**32**

votes

**4**answers

2k views

### Integer-valued factorial ratios

This historical question recalls
Pafnuty Chebyshev's estimates for the prime distribution function. In his derivation
Chebyshev used the factorial ratio sequence
$$
...

**29**

votes

**4**answers

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 ...

**26**

votes

**0**answers

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} ...

**21**

votes

**6**answers

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 ...

**16**

votes

**3**answers

1k views

### Recursions which define polynomials

There are many examples (Somos sequences, special polynomials related to rational solutions of the Painleve equations) when a recurrence relation, which a priori produces a sequence of rational ...

**15**

votes

**4**answers

971 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 ...

**13**

votes

**1**answer

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 ...

**11**

votes

**1**answer

194 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:
...

**11**

votes

**2**answers

753 views

### Positivity of sequences via generating series

There are different ways of showing that a given sequence $a_0,a_1,a_2,\dots$
of integers, say, is nonnegative. For example, one can show that $a_n$ count
something, or express $a_n$ as a (multiple) ...

**10**

votes

**0**answers

144 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 ...

**9**

votes

**7**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 ...

**7**

votes

**3**answers

1k views

### Recent work on hypergeometric functions

Does anyone know of a monograph/survey on the modern history of (basic or elliptic) hypergeometric functions and their applications?
I haven't had much time to search the literature, and because it ...

**6**

votes

**3**answers

684 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 ...

**6**

votes

**2**answers

684 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 ...

**6**

votes

**1**answer

831 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) ...

**6**

votes

**1**answer

199 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 ...

**6**

votes

**1**answer

186 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 ...

**6**

votes

**0**answers

302 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 ...

**5**

votes

**0**answers

382 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 ...

**4**

votes

**3**answers

183 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} ...

**4**

votes

**3**answers

546 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 ...

**4**

votes

**2**answers

369 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 ...

**4**

votes

**1**answer

81 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 ...

**4**

votes

**0**answers

379 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) ...

**3**

votes

**1**answer

932 views

### A (known?) hypergeometric identity

Incidentally I've obtained a hypergeometric identity that I've not seen before:
$${}_3F_2(-m,-n,m+n; 1, 1; 1) = \frac{m^2+n^2+mn}{(m+n)^2} {\binom{m+n}{m}}^2$$
So, I wonder if it is well-known and ...

**3**

votes

**1**answer

81 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 ...

**3**

votes

**1**answer

749 views

### Is there a closed form for this hypergeometric expression?

I am trying to compute the number of distinct ways a $4n$ $\times$ $4n$ chessboard can be colored black and white, with exactly half the squares black and half the squares white. By distinct, I mean ...

**3**

votes

**1**answer

505 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}
...

**3**

votes

**2**answers

392 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

54 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; ...

**3**

votes

**0**answers

80 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 ...

**3**

votes

**0**answers

469 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 ...

**3**

votes

**0**answers

102 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 ...

**2**

votes

**2**answers

426 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 ...

**2**

votes

**4**answers

995 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 ...

**2**

votes

**1**answer

251 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 ...

**2**

votes

**1**answer

146 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 ...

**2**

votes

**1**answer

391 views

### “Closed” form for Motzkin and related numbers

I wonder whether it is impossible to write the nth Motzkin number as a sum of a fixed number of, say, hypergeometric terms. To illustrate what I mean: $n!+(2n)!$ is not a hypergeometric term, but it ...

**2**

votes

**1**answer

199 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

**1**answer

53 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

**1**answer

142 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 ...

**2**

votes

**1**answer

232 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 ...

**2**

votes

**1**answer

435 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)=
...

**2**

votes

**1**answer

231 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 ...

**2**

votes

**0**answers

166 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 $n \in N \cup {0}$?

**2**

votes

**0**answers

225 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 ...

**1**

vote

**2**answers

200 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!}
...

**1**

vote

**1**answer

72 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 ...

**1**

vote

**1**answer

165 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 ...

**1**

vote

**1**answer

194 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 ...