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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39 views

How can I show that this product is equal to a product of Gamma functions? [migrated]

$\prod_{n=0}^{x-1}\left( 1+\frac{a}{an+b}\right) = \frac{\Gamma\left(\frac{a}{b}\right)\Gamma\left(x+\frac{a+b}{b}\right)}{\Gamma\left(\frac{a+b}{b}\right)\Gamma\left(x+\frac{a}{b}\right)}$ I found ...
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117 views

A recurrence formula for the Legendre function $P_\mu^\nu(x)$

Im looking for a recurrence formula of type: $$(\mu-\nu) x P_\mu^\nu(x) + P_{\mu-1}^\nu(x)=?, \quad \mu,\nu\in \mathbb R$$ where $P_\mu^\nu(x)$ is the Legendre function of the first kind (solution to ...
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Proving that a quotient of hypergeometric functions is smaller than a certain function

Im trying to prove that $\forall w \in (0,1), \forall k \in \left(0,\frac{1}{5}\right)$: $$h_k(w) = \left[\frac{_2F_1\left(\frac{3}{2},1+\frac{1}{k};\frac{1}{2}+\frac{1}{k};\frac{1-w}{1+w} \right) }{...
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1answer
84 views

Integral expressions for Bessel-like power series

I'm interested in power series of form $$f(z)=\sum_{k=0}^\infty \frac{z^k}{(k!)^\alpha}.$$ When $\alpha=1$, this becomes $\exp(z)$. For $\alpha=2$ this is a Bessel function and for larger integer $\...
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60 views

Asymptotic expansion of hypergeometric 2F2

I would like to find an asymptotic expansion for the hypergeometric function $$ _{2}F_{2}\left(a,b;c,d;z\right),\quad a,b,c,d\in\mathbb{R}. $$ The parameters are fixed. $z$ is real and $z\rightarrow ...
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109 views

Trying to bound the generalized hypergeometric function ${}_2F_3(x+1,x+1;1,1,1;\alpha)$ as $x\to \infty$?

(See also edit below)... I am trying to get a nice, explicit, bound on the hypergeometric function $$ {}_2F_3(a_1,a_2;b_1,b_2,b_3;\alpha), $$ in the case of a large parameter. In particular I am ...
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108 views

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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1answer
344 views

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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200 views

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 $...
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79 views

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 \...
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183 views

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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71 views

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 ...
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129 views

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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Series expansion for Gaussian hypergeometric function valid for branch points?

I need to use series expansions at $ r = 0 $ and $ r \to \infty$ for the function $ r \mapsto {}_2 F_1\left(a,a+2;2a, r^2 +1\right) $ with $ r \in (0, \infty) $ and where $ a \in \mathbb{C},~ Re(a) &...
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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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160 views

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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272 views

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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125 views

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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119 views

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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130 views

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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318 views

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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106 views

“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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2answers
124 views

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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79 views

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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526 views

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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259 views

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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168 views

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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1answer
124 views

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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98 views

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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110 views

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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301 views

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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1answer
173 views

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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1answer
314 views

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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89 views

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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1answer
370 views

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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346 views

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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1answer
239 views

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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56 views

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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117 views

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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1answer
292 views

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\...
4
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1answer
297 views

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 ...
3
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1answer
103 views

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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1answer
116 views

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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86 views

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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119 views

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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102 views

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 ...