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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2answers
70 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 ...
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1answer
65 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 ...
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1answer
110 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 ...
4
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3answers
157 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} ...
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2answers
115 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 ...
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0answers
66 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 ...
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1answer
132 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
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1answer
40 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 ...
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0answers
99 views

On the Brioschi-like quintic $v^5 - 5d v^3 + 10 d^2 v - d^2 =0 $

The general quintic can be transformed in radicals using a rational Tschirnhausen transformation to the one-parameter Brioschi quintic, $$u^5 - 10c u^3 + 45 c^2 u - c^2 = 0\tag{1}$$ which can be ...
4
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1answer
74 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 ...
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1answer
97 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 ...
4
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1answer
149 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
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1answer
132 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 ...
10
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1answer
163 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: ...
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0answers
77 views

How to simplify this Kampé de Fériet function?

I was dealing with a convolution type integral $$ \int^z_0 t^m {}_0F_1(;1;-t) \: {}_2F_3\Big( 1,1;2,m,m+1 ; -a t\Big) \:\mathrm{d}t $$ By applying one of the identities in Exton's book, the solution ...
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1answer
121 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} ...
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2answers
188 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!} ...
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0answers
63 views

When do the even part and odd part of a hypergeometric like function have common nonnegative real root?

I encountered the following problem in my research project: Let $$f(x)=\sum_{k=0}^{\infty} a_{k} x^{k}$$ We can separate the even part $p(x^2)$ from the odd part $x q(x^2)$ and write ...
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0answers
72 views

Sum over Hypergeometric function 1F2

I would be very grateful for any ideas to find a closed form for the sum: $$ \sum^\infty_{k=0} \frac{z^k}{\Gamma(1+k) \Gamma(k+m+1)} {}_1F_2\left(1;1+k,m+k;z\right) $$ where $m\in\mathbb{N}$ and ...
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0answers
79 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 ...
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1answer
158 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
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1answer
228 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 ...
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2answers
164 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
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0answers
294 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
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2answers
311 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
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1answer
459 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} ...
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1answer
396 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 :)
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6answers
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 ...
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0answers
175 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
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2answers
346 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
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0answers
413 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
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1answer
144 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
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0answers
91 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
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0answers
174 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
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1answer
381 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)= ...
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0answers
295 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
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1answer
222 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
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1answer
486 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.
2
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2answers
391 views

Logarithm of a hypergeometric series

I am sorry if the answer to my question is well-known. I am quite new in this topic, so it will be also nice to have a reference, if it exists. I was wondered if there exists a nice closed formula ...
4
votes
3answers
505 views

Asymptotic bounds for a confluent hypergeometric function $F_{1}[;1;x]$

I know that for infinite series and $|z|<1$ there exists a confluent hypergeometric expression $ \sum_{k=0}^{\infty} \frac{z^k}{k!k!} = F_{1}[;1;z] $ This is not very helpful though, and I 'd ...
2
votes
0answers
220 views

Upper bounds on hypergeometric function 3F2

Are there any existing bounds on the hypergeometric function 3F2(-m,-n,1/2;1,1;4) where m,n>=1? Or any thoughts how one can be obtained? Most of the inequalities I've seen apply to the cases whose ...
6
votes
2answers
669 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
4answers
915 views

Closed-form for modified formal power series

This question have been driving me crazy for months now. This comes from work on multiple integrals and convolutions but is phrased in terms of formal power series. We start with a formal power ...
5
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1answer
795 views

A 2F1 Hypergeometric identity from a Feynman integral

Using two different approaches to evaluating the dimensionally regularized ($d=4-2\epsilon$ dimensional Euclidean space), equal mass ($x=m^2$), 2-loop vacuum Feynman diagram $$ \begin{align} I(x) ...
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0answers
488 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 ...
13
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5answers
2k 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 ...
29
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4answers
2k 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 ...
2
votes
1answer
244 views

hypergeometric closed form for z=1/4,-1/3

There exist the linear identities for the 2f1 hypergeometric function where z is either -1, 1, or 1/2 using the quadratic transdormations it is easy to derive new identities in terms of gamma ...
4
votes
2answers
366 views

Analytic continuation of $_4F_3(1)$

The Gauss theorem $${_2F_1}(a,b;c;1)=\frac{\Gamma(c-a)\Gamma(c-b)}{\Gamma(c)\Gamma(c-a-b)}$$ allows to compute the analytic continuation of ${_2F_1}(a,b;c;1)$ for $a+b>c$ when the series ...
5
votes
0answers
377 views

Relation between two hypergeometric series

EDIT: Context for this investigation can be found in one of my other MO posts, "Pythagorean Theorem for Right-Corner Hyperbolic Simplices?" -- I'm investigating a function that has led me to this ...