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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25 views

The infinite product and the Polya-Laguerre class

Is it possible to write the following equations? $$_{2} F_{1}(a+b,c;z)=e^{\gamma z} \prod_{k=1}^{+ \infty}(1+\alpha_{k}z) \mbox{ where } \alpha_{k},\gamma \geq 0 \mbox{ and } \sum \alpha_{k} \mbox{ is ...
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Are hypergeometric series not taught often at universities nowadays, and if so, why?

Recently, I've become more and more interested in hypergeometric series. One of the things that struck me was how it provides a unified framework for many simpler functions. For instance, we have $$ \...
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Distribution of iid hypergeometric random variables conditioned on the sum

Let $X_1,X_2,\ldots,X_n$ be iid random variables with hypergeometric distribution. To be specific, $$ \mathrm{Prob}(X_1=i) = \frac{\binom{N}{i}\binom{M-N}{m-i}}{\binom{M}{m}}.$$ Let $S=X_1+\cdots+X_n$....
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Hypergeometric function with changed argument [closed]

I have the hypergeometric function $_2F_1 (a, b,c, p\cdot z)$, where $p$ is a parameter and $z$ is the independent variable. I would like to know how the former function is related to the standard ...
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When exactly is the principal AGM equal to the optimal AGM?

Definitions Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
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41 views

Solving a differential equation related to the hypergeometric differential equation

I need to solve the following equation: $x*(1 - s*x) y''[x] + y'[x] + r*y[x] =0,$ where $s$ and $r$ are two parameters. It would seem that is similar to the hypergeometric differential equation, but ...
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Applying Tannery's theorem to generalised hypergeometric functions

I am thinking about applying Tannery's theorem to some generalised hypergeometric functions, which seems to be a standard method to derive various formulæ. For example, \begin{eqnarray} \lim_{n\to+\...
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Totally rational hypergeometric evaluations

This is a followup to the question at which rational points does the Hypergeometric function take rational values asked 10 years ago by Eugene Starling, and is more a challenge than a question. Let $F(...
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High-order associated Legendre polynomials

In a similar spirit (yet not the same) as in the question posted here, I am interested in finding the asymptotic expression for $$P_{a+ib}^{-c}(x)$$ for $c\sim b\gg a$ and for finite $x$. I would be ...
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Log of the confluent hypergeometric function of the second kind

Let $U$ be the confluent hypergoemetric function of the second kind, as denoted in this wikipedia page. For given $a,b$ positives reals, i'm looking for a function $f_{a,b}$ such that, for all $t$ ...
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Higher-order asymptotics of generalized hypergeometric function

I have a question about higher-order asymptotics of generalized hypergeometric functions. According to https://dlmf.nist.gov/15.4 the following is well known: $$ _2F_1(a,b;a+b;z)\sim -\frac{\Gamma(a+b)...
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References for generalized confluent hypergeometric differential equation

According to Wolfram, a generalization of the confluent hypergeometric differential equation is given by: $$y''+\left(\frac{2R}{x}+2F'+p\frac{H'}{H}-H'-\frac{H''}{H'}\right) y'+\left[\left(p\frac{H'}{...
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Two questions on asymptotic expansion of confluent hypergeometric functions for real variable $x, |x| \to \infty$

I'm looking into the asymptotic expansion for confluent hypergeometric function $_1F_1(a;b;z) \equiv M(a;b;z)$ and I've two quick questions regarding its asymptotic behavior for real values $x,$ i.e. ...
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On expressions of hypergeometric functions of two variables

I'm looking for a (confluent) hypergeometric function of two variables that can be expressed in terms of a weighted infinite sum of two Whittaker functions such as: $$ \sum_{n=0}^\infty (\text{n-...
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Integral representation of product of two Whittaker functions

Does anyone know anything about the following formula involving special functions: $$ W_{\kappa,\mu}(z)W_{\lambda,\mu}(w)=\frac{e^{-(z+w)/2}(zw)^{\mu+1/2}}{\Gamma(1-\kappa-\lambda)}\int_0^\infty e^{-t}...
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Estimating the alternating sum $\sum_{j \ge 1} (-1)^j e^{-j^2} j^k$

I have been trying to get a lower bound on the following alternating sum but without much success: $$ \sum_{j=1}^T (-1)^j e^{-j^2} j^k . $$ For small values of $k$, this is easy because the first term ...
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1answer
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Asymptotic expansion of hypergeometric function ${}_3F_2$ for large parameters

I encountered the following hypergeometric function in my research: $${}_3F_2(2,1+n,1+n;1,2+n;z)$$ where $0<z<1$. I'm interested in its behavior for large $n$. Semilog plot suggests ...
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Integral of Legendre's function

Is there any formula for computing the following integral $$\int_a^1(P^m_l)^2(x)\,dx \, ,\text{with} -1<a<1$$ where $P^m_l$ is the associated Legendre's function (of the first kind) of order $m\...
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1answer
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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 ...
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What is the asymptotics of the Fourier transform of $\exp(-x^4)$ for large wave numbers?

The Fourier transform of $\exp(-x^4)$ has an analytical expression, it's the difference of two generalized hypergeometric functions: $\int d x \ e^{-x^4} e^{ikx} = 2 \ \Gamma(\frac{5}{4}) \ _0F_2(;\...
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Upper bounding the sum with hypergeometric and binomial probabilities

Could you please help me upper bound this tricky expression: $$P(A)=\sum_{i=0}^n{\left( 1 - \dfrac{\binom kq \binom {n-k}{i-q}}{\binom {n}{i}} \right)}^I \binom ni p^i {(1-p)}^{n-i}$$. So far I only ...
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Tightness of matrix hypergeometric bound

In Ratnarajah–Vaillancourt–Alvo (link), the authors write (on pg 3) that the following inequality for the Hypergeometric function of matrix argument holds: $${}_0F_1(b; X) < {}_0F_0(X/b)$$ where $b$...
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1answer
111 views

Conformal mapping between two right-angled triangles

I want to derive a conformal mapping $f\!:\!A\!\to\! B$ where $A=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq x \}$ and $B=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq \frac{...
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1answer
681 views

Proof of certain $q$-identity for $q$-Catalan numbers

Let us use the standard notation for $q$-integers, $q$-binomials, and the $q$-analog $$ \operatorname{Cat}_q(n) := \frac{1}{[n+1]_q} \left[\matrix{2n \\ n}\right]_q. $$ I want to prove that for all ...
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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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126 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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1answer
112 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 ...
3
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2answers
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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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2answers
171 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
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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 ...
4
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1answer
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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 $...
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1answer
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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 \...
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1answer
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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 ...
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1answer
139 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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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}^\...
7
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2answers
172 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{...
6
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3answers
292 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\...
12
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145 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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2answers
140 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{...
2
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2answers
142 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^{...
2
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1answer
428 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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1answer
117 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 ...
2
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2answers
192 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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1answer
88 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 ...
3
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4answers
593 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 \...