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{equati …

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!} =\frac{(-m …

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 …

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 …

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 equa …

The confluent hypergeometric function ${}_1F_1(a;b;z)$ has a natural partner in the Tricomi function $U(a,b,z)$, which provides a second, linearly independent solution to the confl …

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 …

I'm having a bit of trouble with the integral $$ \int_0^1 e^{-\frac{z^2}{2}u}\frac{u^{m-\frac{1}{2}} (1-u)^{n/2} }{ \left(1+ \left(s^2-1\right)u\right)^{m+\frac{n}{2}+1}} du. $$ ( …

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 are …

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],[]] …

In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometri …

Define an elliptic modulus $k_n$ such that, $$\frac{K'(k_n)}{K(k_n)}=\sqrt{n}$$ This is the case $m=2$ of the more general, $$\frac{\;_2F_1(\frac{1}{m},1-\frac{1}{m},1,1-x)}{\;_ …

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 …

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 p …

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 mo …