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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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<...
A.Hossem's user avatar
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454 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{\...
Kane's user avatar
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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 ...
Aaron Hendrickson's user avatar
1 vote
1 answer
131 views

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.
Joan Verdera's user avatar
1 vote
1 answer
101 views

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))...
Tyma Gaidash's user avatar
7 votes
1 answer
277 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}^{\...
Hugo Chapdelaine's user avatar
1 vote
0 answers
60 views

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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$\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} z^{2k}$ 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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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/
user338431's user avatar
5 votes
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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 ...
TheTwistedSector's user avatar
4 votes
1 answer
438 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 $...
Mathstudent's user avatar
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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 ...
Student's user avatar
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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, ...
Z. Alfata's user avatar
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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 + ...
PolyPhys's user avatar
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1 answer
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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_{...
user338431's user avatar
1 vote
0 answers
90 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 - \...
Sidharth Ghoshal's user avatar
1 vote
0 answers
170 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\...
Max Muller's user avatar
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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 ...
Per Alexandersson's user avatar
1 vote
0 answers
98 views

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 ...
Twi's user avatar
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3 votes
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261 views

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\...
Henri Cohen's user avatar
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9 votes
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420 views

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) ...
naf's user avatar
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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 ...
queen28's user avatar
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71 views

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}_{...
Sasha's user avatar
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4 votes
2 answers
428 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=\,-...
Marco Cantarini's user avatar
47 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
146 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 $S=X_1+\cdots+X_n$....
Brendan McKay's user avatar
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1 answer
148 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 ...
gustavo's user avatar
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9 votes
0 answers
318 views

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 ...
Wane's user avatar
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0 answers
65 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 ...
gustavo's user avatar
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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(...
Henri Cohen's user avatar
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1 vote
1 answer
151 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)...
Predrag Punosevac's user avatar
2 votes
0 answers
51 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
186 views

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. ...
Stat_math's user avatar
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11 votes
1 answer
475 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)}\...
Y.Okuyama's user avatar
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7 votes
0 answers
307 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 ...
nichehole's user avatar
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2 votes
1 answer
223 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 ...
Bullmoose's user avatar
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2 votes
0 answers
132 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\...
rihani's user avatar
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3 votes
1 answer
311 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
625 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(;\...
Sara's user avatar
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0 answers
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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 ...
abs135's user avatar
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3 votes
1 answer
277 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{...
niran90's user avatar
  • 167
17 votes
1 answer
859 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 ...
Per Alexandersson's user avatar
2 votes
2 answers
418 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 ...
Z. Alfata's user avatar
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3 votes
0 answers
39 views

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) }{...
Rafa's user avatar
  • 191
2 votes
2 answers
229 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 $\...
MCH's user avatar
  • 1,304
0 votes
1 answer
502 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 ...
axl's user avatar
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3 votes
2 answers
369 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 ...
ManUtdBloke's user avatar
0 votes
2 answers
478 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 ...
Thoughtful's user avatar
12 votes
1 answer
496 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 ...
Luis Ferroni's user avatar
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5 votes
1 answer
549 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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