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Trying to figure out if the double sum $ $ f(x,y)=\sum_{n=0}^{\infty}\sum_{m=n}^{\infty}\dfrac{(a)_n\,(b)_n\,(1)_m}{(a)_m\,(b)_m}\dfrac{x^n\, y^m}{n!\,m!}, \;\; \text{where} \;\; (z)_n=z(z+1)\cdots(z+n-1), $ $ is a special case of the generalized hypergeometric function (of two variables). For instance, the Kampé de Fériet function seems to have a similar form, but it looks like Mathematica doesn’t support this function yet, alas. ItRead more

Recently I am considering a geometric question, which is reduced to the following problem. Given $ L<0$ , let $ a\in [L/2,0]$ and $ b=L-a$ . For any $ c>0$ , let $ p,1-p$ solve $ $ x^2-x+c^2=0,$ $ and $ q,1-q$ solve $ $ x^2-x-c^2=0.$ $ Using Gauss hypergeometric functions, we define \begin{align*} f_a(c)&=(a^2-a)\left(\frac{F(1+p,2-p;2;a)}{F(p,1-p;1;a)}+\frac{F(1+q,2-q;2;a)}{F(q,1-q;1;a)}\right)\Read more

Trying to simplify the sum $ $ \sum_{n=0}^{\infty}\dfrac{z_1^n}{n!} {}_{1}F_{2}(1;a+n,1-a+n;z_2), $ $ where $ a\in(0,1)$ , $ z_1,z_2>0$ , and $ {}_{1}F_{2}$ denotes the appropriate version of the generalized hypergeometric function. The obvious Sum[(Subscript[z, 2])^n/n!*HypergeometricPFQ[{1}, {a + n, 1 – a + n},Subscript[z, 1]], {n, 0, Infinity}] does not work. Any suggestions? I suspect the resultRead more

I am attempting to compute the (Kummer’s) confluent hypergeometric function (see also here) \begin{align} {}_1F_1\left(\frac{n}{2};n +\frac{3}{2};z\right) \end{align} for real $ z\lt0$ and many consecutive values of integer $ n\geq0$ . Since the first two arguments scale with $ n$ at different rates, the usual recurrence relations don’t apply. Additionally, it does not seem to fitRead more

How to find indefinite integrals for the rational trigonometric functions of the form $ \Lambda^{pq}_m = \int \frac{\cos^p x\sin^q x}{(a+b\sin x +c\cos x)^m} dx$ where $ (p,q,m)>0$ using Mathematica? Any suggestions for preventing the results in terms of AppellF1 function and to obtain simpler summation functions for the ranges $ 0 \leq p \leq 5$Read more

Let $ 0<\alpha< 1$ . Can the function $ \qquad e^{-\alpha x}U(a,b,x),$ where $ U(a,b,z)$ is the hypergeometric U function be expressed as a Meijer-G function? The closest I found was here which corresponds to $ \alpha=1$ .Read more

I need to numerically compute the hypergeometric function $ $ _2F_1(k,1,c,z) $ $ where $ k$ is an integer, $ c>2$ is a real number and $ |z|<1$ , using the integral representation $ $ {}_2F_1[a,b;c;z] = {i \, \Gamma(c) \, e^{i\pi (b-c)} \over \Gamma(b) \Gamma(c-b) 2 \sin (\pi(c-b))} \int^{(1+)}_{0} t^{b-1} (1-t)^{c-b-1}(1-t z)^{-a} dt\,, $Read more