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.
250
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Analytic continuation of ${_0F_{\beta-1}}(-;\mathbf 1;H^\beta)$ in $\beta$
I am working with series of the form
$$
\sum_{k=0}^\infty\left(\frac{H^k}{k!}\right)^\beta,
$$
where $\beta\in(0,1)$. By the ratio test this series converges for all $\beta>0$. If $\beta\in\Bbb N$ ...
0
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how to derive this elliptic integral?
I am reading the article arXiv: 2207.09961, there are some interesting elliptic integrals, i.e. the formula (3.7) and (3.8). You can also see this image
where $p_0(z)=\sqrt{-Q_0(z)}$ and $Q_0(z)=-\...
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0
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Infinite series involving generalised hypergeometric functions
I've recently stumbled into hypergeometric functions while trying to evaluate the integral:
$$
\int \exp \big( x^2 + bx + c \big) {\rm erf} ( x ) \operatorname{d\!}x
$$
Essentially, working from an ...
- 19
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Expanding 3F2 functions in terms of polylogarithms
I need to expand hypergeometric functions in the form of ${}_3F_2(1, 1, k; m, n)$ and ${}_3F_2(1, 1, 1; m, n)$ with $k < m \le n$ and $k, m, n \in \mathbb{Z}$, in terms of polylogarithms.
The ...
- 1
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2
answers
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Signs of Kravchuk matrix asymptotically produce a large circular region with hyperbolic sinks. Why?
The Kravchuk matrix of dimension $n+1$ is such that its entries satisfy $$K_{i,j}^{(n+1)}=[x^i](1+x)^{n-j}(1-x)^j\quad\forall0\le i,j\le n.$$ It enjoys properties such as involution and has various ...
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On the solutions of $_2F_1(\alpha, \beta; \gamma, z) = \Lambda$
More General Question
Let
$$F(\alpha,\beta;\gamma;z) = \sum_{n=0}^{+\infty} \frac{(\alpha)_n(\beta)_n}{(\gamma)_nn!}z^n, \quad |z| < 1, \quad (x)_n = x(x+1)...(x+n-1), \quad (x)_0 = 1$$
be the ...
- 201
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"Symmetrize" a (balanced) hypergeometric 4F3
Let $a,b,c,d$ be positive integers such that $a+b+c+d=2^{n}$ with $n \ge 2$.
Denote
$$
N \equiv a+b+c+d=2^{n}
$$
Consider the balanced hypergeometric series
$$
\frac{\operatorname{\Gamma}\left( \frac{...
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Can the Taylor shift of a hypergeometric function always be expressed as a nontrivial transformation of another hypergeometric function?
Given a function $F(z)$, the function $F(z-m)$ is called
the Taylor shift of $F(z)$. If $F(z)$ is a (general) hypergeometric
function, is $F(z-m)$ some simple transformation of a hypergeometric
...
- 9
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Is $\sum\limits_{k=0}^{\infty}\frac{1}{(mk)!^{n+1}}$ irrational?
I was using Wolfram Alpha for things, and I came across $I_{0}(2)$. For fun I tried asking Wolfram Alpha if the number was irrational, but said it's unknown. I believe this is an error, as its ...
- 101
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votes
3
answers
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Computing $_2F_2(a,a,a+1,a+1,z)$ (hypergeometric function)
Trying to implement the derivative of the gamma incomplete function, I encountered the hypergeometric function $_2F_2(a,a,a+1,a+1; z=-x)$, where $x$ would always be a positive real (and thus $z$ a ...
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1
answer
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Asymptotic analysis of an expression involving a Fox's H function
One of the performance metrics calculated in the analysis of telecommunications systems is the ergodic channel capacity, $C_{\rm erg}$. During one of my studies, I found the expression below for such ...
2
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2
answers
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Asymptotic behavior of a hypergeometric function
Can anybody see how to deduce an asymptotic formula for the hypergeometric function
$$ _3F_2\left(\frac{1}{2},x,x;x+\frac{1}{2},x+\frac{1}{2} \hskip2pt\bigg|\hskip2pt 1\right), \quad\mbox{ as } x\to\...
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Is the Gauss hypergeometric function $_2F_1(a,b;c,z)$ univalent in $\left|z - \frac{1}{2} \right| < \frac{1}{2}$?
Consider the Gauss hypergeometric function
$$_2F_1(a,b;c,z) = \sum_{n=0}^{+\infty} \frac{(a)_n(b)_n}{(c)_nn!}z^n, \quad |z| < 1, \quad (x)_n = x(x+1)\cdots(x+n-1), \quad (x)_0 = 1$$
The Encylopedia ...
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0
answers
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Ask for a generating function or an explicit expression of a triangle of positive integers
Preliminaries
I encountered the following triangle of positive integers:
$c_{n,k}$
$n=1$
$n=2$
$n=3$
$n=4$
$n=5$
$n=6$
$n=7$
$n=8$
$k=0$
$1$
$3$
$15$
$105$
$315$
$3465$
$45045$
$45045$
$k=1$
$5$
$...
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1
answer
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Ask for references or proofs of two explicit formulas for special Gauss hypergeometric functions
Can one supply related references or detailed proofs of the following two explicit formulas?
$$
{}_2F_1\biggl(2\alpha+1,2;\alpha+3;\frac{1}{2}\biggr)
=2\frac{\alpha B(1/2,\alpha)-1-\alpha}{(1-\alpha) ...
- 706
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votes
0
answers
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Integral representation of a quotient of odd and even parabolic cylinder functions
In some work on nonlinear splines in space, the following expression arises:
$$\frac{e^{-\mathrm{i}\pi/4} \; y_2 \left( \frac{\mathrm{i}}{4 \alpha} -\frac{1}{2} ; e^{\mathrm{i}\pi/4} \sqrt{\alpha} s \...
5
votes
1
answer
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What is the surface area of the finite part of the Cayley nodal cubic surface?
The Cayley nodal surface is defined by the equation $x^2+y^2+z^2-2xyz=1$. The finite part of the surface is the tetrahedral part bounded by the 4 nodes $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, $(-1,-1,1)$....
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1
vote
1
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How to calculate this limit (if exist)?
I have just asked the calculation of the following summation see here $$S(a,b,m,n_1,n_2)=\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k}, $$
which is motivated by the calculation of the ...
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votes
1
answer
335
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How to calculate this summation $\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k} $?
Question: How to calculate this summation $S=\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k} $? Where $m<n_1,m<n_2$
Remark1: When $a=b$, I know the above summation $S=a^m\sum_{k=0}^m {...
- 57
1
vote
0
answers
64
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polynomial approximation of hypergeometric function 2F1
I have the following function $T(k_1,k_2)$ resulting from multiphoton transition matrix elements calculations:
$T(k_1,k_2)=\gamma^{-k_2}\sum_{j=0}^{k_1}(j+2)_{l+1}\binom{k_1}{j}(k_1+1)_3(\gamma-1)^{j}{...
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1
answer
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Closed form for ₄F₃(n,n,n,2n;1+n,1+n,1+n;−1)
For positive integer $n$ the following value of a hypergeometric function
$$_4F_3(n,n,n,2n,1+n,1+n,1+n,-1)$$
based on the first few terms looks like
$$ R_1(n) + R_2(n) \pi^2$$
where $R_{1,2}(n)$ are ...
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0
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Fractional partial derivatives and integrals of De Bruijn's $H_t(z)$-function. Does a simpler form exist for the $z$ derivative/integral?
$\newcommand\KummerU{\text{KummerU}}$
$\newcommand\Hypergeom{\text{Hypergeom}}$
During 2018/2019, the polymath 15 project managed to successfully reduce the upper bound of the De Bruijn-Newman ...
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0
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Classification of Hypergeometric Functions: How Fundamental is the Order?
Horn classified the hypergeometric functions of two independent variables into 34 different convergent functions, where 14 are complete and 20 are confluent. However, Carlson notes that linear ...
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Why does a given image value of a function $x H^3(x)$ where $H(x)$ contains hypergeometric functions have preimage value?
Prof. Edward B. Bender is a famous combinatorial mathematician. In his paper of A Survey of Asymptotic Behaviour of Maps published in Journal of Combinatorial Theory, there is a part of analyzing the ...
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votes
1
answer
140
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Does any Hyper-geometric function can be analytically continuated to the whole complex plain except $e^{\pm i\pi/3}$ [closed]
We all know that the famous Hypergeoemtric function $_2F_1$ has an integral form as follows:
$$_2F_1(a,b,c;z)=-\frac{e^{-i\pi c} \Gamma(c)}{4\Gamma(b)\Gamma(c-b)\sin\pi b\sin\pi(c-b)}\int_P t^{b-1}(1-...
- 123
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votes
2
answers
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Is the Gauss hypergeometric series ${}_2F_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)$ an elementary function?
For $\alpha,\beta\in\mathbb{C}$ and $\gamma\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}$, Gauss' hypergeometric function ${}_2F_1(\alpha,\beta;\gamma;z)$ can be defined by the series
\begin{equation}\...
- 706
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votes
1
answer
103
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Nonnegativity of q-hypergeometric series
What are methods for proving nonnegativity of q-hypergeometric functions? Specifically, I have a function of the type 4-phi-3, it is a terminating series:
$$
{}_{4}\phi_3\left(\begin{matrix} q^{-i_1},...
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votes
1
answer
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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 $n\geq 0$:
$$
(-1)^{...
- 1,410
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votes
1
answer
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views
Generalization of identity for terminating hypergeometric function
Let ${}_2F_1(a,b;c;z)$ be the ordinary hypergeometric function for $z \in \mathbb{C}$
\begin{equation}
{}_2F_1(a,b;c;z) = \sum_{k=0}^{\infty} \frac{z^k}{k!} \frac{(a)_{k} (b)_k}{(c)_k}\,,
\end{...
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Zeros of hypergeometric functions with complex variables
Let $z$ be a complex number and let $a,b,c > 0$. I would like to know the zeros of the following hypergeometric function:
$$_{2}F_{1} (a,b; c :z )=\sum_{k=0}^{+ \infty} \frac{(a)_{k}(b)_{k}}{(c)_{...
5
votes
1
answer
719
views
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....
- 63
1
vote
2
answers
332
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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 $\...
- 29
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votes
1
answer
106
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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
1
answer
359
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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{\...
- 43
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votes
0
answers
253
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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 ...
1
vote
1
answer
83
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.
- 11
1
vote
1
answer
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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
$$\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}=\sum_{m=0}^\infty z^{m+a}\sum_{n=0}^\infty\frac{(1-(b-m))...
- 125
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votes
1
answer
253
views
On a certain double integral appearing in the Fourier series coefficients of $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series
The following integral appears naturally within the computation of the Fourier series coefficients of a real analytic $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series:
\begin{align*}
\int_{-\infty}^{\...
- 7,441
1
vote
0
answers
53
views
Integrability of an alternating series with hypergeometric coefficients
during my research, I came up with the following series
$$
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)...
- 11
3
votes
2
answers
246
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{...
- 630
0
votes
1
answer
90
views
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/
- 31
5
votes
0
answers
195
views
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 ...
- 153
4
votes
1
answer
303
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 $...
- 59
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votes
1
answer
101
views
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 ...
- 591
1
vote
0
answers
56
views
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, ...
- 630
2
votes
0
answers
37
views
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 $$F_B^{(n)}(a_1, \ldots, a_n, b_1, \ldots, b_n, c; x_1, \ldots x_n) = \sum_{k_1, \ldots, k_n = 0}^\infty \frac{1}{(c)_{k_1 + \ldots + ...
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3
votes
1
answer
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views
6-j symbols and hypergeometric series
What’s the correct formula for $_{4}F_{3}(a,b,c,d;e,f,g;1)$ where $a+b+c+d-e-f-g=-1$?
The Wolfram Alpha formula involves $6j$ symbols and makes no sense for some specific cases. For example, $_{4}F_{...
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1
vote
0
answers
84
views
Any known relations to this doubly exponential constant?
Two years back I was working with lacunary series. In my explorations I had derived that the following series is periodic with period 1:
$$ f(x)= \sum_{n=0}^{\infty} \left[ e^{2^{x+n}} \right] + x - \...
- 1,450
1
vote
0
answers
152
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\...
- 3,939
5
votes
1
answer
417
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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