I tried to solve it via the standard method, i.e.,...
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Is a basic hypergeometric function ${}_2\phi_1(a, b; c; q, z)$ a meromorphic function in $z$?

Here a basic hypergeometric function is the analytic continuation of the basic hypergeometric series (or called the $q$-hypergeometric series) $$ {}_2\phi_1(a, b; c; q, z) = \sum^{\infty}_{n = 0} \...
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128 views

Relation between different basis of solutions of an ODE

Take a linear ordinary differential equation of the form : $$ \sum_{k=0}^n p_{n-k}(z)(z (z-1))^k \partial_k f = 0 $$ Where $p_i$ is a polynomial fraction of degree $i$, without zeros at $0$ or $1$, $...
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Sum of squared hypergeometric polynomials

$\sum_{m=1}^\infty \frac{1}{m} \bigg[{}_2F_1(-m,m,2,u)\bigg]^2 = \frac 1 4 -\frac 1 2 \log u$ I have very strong numerical support that this is true when $0<u\le1$. Can anyone help proving or ...