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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A curious $q$-series identity on a truncated Euler function

Recall that a $q$-Pochhammer symbol is defined as $$ (x)_n = (x;q)_n := \prod_{l=0}^{n-1}(1-q^l x). $$ I found the following curious $q$-series identity that seems to hold for any $n\geq 0$: $$ (-1)^{...
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Generalization of identity for terminating hypergeometric function

Let ${}_2F_1(a,b;c;z)$ be the ordinary hypergeometric function for $z \in \mathbb{C}$ \begin{equation} {}_2F_1(a,b;c;z) = \sum_{k=0}^{\infty} \frac{z^k}{k!} \frac{(a)_{k} (b)_k}{(c)_k}\,, \end{...
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2 answers
135 views

Zeros of hypergeometric functions with complex variables

Let $z$ be a complex number and let $a,b,c > 0$. I would like to know the zeros of the following hypergeometric function: $$_{2}F_{1} (a,b; c :z )=\sum_{k=0}^{+ \infty} \frac{(a)_{k}(b)_{k}}{(c)_{...
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Appearances of the basic hypergeometric series ${}_0\phi_1(;z;q;q^l z)$

On wikipedia one can find the general definition of a unilateral basic hypergeometric series ${}_r\phi_s$. The special case ${}_0\phi_1(;z;q;q^l z)$ has the expansion $$ {}_0\phi_1(;z;q;q^l z) = \sum_{...
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Any name for this special function?

We know $$ \sum_{m=0}^\infty \frac{x^m}{(a-m)!m!} = \frac{1}{a!}(1+x)^m $$ where we understand the factorial as Gamma function $\Gamma(x)$ such that it is divergent if the argument is negative integer....
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2 answers
224 views

Power series of ratio of Gamma functions

Let $a>1$ and define $G_a(x)=\sum\limits_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$ where $\Gamma$ is the Gamma function. This series is convergent on $\...
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Antiderivative of $f\left( \xi \right) =\xi^{a}\left( b\xi ^c+K\right) ^d$

I need to know the primitive function (Antiderivative) of this function: $$f\left( \xi \right) =\xi^{a}\left( b\xi ^c+K\right) ^d$$ where $K$ is an integration constant, $d=-\frac{1}{2p}$ with $p<...
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3 votes
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213 views

Generating function of the product of Legendre polynomials

The generating function of the product of Legendre polynomials for the same $n$ is given by \begin{aligned} \sum_{n=0}^{\infty} z^{n} \mathrm{P}_{n}(\cos \alpha) \mathrm{P}_{n}(\cos \beta)&=\frac{\...
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Functional inverse of $z=1+w+\cdots+w^{n-1}$

Migrated from the MSE. I am interested in the functional inverse of $$ z=1+w+\cdots+w^{n-1},\quad w\geq0,\ n>1. $$ This function is strictly increasing on $w\geq0$ and thus admits an inverse. By ...
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1 answer
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Estimates for the absolute value of the hypergeometric function ${}_2F_1(2-n,n+2,2;x)$ on $[0,1]$

I would like to know whether the estimate $$ |{}_2F_1(2-n,n+2,2;x)| \le \frac{n}{2}, \quad x \in [0,1] $$ holds and in that case where to find a reference.
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Simplification of $\sum_{m=0}^\infty \text B_z(m+a,b-m)x^m,\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}$ in terms of Kampé de Fériet function

Here is the goal sum where the Pochhammer Symbol with the Incomplete Beta function series $$\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}=\sum_{m=0}^\infty z^{m+a}\sum_{n=0}^\infty\frac{(1-(b-m))...
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231 views

On a certain double integral appearing in the Fourier series coefficients of $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series

The following integral appears naturally within the computation of the Fourier series coefficients of a real analytic $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series: \begin{align*} \int_{-\infty}^{\...
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Integrability of an alternating series with hypergeometric coefficients

during my research, I came up with the following series $$ f(t):=\sum_{k=0}^\infty \frac{\left(-t^2\right)^k}{(k!)^2}{}_3F_{2}\left(\left(-k,-k,-k\right);\left(1,\frac{1}{2}-k\right);\frac{1}{4}\right)...
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3 votes
2 answers
214 views

$\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} z^{2k}$ is an elementary function

I try to calculate the following series \begin{align*} S_{n,m}(z)=\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} \, z^{2k}, \end{...
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1 answer
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Hyper geometric series reference

Can someone point out a reference for the proof of this identity? Thanks in advance. https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric4F3/03/03/01/0002/
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Linear differential equation hypergeometric series

I am studying the following series $$\sum_{d\geq 1}\left(\frac{x^d}{d \, h^d} \sum_{k=1}^d (-1)^{d-k} \frac{1}{(k-1)! \, (d-k)!}\prod_{i=1}^d G((k-i)h)\right).\tag{*} $$ It's proven that (*) satisfies ...
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Is there a special function for $\sum_{m=0}^{\infty} x^m/(m!)^s$?

Is there a special function for the following series? $$\sum_{m=0}^{\infty} {x^m \over (m!)^s}$$ Here, $s$ is a positive real number. When, $s$ is an integer, $s=n \in \mathbb{Z}$, this series can be ...
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3 votes
1 answer
192 views

Proof identity for hypergeometric series 2F1(a,b;c;x)

I would like prove the following identity: $$ _2F_1(a,b;c,x) = \frac{c+(a−b+1)x}{c} {} _2F_1(a+1,b;c+1;x) - \frac{(a+1)(c−b+1)x}{c(c+1)} {} _2F_1(a+2,b;c+2;x). $$ I've tried this so far: I know that $...
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1 answer
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How to show the following inequality $_2F_1(5.5, 1, 5;-|x|^2]>0$? [closed]

Consider the function $_2F_1(5.5, 1, 5;-|x|^2]$ for $x\in \mathbb{R}^n.$ I want to show that this function is positive. I checked that it does not have any roots so can I conclude the inequality by ...
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Identities on the Whittaker function $W_{-\kappa,\mu}(z)$?

As in (for example) [Magnus, W., Oberhettinger, F., Soni, R.: Formulas and Theorems for the Special Functions of Mathematical Physics, Grundlehren der mathematischen Wissenschaften 52, Springer, ...
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2 votes
0 answers
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Reduction of the general Lauricella hypergeometric function $F_B$ for identical parameters and variables

The Lauricella function $F_B^{n}$ of $n$ variables is defined as $$F_B^{(n)}(a_1, \ldots, a_n, b_1, \ldots, b_n, c; x_1, \ldots x_n) = \sum_{k_1, \ldots, k_n = 0}^\infty \frac{1}{(c)_{k_1 + \ldots + ...
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6-j symbols and hypergeometric series

What’s the correct formula for $_{4}F_{3}(a,b,c,d;e,f,g;1)$ where $a+b+c+d-e-f-g=-1$? The Wolfram Alpha formula involves $6j$ symbols and makes no sense for some specific cases. For example, $_{4}F_{...
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0 answers
81 views

Any known relations to this doubly exponential constant?

Two years back I was working with lacunary series. In my explorations I had derived that the following series is periodic with period 1: $$ f(x)= \sum_{n=0}^{\infty} \left[ e^{2^{x+n}} \right] + x - \...
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1 vote
0 answers
143 views

Asymptotics of $\sum \frac{d(n)}{n}$ with generating functions

We can determine the asymptotics of partial sums involving the divisor function accurately by means of, for example, the hyperbola method: $$\sum_{n\leq N}\frac{d(n)}{n}=\frac{1}{2}(\log(N))^{2}+2\...
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5 votes
1 answer
287 views

Hypergeometric function evaluation 4F3

I need to show that for $m$ being non-negative integer, the hypergeometric function ${}_4F_3$ below evaluates to $-1/2$ independent of $m$. This is Mathematica notation, but we have 4 and 3 sets of ...
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1 vote
0 answers
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Asymptotic behavior of hypergeometric function ${}_{3}F_{2}(a,b-n,c-n;d-n,e-n;1)$ for $n\to\infty$

Suppose $a,b,c,d,e\in\mathbb{R}$ are such that $d+e-a-b-c>0$ and $d,e\notin\mathbb{Z}$. I would like to know whether it is possible to deduce an asymptotic formula for the sequence given by the ...
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3 votes
1 answer
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Periodic Gauss hypergeometric function

I stumbled upon the following identity, which I have not tried to prove, but seems true: the function $$f(t):={}_2F_1(1/2,2t;1-t;4)$$ is periodic of period 1, and more precisely $$f(t)=\dfrac{1+2e^{-2\...
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9 votes
0 answers
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What is the precise definition of "Hypergeometric motives over $\mathbb{Q}$"?

The question is as in the title, but here is some background: Section 4 of this paper by Beukers, Cohen and Mellit is called "Hypergeometric motive over $\mathbb{Q}$" but no actual (pure) ...
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5 votes
1 answer
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Relationship between Lambert $W$ function and Hypergeometric function

The Lambert $W$ Function is defined in this Wikipedia entry, while the Hypergeometric Function is defined in this other Wikipedia entry. There exists also a multivariate generalization which solves ...
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Uniform bound on a certain family of hypergeometric functions

We have the following problem, which we can't solve. Let $a \in \mathbb{C}$ be fixed, with real part $1/2$ and imaginary part $\neq 0$. We consider parameters $n \in \mathbb{Z}$ and $k \in \mathbb{Z}_{...
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0 votes
0 answers
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Upper bound an integral

I am trying to upper bound the following integral $$\int_{0}^{1} \left| (1-(x-a^2))^n - (1-(x-b)^2)^n \right|dx,$$ where $x \in [0,1]$, $a$ and $b$ are fixed constants, and $n$ is a possitive integer. ...
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3 votes
2 answers
342 views

About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions

NOTE: I post this question on math.stackexchange but nobody answered, so I try here. For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-...
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43 votes
6 answers
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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 $$ \...
6 votes
0 answers
106 views

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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-2 votes
1 answer
115 views

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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9 votes
0 answers
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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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0 answers
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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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5 votes
0 answers
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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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1 vote
1 answer
120 views

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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2 votes
0 answers
38 views

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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4 votes
0 answers
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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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11 votes
1 answer
287 views

Integral representation of product of two Whittaker functions

Does anyone know anything about the following formula involving special functions: $$\begin{multline*} W_{\kappa,\mu}(z)W_{\lambda,\mu}(w)=\frac{e^{-(z+w)/2}(zw)^{\mu+1/2}}{\Gamma(1-\kappa-\lambda)}\...
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  • 373
7 votes
0 answers
210 views

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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  • 131
2 votes
1 answer
113 views

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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2 votes
0 answers
102 views

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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2 votes
1 answer
161 views

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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11 votes
0 answers
369 views

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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0 votes
0 answers
42 views

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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3 votes
1 answer
166 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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17 votes
1 answer
785 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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