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 ...

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3answers
135 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 ...
0
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0answers
40 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. ...
5
votes
4answers
251 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} ...
14
votes
2answers
1k views

Binomial supercongruences: is there any reason for them?

One of the recent questions, in fact the answer to it, reminded me about the binomial sequence $$ a_n=\sum_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2, \qquad n=0,1,2,\dots, $$ of the Apéry ...
9
votes
8answers
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 ...
2
votes
0answers
40 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
0answers
59 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
1answer
123 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 ...
4
votes
2answers
116 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 ...
31
votes
5answers
3k views

Groups, quantum groups and (fill in the blank)

In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...
12
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1answer
230 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 ...
3
votes
0answers
134 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 ...
18
votes
4answers
1k views

Are the q-Catalan numbers q-holonomic?

The generating function $f(z)$ of the Catalan numbers which is characterized by $f(z)=1+zf(z)^2$ is D-finite, or holonomic, i.e. it satisfies a linear differential equation with polynomial ...
6
votes
1answer
213 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 ...
2
votes
1answer
104 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 ...
23
votes
6answers
3k views

Why are hypergeometric series important and do they have a geometric or heuristic motivation?

Apart from telling that the hypergeometric functions (or series) are the solutions to the (essentially unique?) fuchsian equation on the Riemann sphere with 3 "regular singular points", the wikipedia ...
0
votes
0answers
136 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
0answers
200 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
0answers
63 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
3answers
716 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
0answers
65 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; ...
1
vote
3answers
201 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 ...
6
votes
1answer
216 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 ...
1
vote
1answer
198 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 ...
3
votes
1answer
88 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 ...
26
votes
0answers
1k views

Curious $q$-analogues

Consider the Fibonacci polynomials $$F_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \right\rfloor }\binom{n-j}{j} x^{n - 2j} $$ and the Lucas polynomials $$L_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} ...
1
vote
0answers
545 views

Limit of two hypergeometric functions (2F1)

Hi, Does anyone know whether there is a known function/distribution that corresponds to the limit: $\lim_{\epsilon\rightarrow0^+} \mathfrak{Re}\left[f(x+i\epsilon) - f(x-i\epsilon)\right]$ when ...
1
vote
0answers
87 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 ...
6
votes
2answers
701 views

Closed form or/and asymptotics of a hypergeometric sum

Dear mathematicians, I am a computer scientist wandering in the deep sea of combinatorics and asymptotics to pursue a recent interest in average case analysis of algorithms. In doing so, I designed ...
2
votes
1answer
260 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}} ...
2
votes
1answer
65 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 ...
2
votes
1answer
155 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 ...
1
vote
1answer
112 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 ...
3
votes
2answers
437 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 ...
11
votes
1answer
210 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
1answer
128 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
2answers
216 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
0answers
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
1answer
226 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
1answer
235 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
2answers
210 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
0answers
434 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
2answers
408 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
1answer
532 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
1answer
481 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 :)
1
vote
0answers
218 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
0answers
554 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
1answer
149 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
0answers
106 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
0answers
187 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 ...