**4**

votes

**0**answers

58 views

### An identity of complicated combinations of gamma functions (related to hypergeometric functions)

Can somebody help me in proving the following equation?
\begin{align*}&\textstyle \sum _{d=0} ^{n} \frac{1}{d!(n-d)!} \frac{\Gamma (b+d) \Gamma (b+n-d) \Gamma (c-n+d) \Gamma (c-b+1-n + 2d) ...

**1**

vote

**1**answer

143 views

### How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case

After multiple plots I noticed that function $h(x)= (1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$), for $x\ll 1$ (specifically $0<x<0.1$) and ...

**1**

vote

**1**answer

100 views

### Approximation of $ _2F_1((b-1)a,b;ba;x) $

Is there any (simple) approximation of this Hypergeometric function: $ _2F_1((b-1)a,b;ba;x) $, where $0<x<1$ and $b>a>1$.
Thanks!

**0**

votes

**0**answers

41 views

### Equivalent of Lauricella $F_D$ on an elliptic curve?

Lauricella's hypergeometric function $F_D$ is related to (weighted) configurations of points on $\mathbb{P}^1$. I am looking for generalizations to weighted point configurations on an elliptic curve. ...

**2**

votes

**0**answers

54 views

### Does there exist duality/symmetry between Fuchsian differential equations, like in the case of confluent hypergeometric?

My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for ...

**4**

votes

**0**answers

61 views

### Identities for ${~}_3\phi_1$?

I am looking for some source of summation formulas for the $q$-hypergeometric function ${~}_3\phi_1$ in the sense of Gasper-Rahman book. For some reason, the book focuses heavily on ${~}_{r+1}\phi_r$ ...

**2**

votes

**1**answer

130 views

### Are binomial coefficients $F_1$ analogs of $q$-binomial coefficients?

This is a mostly philosophical question. Is it fair to think of usual binomial coefficients and their identities as an $F_1$ case of $q$-binomial coefficients and identities? Here $F_1$ is the field ...

**3**

votes

**0**answers

137 views

### System of linear ODEs with hypergeometric coefficients

For quite some time I have been trying to solve the following system of differential equations for the two functions $G$ and $H$ defined on the interval $[0,1]$:
$$
\begin{align}x ...

**2**

votes

**1**answer

108 views

### hypergeometric at nearest singularity

Reference request. A prototype case:
In
$$
{}_2F_1\left(\frac{1}{12},\frac{5}{12};\frac{1}{2};x\right) =
A\log\left(\frac{1}{1-x}\right) + B + o(1),
\qquad x \to 1^-
$$
what can we say about the ...

**12**

votes

**1**answer

239 views

### A hypergeometric puzzle

$$
143\,\sqrt {3}\;{\mbox{$_2$F$_1$}\left(\frac{1}{2},\frac{1}{2};\,1;\,{\frac {3087}{8000}}\right)}=
40\,\sqrt {5}\;
{\mbox{$_2$F$_1$}\left(\frac{1}{3},\frac{2}{3};\,1;\,{\frac ...

**4**

votes

**2**answers

125 views

### Sharp upper bounds on hypergeometric function ${}_2F_1[a,b,c;z]$ when $|z|\geq1$

Generally, hypergeometric function ${}_2F_1[a,b,c;z]$ is defined using Gauss series ${}_2F_1[a,b,c;z]=\sum_{n=0}^\infty\frac{(a)_n(b)_n}{(c)_nn!}z^n$ with $|z|<1$, and there seems to be a lot of ...

**0**

votes

**0**answers

142 views

### a question on sum of Gaussian binomial coefficients

I was trying to calculate something and at some point I get the following sum:
\begin{equation}
\sum_{t=0,t \text{ even}}^{s}{s+3n \brack s-t}\sum_{i = 0}^{t/2}q^{2i^2}{t/2+2n-i \brack t/2-i}{n ...

**2**

votes

**0**answers

229 views

### closed form expression for an infinite series

Is there any closed form expression for the infinite sum $\sum_{n \geq 0}q^{n(n+1)/2}(1+q)(1+q^2)\cdots(1+q^n)u^n$ where both $q$ and $n$ are variables and $n \in N \cup {0}$?

**0**

votes

**0**answers

73 views

### Bessel function stable recurrence relation

I want to compute hypergeometric 1f1 function using
I can't use direct computation of Bessel functions due to complexity. I want to use BesselJ recurrence relation:
But forward recursion is ...

**6**

votes

**3**answers

726 views

### Combinatorial identity involving the square of $\binom{2n}{n}$

Is there any closed formula for
$$
\sum_{k=0}^n\frac{\binom{2k}{k}^2}{2^{4k}}
$$
?
This sum of is made out of the square of terms $a_{k}:=\frac{\binom{2k}{k}}{2^{2k}}$
I have been trying to verify ...

**3**

votes

**1**answer

121 views

### Limit of a hypergeometric integral

Let $n,N,T$ be positive integers, with $N=\binom{n}{2}$, and $3\leq n\leq T\leq N$. Define:
$$P(z):=z^{N+1-T}\int_0^1\frac{(1-t^2)^{n-2}}{(1-(1-z)t)^{N+1}}{}_2F_1\left[-T,N+1,N+2-T; ...

**0**

votes

**3**answers

141 views

### Hypergeometric sum specific value

How to show?
$${}_2F_1(1,1;\frac{1}{2}, \frac{1}{2}) = 2 + \frac{\pi}{2} $$
It numerically is very close, came up when evaluating:
$$ \frac{1}{1} + \frac{1 \times 2}{1 \times 3} + \frac{1 \times 2 ...

**3**

votes

**1**answer

91 views

### proof that the schwarz map defined as ratios of gauss hypergeometric functions is univalent

The ratio of two linearly independent solutions of the Guass hypergeometric differential equation defines a map from the upper half plane to a Schwarz triangle. Everything I read tells me that this ...

**1**

vote

**1**answer

217 views

### Asymptotic form of the Gauß Hypergeometric function 2F1 for three parameters approaching infinity

I am trying to find the leading order expression in an expansion for large $\Delta$ of ${}_2F_1\left(\frac{\Delta}{2},\frac{\Delta+1}{2},\Delta,z^{-2}\right)$, where $z\in\mathbb{C}$.
The only ...

**5**

votes

**4**answers

255 views

### Exponential of a specific hypergeometric series

This is motivated by this question.
Let $f$ be the hypergeometric series
$ f(x) = 2 x \, _{4}F_3([1, 1, 4/3, 5/3], [2, 2, 2], 27 x) $
which is explictly given by
$ f(x) = \sum_{n \geq 1} ...

**1**

vote

**3**answers

210 views

### A definite integral of hypergeometric function 2F1

I am wondering whether there exists a closed form for the definite integral
$$F(x)=\int_0^1t^{-a}(1-t)^{N}(1-xt)^{-a}{}_2F_1(-a,k-a-1/2,k-a;4xt(1-xt))dt,$$
where $a\in(0,1)$ and $N,k$ are positive ...

**1**

vote

**0**answers

88 views

### Quadratic transformation of hypergeometric function 2F1

I want to know whether there is some transformation between
$_2F_1(a,b;c;x)$ and $_2F_1(a',b';c';x(1-x))$.
Here is an example called the Kummer quadratic transformation, which may be known to most of ...

**2**

votes

**1**answer

292 views

### About large z behavior of hypergeometric function $_2F_1(1/2,1/2,1;z)$

The hypergeometric function $_2F_1(\large \frac{1}{2},\frac{1}{2},1;\frac{1-\frac{u}{\Lambda^2}} {2} \large)$ at large $\mid u\mid$ can be approximated by
$$ -\frac{\Lambda}{\pi} \sqrt{\frac{2}{u}} ...

**1**

vote

**1**answer

66 views

### Fast numerical approximation of Lauricella series of the fourth kind for real variables and real parameters

I'm looking for a method to efficiently compute a numerical approximation of
$$F^n_D(x_1,\ldots,x_n) = \sum_{m=0}^{\infty} \sum_{i_1 +\ldots+i_n=m}\frac{(a)_{m}(b_1)_{i_1}\ldots ...

**6**

votes

**1**answer

218 views

### Logarithm of the hypergeometric function

For $F(x)={}_2F_1 (a,b;c;x)$, with $c=a+b$, $a>0$, $b>0$, it has been proved in [1] that $\log F(x)$ is convex on $(0,1)$.
I numerically checked that with a variety of $a,\ b$ values, $\log ...

**1**

vote

**1**answer

113 views

### $q$-differential equation for the Rodgers polynomials?

The Rodgers polynomials $C_{\alpha,q}$ are a particular family of well-known $q$-hypergeometric function. For example, a description can be found here on Wikipedia. For the special case of $q=1$, we ...

**6**

votes

**1**answer

222 views

### Conjectured bound on Kummer's function (confluent hypergeometric function)

I believe the following bound to be correct:
${}_1F_1(-a;1+a;-z) a z^{-a} \gamma(a,z) \leq 1$
for real-valued a > 0 and z $\geq 0$. $\gamma(a,z)$ is the lower incomplete gamma function.
Apart from ...

**2**

votes

**1**answer

157 views

### A definite integral related to hypergeometric function

I obtained the following integral when looking for a probability density function:
$$\int_0^1 x^{\alpha-1} \,(1-x) ^{-A}\, {}_2F_1 (1-A, \alpha -1-A, \alpha -A, x) \,dx$$
Can anyone please give me ...

**11**

votes

**1**answer

217 views

### An elementary expression for $_3F_2(1,1,9/4;2,2;-1)$

Consider the following series:
$$S=\sum_{n=1}^\infty\frac{(-1)^n\ \Gamma\left(\frac{5}{4}+n\right)}{n^2\ \Gamma(n)}.$$
It can be expressed in terms of a hypergeometric function:
...

**0**

votes

**1**answer

130 views

### Find a generalized hypergeometric-based function yielding certain ratios of fifth-degree polynomials

Find a (presumably, generalized hypergeometric-based function $f(n,a,k)$), yielding for $n=1, a=\frac{1}{2}$,the rational function (ratio of fifth-degree polynomials)
\begin{equation}
...

**1**

vote

**2**answers

220 views

### Hypergeometric identities

Let $m,k$ be positive integers with $k\le m$. Does anyone know some hypergeometric identities that imply
$$\sum_{j=0}^k\frac{(-1/2)_{k-j}(m+1)_j(-m)_j}{(1/2)_j(k-j)!j!}
...

**4**

votes

**0**answers

89 views

### Is there a way to express hypergeometric identities in terms of D-modules?

A hypergeometric function is a solution to
(*) w'' + p(z) w' + q(z) w = 0
where q has at most simple poles and q has at most double poles at 0,1,infty.
That differential equation is equivalent ...

**1**

vote

**1**answer

248 views

### principal specialization of projective Schur functions

Is there a nice/known formula for the (some) principal specialization of a projective (P or Q) Schur polynomial? That is, I am talking about a projective analogue of Stanley's hook content formula ...

**2**

votes

**1**answer

236 views

### growth of energy of eigenfunctions on hyperbolic surface

I am looking to the behavior of eigenfunctions associated to small eigenvalues on a degenerating hyperbolic surface.
Let $(\Sigma_n, h_n)$ a sequence of compact surfaces with area equal to $1$ and ...

**0**

votes

**2**answers

214 views

### Series representation of ratio of two Meijer G-functions

Let me use the notation from Maple http://www.maplesoft.com/support/help/Maple/view.aspx?path=MeijerG for the Meijer G-function. Then let me define,
$f_+(x) = MeijerG( [[+1/2],[]], [[0,0],[]], x )$
...

**4**

votes

**0**answers

479 views

### hypergeometric function $_2F_1(-n;-r;1;2)$

The hypergeometric function $_2F_1(-n;-r;1;2)$ appears in many different situations. For instance, it counts the number of integer points within a sphere in the $l_1$ norm, i.e.,
$$_2F_1(-n;-r;1;2) ...

**1**

vote

**2**answers

438 views

### Hypergeometric sum 3F2 at 1

Is there a closed-form sum for the hypergeometric series $_3F_2(a+c, c+d, 1; c+1, a+b+c+d \mid 1)$ where $a, b, c, d$ are all positive and not necessarily integers?
Update: The motivation for this ...

**3**

votes

**1**answer

546 views

### Pochhammer symbol of a differential, and hypergeometric polynomials

I have a minor result which I'm sure has come up somewhere before but I can't seem to find it.
Consider a confluent hypergeometric function of the form
$$\newcommand{\ff}{{}_1F_1}
...

**0**

votes

**1**answer

495 views

### The zeros of the digamma function

I wonder what work have been done on the zeros of the digamma function and on the values of the gamma function at such points (on the negative real axis). Any help please :)

**9**

votes

**8**answers

1k views

### A good reference to grok hypergeometric functions?

When I was introduced during my degree to special functions, I made friends with a number of nice functions - Laguerre, Legendre, Hermite, Bessel, and whatnot - but I made only a passing acquaintance ...

**1**

vote

**0**answers

229 views

### solving a sum of Hypergeometric function 2F3

I am trying to find a closed form solution for the two sums given by
$$\sum^n_{k=0}\frac{y^k}{k!}(-a+n)_k \left(\frac{2}{z} \right)^k {}_2F_3\left({-\frac{k}{2}, \frac{1 - k}{2}}; {-k, -a+n,1+a-k-n}; ...

**3**

votes

**2**answers

451 views

### Product of Hypergeometric Functions

I am looking for the product of Gaussian hypergeometric functions of the form
$_2F_1(a,b;c;\lambda z) _2F_1(d,e;f;\phi z)$. I would like, but do not expect, an answer of the form ...

**3**

votes

**0**answers

577 views

### Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions?

I am trying to somewhat simplify a series, whose coefficients feature generalised hypergeometric functions ${}_1F_2(1;a,a+1;z)$. I was unable to find useful functional relations for this specific ...

**2**

votes

**1**answer

154 views

### Differential finiteness of power series with hypergeometric coefficients

Let $f$ be a formal (or convergent near the origin, as you wish) power series with variables $x_1$, ... , $x_n$ and complex coefficients.
$$ f = \sum_I f_I \mathbf x^I $$
This power series is said to ...

**3**

votes

**0**answers

107 views

### Proving non-negativity of a hypergeometric type sum

I am trying to prove the following inequality:
$$
\sum\limits_{k=0}^{m}\frac{(a)_k(a+\mu)_{m-k}}{(c)_k(c+\mu)_{m-k}}\binom{m}{k}(m-2k+\mu)\geq{0},
$$
where $(a)_0=1$, $(a)_k=a(a+1)\cdots(a+k-1)$ is ...

**0**

votes

**0**answers

189 views

### Convert 2F1 to polynomial

Is there any transformation to convert each of the following versions of ${}_2F_1$ to a polynomial?
The first one is
$${}_2F_1\left(\frac{1-a}{2}, -\frac{a}{2}; b;\frac{4z}{(1+z)^2} \right), \quad ...

**2**

votes

**1**answer

512 views

### Quadratic Transformation of the Hypergeometric Function 2F1

The function ${}_2F_1\Big(\frac{a-b}{2},\frac{a+b-1}{2};c;y\Big)$ can be transformed (as reported by A. Erdélyi) by the following formula
${}_2F_1\Big(\frac{a-b}{2},\frac{a+b-1}{2};c;y\Big)=
...

**7**

votes

**0**answers

309 views

### Polynomials with presumably positive coefficients

The $q$-Pochhammer symbol
$(q) _ 0:=1$ and $(q) _ n:=\prod _ {j=1}^n (1-q^j)$ for $n > 0$ is clearly a polynomial in $q$ which has both positive and negative coefficients when $n>0$.
The ...

**2**

votes

**1**answer

242 views

### is this bound on a kummer function known?

Is it already known that ${}_1F_1(a;b;x) \leq \Gamma(b)(1+|x|)^{-a}$ when $a$ is an integer, $a <0,$ and $b>0?$ If it is, what is a reference?
my proof:
Since the Kummer function can be ...

**0**

votes

**1**answer

569 views

### infinite series with Hypergeometric functions

Can we get a closed form for the series
$\sum^\infty_{k=0} \frac{ t^k}{k!} \Gamma(k+a)\Gamma(k+\frac{1}{2}){}_2F_1(k+a,k+\frac{1}{2};n+1,x)$
any hints or clues are welcomed.