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
9
votes
1
answer
745
views
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 $\frac{1}{\pi^2}$.
$$
\sum_{n=0}^{\infty}\frac{(2n)!^4}{2^{8n}n!^8}\frac{8n^2+10n+3}{(n+1)^3}=\...
user574597
3
votes
4
answers
499
views
Closed form for a hypergeometric sum
Sorry if this is too simple but, I came across the following sum
$$\sum_{k=0}^n (-1)^{n-k} \frac{(2(n-k)-1)!!}{(2k)!!}x^{2k}$$
where $n!!$ is the double factorial.
I am asking if it has some closed ...
0
votes
0
answers
88
views
Solutions of Appell system of type $F_4$
The Appell's hypergeometric function of type $F_4$ is defined as:
$$
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,...
- 635
11
votes
2
answers
739
views
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:
$$
\sum_{k,\ell,m}
(-1)^k
\frac{(k+\ell)!(a+b-k-\ell)!^2(a+b-m)!}{k!(a-k)!\ell!(b-\ell)!m!(c-m)!(a+b-k-\ell-...
- 23.7k
9
votes
1
answer
1k
views
Nonnegativity of an alternating combinatorial sum
Let $u,a,b,n$ be nonnegative integers such that $n\le a+b$.
Define the quantity
$$
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
2
votes
1
answer
279
views
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 ...
- 21
19
votes
0
answers
406
views
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 ...
- 331
2
votes
0
answers
148
views
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 ...
- 329
1
vote
0
answers
73
views
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 ...
0
votes
0
answers
75
views
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 ...
- 1
6
votes
1
answer
336
views
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 ...
- 63
-1
votes
1
answer
185
views
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
267
views
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 ...
- 635
10
votes
1
answer
604
views
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.
...
- 18.1k
0
votes
0
answers
94
views
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 ...
- 439
13
votes
2
answers
2k
views
A curious hypergeometric series related to Riemann's zeta function
Last week I found the following curious hypergeometric identity:
$$\sum_{k=1}^\infty\frac{(-1)^k(560k^4-640k^3+408k^2-136k+17)}{(2k-1)^4 k^5 \binom{2k}k\binom{3k}{k}}=180\zeta(5)-\frac{56}3\pi^2\zeta(...
- 18.1k
4
votes
1
answer
558
views
How to prove this combinatorial equality on $\sum_{k=0}^{n} \binom{k}{m}(-1)^{k}$?
I have been trying to prove the equality
$$\sum_{k=0}^{n} \binom{k}{m}(-1)^{k} = \left(\frac{-1}{2}\right)^m[\text{$n$ is even}] + \frac{(-1)^n}{2}\sum_{k=1}^{m}\binom{n+1}{k}\left(\frac{-1}{2}\right)^...
- 59
7
votes
1
answer
363
views
Are there q-analogs of GKZ (Gelfand-Kapranov-Zelevinsky) hypergeometric differential equations?
GKZ (Gelfand-Kapranov-Zelevinsky) hypergeometric differential equations are generalisations of standard hypergeometric equations/functions and important for example for mirror symmetry. It is system ...
- 26.4k
2
votes
1
answer
118
views
On the vanishing of a polynomial with coefficients related to the Gauss hypergeometric function
I recently asked a question about resolving a conjecture I had on the solution set of the ODE
$$(a^2-1)Q''(a) + \left(3a-\frac{d-1}{a} \right)Q'(a) + \frac{3}{4}Q(a)=0,$$
obtained by looking for ...
- 946
3
votes
0
answers
200
views
A closed formula for a sum involving hypergeometric functions
Can we find a closed formula for this sum:
$$\sum_{p,q\geq 0} (p+q+1)r^{p+q} \frac{{}_1F_1(1+p;2+p+q;r^2)}{{}_1F_1(1+p;2+p+q;1)}$$
where
$$_1F_1(a;c;z) = \sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n n!} z^...
- 571
4
votes
0
answers
220
views
Solving the quintic using the eta quotients $\frac{\eta(\tau)}{\eta(2\tau)},\,\frac{\eta(\tau)}{\eta(3\tau)},\,\frac{\eta(\tau)}{\eta(4\tau)},$ etc?
I. Reduced quintics
The general quintic can be reduced to the one-parameter forms,
$$x^5+5x+\alpha=0\\[5pt]
x^5+5\alpha x^2-\alpha=0$$
for some generic alpha. The first is the Bring form and there are ...
- 13.1k
3
votes
0
answers
205
views
Analytic functions and Hyperfunction as TVS
I have several related questions on Analytic functions and Hyperfunction as topological vector spaces (I am mainly interested in questions 4,6,10):
For an open set $U\subset \mathbb C^n$ we can ...
- 2,679
5
votes
1
answer
300
views
A definite integral of a hypergeometric series related to the enumeration of fusenes
If my calculations are correct, the number of (not necessarily planar) fusenes of perimeter $2n$ grows asymptotically like $\mu^n$ where
\begin{equation}
\mu = \frac{6}{\int_0^{1/6} H(t)\mathrm{d}t} = ...
- 4,082
3
votes
2
answers
576
views
An Integral invoving products of modified bessel functions
I am a physicist working on a problem where the following integrals are concerned:
$$\int_0^\infty k^{l+1} e^{-p^2k^2}I_\mu(k)K_{l-\mu}(k) \, dk$$
$$\int_0^\infty k^{l+1} e^{-p^2k^2}(K_\mu(k))^2 \, dk,...
- 41
1
vote
0
answers
212
views
Hypergeometric sheaves on $\mathbb{A}^{1}_{E}$
Let $m, n$ be non-negative integers. Assume that $\boldsymbol{\chi} = \left( \chi_i \right)_{1 \leq i \leq m}$ and $\boldsymbol{\eta} = \left( \eta_j \right)_{1 \leq j \leq n}$ are two collections of ...
7
votes
1
answer
328
views
A contiguous ${}_3F_2(1)$ hypergeometric identity?
I stumbled upon a curious identity that seems to hold for all integers $n\ge3$:
$$\sum_{i=0}^{\lfloor n/3\rfloor}\prod_{j=1}^i{-{n-3j\choose3}\over{n\choose3}-{n-3j\choose3}}
={n\over3}\sum_{i=0}^{\...
- 1,017
0
votes
0
answers
125
views
Series expansion probably related to a modified Bessel function of the first kind
Recently, I came across the following series expansion
$$\sum_{k=0}^{\infty} \frac{(s+2k-1)!}{k!(s+k)!}\left(\frac{x}{2}\right)^{s+2k}$$
It looks similar to a modified Bessel function of the first ...
- 11
9
votes
1
answer
534
views
A hypergeometric series for $\Gamma(1/4)^4/\pi^3$
Sorry if this comes out of the blue. Looking at old notes of mine, I found the identity
$$\dfrac{\Gamma(1/4)^4}{\pi^3}=4+\sum_{n\ge0}\binom{2n+1}{n}^3\dfrac{1}{2^{6n+1}}\;.$$
I cannot remember how I ...
- 14.1k
5
votes
0
answers
374
views
Monstrous moonshine, Dedekind eta function, and the hypergeometric function
I. Monstrous Moonshine
Let $q = e^{2\pi i\tau}$ and $\tau = \sqrt{-d}$ or $\tau = \frac{1+\sqrt{-d}}2$ for positive integer $d$. Given the Dedekind eta function $\eta(\tau)$, consider the known ...
- 13.1k
1
vote
0
answers
227
views
On the Jacobi theta functions and the Borweins' cubic theta functions
The post has been divided into sections to show some patterns, as well as possible evaluations of,
$$_2F_1\big(s,1-s,1,z\big)$$
with $s = \frac12, \frac13, \frac14, \frac16$ for infinitely many ...
- 13.1k
0
votes
1
answer
316
views
Closed form of a Hypergeometric Function ${}_2F_1$ at $z=-8$
How this can be proved?
$$
E = {}_2F_1(-\frac{1}{2}, \frac{1}{3}, \frac{4}{3},-8) = \frac{6}{5} - \frac{\chi}{2}
$$
where
$$
\chi = \frac{6\sqrt{\pi}}{5}\frac{\Gamma(\frac{1}{3})}{\Gamma(-\frac{1}{6})}...
5
votes
1
answer
246
views
Do the zeroes of some hypergeometric functions interlace?
Confluent hypergeometric functions differing from $F={}_1F_1(a,b,z)$ by $\pm1$ in either parameter $a$ or $b$ are called contiguous to $F$. For rational $a, b$, assume I know $z_0$ is a zero of $F$. ...
3
votes
1
answer
221
views
What's the asymptotic behaviour of $_1F_1(a,b,az)$ when $a\to\infty$?
I'm working towards the solution to a problem about involving the Landau-Zener transition, but I'm finding some difficulties. I need to estimate $ \,_1 F_1\left(\frac{\mathrm i}{4\epsilon},\frac12;\...
- 97
2
votes
0
answers
139
views
A strange identity between generalized basic hypergeometric series
Calculating matrix elements in some quantum integrable system, I encountered a strange $q$-series identity for non-terminating basic hypergeometric functions $\phantom{i}_3\phi_2$. It comes from the ...
- 121
1
vote
1
answer
271
views
Proving a quadratic transformation of ${}_2F_1$
For context, please see this MSE post
I want to prove that
$${}_2F_1\left(a,b;a−b+1;z\right)=(1−z)^{1−2b}(1+z)^{2b−a−1}{}_2F_1\left[\frac{a−2b+1}{2},\frac{a−2b+2}{2};a−b+1;\frac{4z}{(1+z)^2}\right]$$
...
- 101
2
votes
1
answer
297
views
Hypergeometric function ratio inequality
Related to a recent question (Concavity of hypergeometric function ratio), I actually only need the following inequality:
$$
\frac{{}_2\mathrm{F}_1\big(\frac{1}{2},\frac{1}{2};c+1\,;x\big)}{{}_2\...
- 401
2
votes
1
answer
371
views
Concavity of hypergeometric function ratio
I would like to show that the function,
$$
f(x) = \frac{{}_2\mathrm{F}_1\big(\frac{1}{2},\frac{1}{2};c+1\,;x\big)}{{}_2\mathrm{F}_1\big(\frac{1}{2},\frac{1}{2};c\,;x\big)}
$$
is concave for $0 < x &...
- 401
0
votes
0
answers
95
views
Parabolic cylinder function ratio inequality
I would like to prove the following inequality involving the parabolic cylinder function $D_{-\nu}(x)$ for $\nu>0$:
$$
\frac{2\hspace{1pt}D_{-\nu+1}(x)}{D_{-\nu}(x)} \;<\; x+\sqrt{x^2+4\hspace{...
- 401
1
vote
0
answers
135
views
Expansion of Gaussian Hypergeometric Function for Large Third Parameter - new formula?
I am looking at the asymptotic expansion of the Gaussian hypergeometric function for large third parameter ($\lambda \rightarrow \infty$), which is given by
$$
_{2}F_{1}(a,b;\gamma+\lambda;z)\ \...
- 1,343
2
votes
1
answer
301
views
Incomplete integral of confluent hypergeometric function
I would like to prove the following: if $a,b,c > 0$, $0<x<1$, $y>0$, then,
$$
\frac{1}{\Gamma(b)}\int_0^{\infty} s^{b-1} e^{-s} \,\mbox{$ {}_1\mathrm{F}_1 (a\,; c\,; x s + y\hspace{1pt})$}\...
- 401
11
votes
0
answers
485
views
How are the hypergeometric motives of WZ-Pairs connected?
If $\small{(F,G)}$ is a WZ-pair and general asymptotic conditions $\lim_{k\rightarrow\infty}\small{G(n,k)=0}$ and $\lim_{n\rightarrow\infty}\small{F(n,k)=0}$ hold, then we have the certified ...
- 3,291
4
votes
0
answers
204
views
Congruences between hypergeometric functions coming from modular forms
Consider the formal power series
$${}_2F_1(1/12,5/12;1;z)^{24}-{}_2F_1(1/12,7/12;1;z)^{24}=-z/3-z^2/2-(320293/559872)z^3-\cdots$$
It follows from a theorem on modular forms that for $p\ge5$ the ...
- 14.1k
0
votes
0
answers
231
views
Parameter derivative of the confluent hypergeometric function
Let $X$ follow a normal distribution with mean $\mu$ and variance $\sigma^2$. Mathematica gives
$$
E[\log|1+X|]=\frac{1}{2}\biggl(-\gamma-\log 2+2\log\sigma-\frac{\partial}{\partial a}{}_1 F_1\biggl(0,...
- 73
2
votes
2
answers
332
views
Behavior of the hypergeometric function near x=1
It is known that the hypergeometric function ${}_2F_1(a, b, c; x)$ defined by the series
$$\sum_{n=0}^\infty \frac{a(a+1)\cdots (a+n-1)\cdot b(b+1)\cdots (b+n-1)}{c(c+1)\cdots (c+n-1) n!}x^n$$
behaves ...
- 393
2
votes
2
answers
592
views
How did Ramanujan find $\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}?$
The formula
$$\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}$$
(in older notation) appears as eq. 38 in Ramanujan's paper Modular equations ...
- 317
1
vote
0
answers
209
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
$...
- 259
0
votes
1
answer
503
views
How to write Tricomi's confluent hypergeometric function in terms of Meijer-G function
I am calculating a closed form expectation and I encountered the Tricomi's confluent hypergeometric function
(aka confluent hypergeometric function of the second kind) given by integral $U\left( a,b,z ...
- 111
1
vote
1
answer
182
views
Factorial series $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$ and hypergeometric functions
For positive integer $D$, define $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$.
For $D \le 6$, sage finds closed form in terms of hypergeometric functions
at algrebraic arguments and fails to find closed ...
- 25.8k
2
votes
0
answers
101
views
When can GKZ setup encompass HMS?
Are there any instances when the Landau-Ginzburg superpotential describing the mirror of a smooth projective Fano variety $X_\Sigma$ is encompassed by a GKZ hypergeometric system? In some sense I am ...
8
votes
1
answer
1k
views
Why don't Zeilberger and Gosper's algorithms contradict Richardson's theorem?
Richardson's theorem proves that whether an expression A is equal to zero is undecidable. A is in this case an expression, constructed from $x,e^x,\sin(x)$ and the constant function $\pi$ and $\ln(2)$ ...
- 157