Questions tagged [hypergeometric-function]
Hypergeometric functions often refer to a family of functions ${}_p F_q$ represented by a corresponding series, where $p,q$ are non-negative integers. The case ${}_2F_1(a,b;c;z)$ is a special case of particular importance; it is known as the Gaussian, or ordinary, hypergeometric function.
1,590 questions
22 votes
3 answers
499 views
Does $\sum_{n=0}^{\infty}\frac{(4n)!^2}{2^{8n}(n+1)^2(2n)!^4}$ has the proposed closed form?
Context While working with the $_4F_3(a,b,c,d;e,f,g;x)$ I arrived to: $$S=\frac{64}{9}\left(\hspace{.1cm} _4F_3(-\frac{3}{4},-\frac{3}{4},-\frac{1}{4},-\frac{1}{4};-\frac{1}{2},-\frac{1}{2},1;1)-1\...
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1 vote
0 answers
52 views
Reduction formula or Whipple/Bailey-type transformation for a double terminating, double balanced Kampé de Fériet function $F_{1:2;2}^{1:3;3}(1,1)$
Context: I have a double terminating, double balanced Kampé de Fériet function: $$F_{1:2;2}^{1:3;3}\left[\left.\begin{matrix} M:\:& A,\;B,\;C\; &;& F,\;G,\;H\\ N:\:& D,\;E\; &;&...
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2 votes
2 answers
113 views
Closed form/finite sum solution to this infinite sum of a Hypergeometric Function?
I've come across the following infinite sum involving a hypergeometric function: $$\left(\frac{1-\theta-\phi}{\sqrt{1-4\theta\phi}}\right)^n\sum_{h=0}^{\infty}\binom{n+h-1}{h}\left(\frac{2\phi}{1+\...
4 votes
0 answers
134 views
How to prove this special case of hypergeometric function identity
I'm trying to see if there are good closed form expressions for $$ \, _2F_1\left(-\frac{1}{2} (2 \nu +1),-\frac{1}{2} (2 \nu +1);\frac{1}{2};z\right) $$ where $\nu \in \{0,1,2,3,\ldots \}$. Using ...
5 votes
1 answer
483 views
Pattern spotted for zeros of polynomials arising from hypergeometric roots
Let $k$ be an integer greater or equal three, and consider the $k-1$ roots of the hypergeometric equation \begin{equation} \, _2F_1\left(1-k,1;\frac{3}{2};x+1\right)=0. \end{equation} Now take these ...
- 109
4 votes
0 answers
118 views
How to solve the second derivative of the following $_3F_2$ hypergeometric function
This problem originated from the solution of following integral: $\displaystyle \begin{aligned}\int_{[0,π]^2}^{}{\ln^2\left(a-\cos x-\cos y\right)\ \mathrm{d}x\ \mathrm{d}y}\end{aligned}$ let $\...
- 437
2 votes
0 answers
112 views
Deriving a hypergeometric identity with argument $1/4$
I have seen this identity in several papers, but no derivation. $ {}_2F_1\!\left(\tfrac{1}{2},\, b;\, \tfrac{5}{2} - 2b;\, \tfrac{1}{4}\right) = \frac{2^{2b} \sqrt{\pi}}{3} \cdot \frac{\Gamma\!\left(...
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5 votes
1 answer
108 views
On reducing this Meijer G-function
One can prove rather straightforwardly, by Mellin transforms, that $$I=\int\limits_{0}^{\infty}\frac{J_{0}^{2}(t)J_{1}(t)}{t}\mathrm{d}t=\frac{1}{2\sqrt{\pi}}G^{1,2}_{3,3}\left(\left.\begin{matrix}\...
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0 votes
0 answers
48 views
A hypergeometric orthogonal rational function
Let $$f_n(x) = {}_3F_2 \left(\left.\begin{matrix} -n, n+2k+2a,k+1 \\ 2k+1,k+a+1+i x\end{matrix}\right| 1 \right),$$ and also $$w(x) = \left| \frac{\Gamma(a+ix)}{\Gamma(a+k+1+ix)} \right|^2.$$ For $k \...
4 votes
0 answers
159 views
Hypergeometric function identity at specific point
How to prove $_2F_1\!\left( -\frac34, \frac34; \frac 14; \frac {\sqrt 3} 2 \right) = \sqrt {\frac {\sqrt 3 - 1} 2} - \frac {\Gamma^2(\frac 14)} {\sqrt \pi 2^{3/4} 3^{5/8}}$? (both approximately equal ...
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2 votes
1 answer
85 views
Inverse Laplace transform of the Gaussian hypergeometric function $_{2}F_{1}(a,b,;c;x)$
I want to calculate the inverse Laplace transform of the Gaussian hypergeometric function $_{2}F_{1}(a+p,b,;c;-\omega)$ in which $p$ is the Laplace variable. The inverse Laplace transform is given by $...
- 1,998
6 votes
0 answers
684 views
About Legendre's like relations generalizations.
Context Being: $$K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2{t}}}=\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{(2n)!^2k^{2n}}{2^{4n}n!^4},\hspace{.5cm}k \in(0,1)\tag{1}$$ and: $$E(k)=\int_{0}^{\pi/2}{\...
- 1,411
0 votes
0 answers
63 views
Asymptotic expansion for third Appell funcion
Edit. Hello, I would like to ask, if there is a literature, or peer reviewed paper, regarding asymptotic expansions for Appells’s function. I've tried it myself and I came up with $F_3\left(\begin{...
- 1
2 votes
1 answer
127 views
Closed form or representation as a hypergeometric series of $\sum _{k=1}^{\infty } \frac{x^{k-1}}{B\left(k,-\frac{k}{n}\right)}$
Working on a problem I came across this series, \begin{equation} \frac{1}{2}+(\frac{n-1}{n})\sum_{k=1}^{\infty}\frac{x^{-\frac{k(n-1)}{n}}}{k(k+1)B\left(k,-\frac{k}{n}\right)} \end{equation} Removing ...
3 votes
2 answers
135 views
Upper bounding the Gauss hypergeometric function outside the unit disk $z=4$
I would like to upper bound the Gauss hypergeometric function with following inputs: ${}_2F_1\left( \frac{a}{2}, \frac{a+1}{2}; b; 4 \right)$ where $a\leq 0$ is an integer and $b\geq 1$ is also an ...
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