Questions tagged [hypergeometric-functions]
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 many parts of mathematics.
311 questions
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Is this series for $1/\pi^2$ known?
Context
While working with $_4F_3(1/2,1/2,1/2,1/2;1,1,1;x^2)$, I have found
the following sum involving $\frac{1}{\pi^2}$.
$$
\sum_{n=0}^{\infty}\frac{(2n)!^4}{2^{8n}n!^8}\frac{8n^2+10n+3}{(n+1)^3}=\...
user574597
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Closed form for a hypergeometric sum
Sorry if this is too simple but, I came across the following sum
$$\sum_{k=0}^n (-1)^{n-k} \frac{(2(n-k)-1)!!}{(2k)!!}x^{2k}$$
where $n!!$ is the double factorial.
I am asking if it has some closed ...
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Solutions of Appell system of type $F_4$
The Appell's hypergeometric function of type $F_4$ is defined as:
$$
F_4 (\alpha,\beta,\gamma,\gamma'; z_1,z_2) = \sum_{m,n = 0}^{\infty}
\frac{(\alpha, m+n)(\beta, m+n)}{(\gamma,m)(\gamma',n)(1,m)(1,...
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Reducing a triple combinatorial sum to a single sum
I conjecture the following identity is true for $a,b,c$ nonnegative integers with $a$ even:
$$
\sum_{k,\ell,m}
(-1)^k
\frac{(k+\ell)!(a+b-k-\ell)!^2(a+b-m)!}{k!(a-k)!\ell!(b-\ell)!m!(c-m)!(a+b-k-\ell-...
- 23.7k
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votes
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Nonnegativity of an alternating combinatorial sum
Let $u,a,b,n$ be nonnegative integers such that $n\le a+b$.
Define the quantity
$$
L(u,a,b,n):=
(u+a+b-n)!\times\sum_{i,k,\ell}\
\frac{(-1)^k\ \ (u+a+b-i)!\ (k+\ell)!\ (a+b-k-\ell)!\ (u+a+b-k-\ell)!}...
- 23.7k
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Identities for hypergeometric functions
In my work I came across the hypergeometric function $_3F_2(a,b,c;a-b+3,a-c+3;1)$. Since I need to study the poles of this function, I would prefer to express it in terms of finite ratios of gamma ...
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On $q=z+2z^5+15z^9+150z^{13}+\cdots$
Let $K$ be the complete elliptic integral of the first kind:
$$K(k)=\int_0^1 \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}$$
and let $k'=\sqrt{1-k^2}$. Then the quantity $q=e^{-\pi K(k')/K(k)}$, also called the ...
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Upper bounds for Gegenbauer functions of non-integer degree
(Cross posted from MSE https://math.stackexchange.com/questions/5073844/)
Let $C_{\nu}^{\lambda} \colon (-1, 1] \to \mathbb{R}$ be the (normalized) Gegenbauer function of order $\lambda \geq 0$ and ...
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vote
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Confluence procedure from Kummer to Hermite-Weber
Let us consider the Gauss hypergeometric differential equation:
$$
x(1-x)y''+\{c-(a+b+1)x\}y'-ab y=0.
$$
For this equation, by setting $x\mapsto x/b$ and then taking the limit $b\to\infty$, we obtain ...
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Asymptotic behavior for Appell series
Could you please help me with finding literature?
Now I am working on Appell's double hypergeometric series $F_3(a,a';b,b';c;x,y)$ and I need to find any reference in literature dealing with the ...
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Does the hypergeometric function ${}_1F_2(n;1+\frac{n}{2},\frac{3}{2}+\frac{n}{2};-\frac{x^2}{4})$ have an elementary form, or other simplified form?
I am interested in an elementary or simplified form of the hypergeometric function $f(n,x)={}_1F_2(n;1+\frac{n}{2},\frac{3}{2}+\frac{n}{2};-\frac{x^2}{4})$ for integer $n\geq1$. I would be satisfied ...
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Can the diagram of relations between special functions be expanded more? [closed]
The problem is to expand a diagram of relations between special functions, similar to the one found in John D. Cook's blog, to include more functions and highlight relationships through Meijer G-...
user555829
4
votes
1
answer
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Hopf-algebraic approach to special functions
In my edition of Andrews', Roy's and Askey's "Special functions", they mention a certain new approach to hypergeometric series which they couldn't include into the book, and which deals with ...
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How to prove the identity $\sum_{k=0}^\infty(22k^2-92k+11)\binom{4k}k/16^k=-5$?
I have found the hypergeometric identity
$$\sum_{k=0}^\infty(22k^2-92k+11)\frac{\binom{4k}k}{16^k}=-5. \tag{1}$$
As the series converges fast, one can easily check $(1)$ numerically by Mathematica.
...
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Recursive formula on dimensions for integrating the Gaussian on d-dimensional hyperboloid
Are there any dimension reduction formulas or recursive formula on dimensions for integrating the Gaussian function on d dimensional hyperboloid by using an integration formula on a d-2 dimensional ...
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