# 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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### Congruences between hypergeometric functions coming from modular forms

Consider the formal power series
$${}_2F_1(1/12,5/12;1;z)^{24}-{}_2F_1(1/12,7/12;1;z)^{24}=-z/3-z^2/2-(320293/559872)z^3-\cdots$$
It follows from a theorem on modular forms that for $p\ge5$ the ...

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### Parameter derivative of the confluent hypergeometric function

Let $X$ follow a normal distribution with mean $\mu$ and variance $\sigma^2$. Mathematica gives
$$
E[\log|1+X|]=\frac{1}{2}\biggl(-\gamma-\log 2+2\log\sigma-\frac{\partial}{\partial a}{}_1 F_1\biggl(0,...

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### Behavior of the hypergeometric function near x=1

It is known that the hypergeometric function ${}_2F_1(a, b, c; x)$ defined by the series
$$\sum_{n=0}^\infty \frac{a(a+1)\cdots (a+n-1)\cdot b(b+1)\cdots (b+n-1)}{c(c+1)\cdots (c+n-1) n!}x^n$$
behaves ...

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### How did Ramanujan find $\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}?$

The formula
$$\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}$$
(in older notation) appears as eq. 38 in Ramanujan's paper Modular equations ...

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### Inverse Laplace transform of the Gaussian hypergeometric function $_{2}F_{1}(a,b,;c;x)$

I want to calculate the inverse Laplace transform of the Gaussian hypergeometric function $_{2}F_{1}(a+p,b,;c;-\omega)$ in which
$p$ is the Laplace variable. The inverse Laplace transform is given by
$...

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### Using Ehrhart polynomials to count primes?

As indicated below, one could use the Ehrhart polynomials of the simplex in number theory.
Here are the questions without context first:
Questions:
The sum $$\sum_{k=0}^t (-1)^k ( \operatorname{...

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### Asymptotic expansion of generalised hypergeometric functions at infinity

I am looking for some resources that are available in public domain that have detailed description of the asymptotic expansion of generalised hypergeometric function ${}_pF_q(z)$ at infinity, i.e., $|...

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### How to write Tricomi's confluent hypergeometric function in terms of Meijer-G function

I am calculating a closed form expectation and I encountered the Tricomi's confluent hypergeometric function
(aka confluent hypergeometric function of the second kind) given by integral $U\left( a,b,z ...

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### Factorial series $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$ and hypergeometric functions

For positive integer $D$, define $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$.
For $D \le 6$, sage finds closed form in terms of hypergeometric functions
at algrebraic arguments and fails to find closed ...

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### When can GKZ setup encompass HMS?

Are there any instances when the Landau-Ginzburg superpotential describing the mirror of a smooth projective Fano variety $X_\Sigma$ is encompassed by a GKZ hypergeometric system? In some sense I am ...

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### Why don't Zeilberger and Gosper's algorithms contradict Richardson's theorem?

Richardson's theorem proves that whether an expression A is equal to zero is undecidable. A is in this case an expression, constructed from $x,e^x,\sin(x)$ and the constant function $\pi$ and $\ln(2)$ ...

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### How to calculate this integral of squared Tricomi hypergeometric function

How to solve this integral
$$
\int_{0}^{\infty}r^2 e^{-\omega r^2}U(-\nu,\frac{3}{2},\omega r^2)^2 \mathrm{d}r
$$
where $\omega>0$ and $\nu \in \mathbb{R} \setminus \left \{ \frac{n-1}{2}\mid n \in ...

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### Computing hypergeometric function at 1

I'm looking to compute
$${}_ 3F_ 2\biggl(\begin{matrix} -m-1/2,\ -m,\ k-m+1/2 \cr
1/2-m,\ k-m+3/2\end{matrix};1\biggr)$$
for $m,k > 0$ are positive integers and $0 < k < m$. I'm wondering if ...

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### Derivation of indefinite integral involving hypergeometric function

I am doing a project on projectile motion and I ended up with this integral:
$$\int \frac{m \left(g - \left(\frac{1}{e^t - g^{\frac{m}{c}}}\right)^{\frac{m}{c}}\right)}{c} \, dt$$
where $g, c,$ and $m$...

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2
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### A closed formula for a sum involving hypergeometric function

Let ${ }_1 F_1(a ; c ; z)$ be Kummer's function defined by the function, and all its analytic continuations, represented by the infinite series $\sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n} \frac{z^n}{n !...

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### A hypergeometric series for $\sqrt3\pi$ with converging rate $1/9$

Recently, I found a (conjectural) new series for $\sqrt3\pi$:
$$\sum_{k=1}^\infty\frac{(8k-3)\binom{4k}{2k}}{k(4k-1)9^k\binom{2k}k^2}=\frac{\sqrt3\pi}{18}.\label{1}\tag{1}$$
The series converges fast ...

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### How to calculate the Integral with confluent hypergeometric function

How to prove this.Thank you in advance
Let $\delta,\beta>0$ How to prove this
\begin{align}
& \int^1_0 \frac{w^{1-\beta}}{(1-w)^{1+\delta}} (-t.s w)^{\frac{-\delta}{2}} e^{-\frac{w}{1-w}(s+t)}...

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### Two particular combinations of Gauss hypergeometric functions

Browsing this site and the web I could not find a reference on the following combinations of Gauss hypergeometric functions $F={}_2F_1$, for which I have reason to believe that they can be simplified ...

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### How can I verify this family of values for hypergeometric functions?

This Wolfram MathWorld page on hypergeometric functions states that
An infinite family of rational values for well-poised hypergeometric functions with rational arguments is given by $$_kF_{k-1}\left[...

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### New $_2F_1$ identity?

The following identity arises in (a new) derivation of the distribution of the estimate $r$ of the binormal correlation coefficient $\rho$. Here formulated in terms of $x = r\rho$.
For $-1 < x < ...

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### $n$th Derivative of $_p F_q(a_1,...,a_p; b_1,...,b_q;x^{-m})$, $p \le q$

Maple seems to suggest the following formula for $n>0$, $p \le q$:
\begin{align}
\frac{d^n}{d x^n} & {}_p F_q (a_1,\ldots,a_p;b_1,\ldots,b_q;1/x) \\[8pt]
= {} & (-1)^n \hspace{1pt} n!\...

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### Log convexity of hypergeometric function for $a,b,c>0$

Prove that:
$$ f(x) = \log\big(
{}_2F_1(a,\,b\,;\,c\,;\,x^{-1})\big),\;\;a,b,c>0
$$
is convex (and decreasing) on $(1,\infty)$.
It actually seems that the stronger result that $f\big((x+1)^{\beta}\...

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### Summation of the following form with non-integer n

I have the following function:
$$ G(z) = \sum_{j = 0}^{\infty} \frac{\Gamma(1 + n) z^j}{j ! (\Lambda + jC)^{n+1}} $$
If $n \quad \epsilon \quad \mathbb{Z}^{+} $, the above function can be ...

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### A Riemann-Liouville differintegral for all entire Dirichlet L-series. Could it be simplified further?

It appears that the well-known relation between entire Dirichlet L-series and the Hurwitz zeta function $\zeta(s,a)$, with $k$ = modulus, $j$ = index of the Dirichlet character $\chi$:
$$(s-1)\,L\left(...

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### Generalized Wigner 3-j symbol and Legendre functions

Let $P_{n}(x)$ the $n-th$ Legendre polynomial. It is well-knonw that $$\int_{-1}^1 P_n(x) P_m(x) P_h(x) \, dx=2\left(\begin{array}{ccc}
n & m & h\\
0 & 0 & 0
\end{array}\right)^{2}\tag{...

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### Hilbert's 13th Problem and series solutions for the reduced sextic, septic, and octic?

I. Reduced equations
One can eliminate 3 terms from the general quintic, sextic, septic, and octic using a Bring-Jerrard transformation to get the reduced forms in radicals,
$$x^5+(x+p) = 0$$
$$x^6+(x+...

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### Simplification of hypergeometric Function

First of all I am not at all a math expert, but I have some working knowledge.
That said, please excuse "dumb" questions.
I am looking at the following process: Assume you are on the 2-...

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### Is the hypergeometric function ${}_1F_2(1;a,a+\frac12;-x^2)$ an elementary function? How about its positivity, monotonicity, and convexity in $x$?

Is the generalized hypergeometric function ${}_1F_2\bigl(1;a,a+\frac12;-x^2\bigr)$ for $a>-1$ and $x>0$ an elementary function?
How about the positivity, monotonicity, and convexity of the ...

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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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### 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 ...

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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 ...

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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
...

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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 ...

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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
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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 ...

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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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### Ask for a generating function or an explicit expression of a triangle of positive integers

Preliminaries
I encountered the following triangle of positive integers:
$c_{n,k}$
$n=1$
$n=2$
$n=3$
$n=4$
$n=5$
$n=6$
$n=7$
$n=8$
$k=0$
$1$
$3$
$15$
$105$
$315$
$3465$
$45045$
$45045$
$k=1$
$5$
$...

2
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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) ...

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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 \...

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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
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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
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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 {...

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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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votes

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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 ...

2
votes

1
answer

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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-...

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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}\...

3
votes

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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},...