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.
1,281
questions
0
votes
0
answers
61
views
${_3F_3}\left({2,a,a\atop 1,1+a,1+a};z\right)=e^z{_3F_3}\left({?,?,?\atop ?,?,?};-z\right)$?
I'm interested in a transformation of the form
$$
{_3F_3}\left({2,a,a\atop 1,1+a,1+a};z\right)=e^z{_3F_3}\left({?,?,?\atop ?,?,?};-z\right).
$$
Many times, hypergeometric functions of the form ${_nF_n}...
- 4,615
0
votes
0
answers
54
views
A Fourier series inverting $\frac{\tan(y)}y$?
$\def\k{\operatorname k}\def\W{\operatorname W}$
Using a Dirac $\delta(x)$ Fourier series and this post
$$\begin{align}
y\cot(y)=\frac1x\implies\frac1{\sec^2(y)-x}=\int_0^\frac\pi2\delta(\tan(t)-xt)dt-...
- 7,279
1
vote
0
answers
133
views
Sum with gamma value.
Context:
Reading this paper: https://arxiv.org/abs/1309.1140.
(On proving some of Ramanujan's formulas for $1/\pi$ with an elementary method). We find on page 26 that the author claims:
$$\sum_{n=0}^{\...
0
votes
0
answers
23
views
Question about the proof of Dougall's 7F6 formula by L.J.Slater
I am reading L.J.Slater"Generalized Hypergeometric Functions". I have a question about the proof of Dougall's 7F6(1) Formula in this book. Equation(2.3.4.3) is\begin{eqnarray}&&(1+a-...
- 1
1
vote
1
answer
180
views
Inverse Fourier transform of $\frac{1}{ib|\omega|^a - \omega}$
I would like to know if there might be a connection between the following inverse Fourier transform (with some physics conventions):
$$f(t) = \int_{-\infty}^\infty d\omega e^{i\omega t}\frac{1}{ib|\...
- 71
0
votes
0
answers
12
views
Hypergeometric Fox H-function representation with Mellin-Barnes-type contour integrals problem.
I need help with the double complex integral (Mellin-Barnes type). I am solving the following integral:
\begin{equation} \int_0^{\infty}x^{b-1}\mbox{}_1{\rm{F}}_1(a,b, -c x)e^{-s x^\delta}e^{-\...
- 51
0
votes
0
answers
25
views
Relation between HyperGeometric function and Gamma function
It seems numerically that this equality holds for $n_1,n_2 \in \mathbb{N}$ :
$\frac{\Gamma (n_1+1) \Gamma (n_2+1)}{\Gamma (n_1+n_2+2)} = \frac{1}{2^{n_1+n_2+2}} \left(\Gamma (n_1+1) \, _2F_1^{(reg)}\...
- 1
3
votes
2
answers
112
views
How to integrate $\int^1_{-1}\frac{\pi}2 e^{ix}\operatorname{sech}\left(\frac{\pi x}{2}\right)\text{ d}x$?
I have an integral I need to integrate, as follows
$$\int^1_{-1}\frac{\pi}2 e^{ix}\operatorname{sech}\left(\frac{\pi x}{2}\right)\text{ d}x$$
This is quite troublesome to do, since I cannot use ...
- 2,733
10
votes
1
answer
235
views
Evaluate $\int_{0}^{1}\frac{K(x)\ln\left(1-x^2\right)}{\sqrt{1-x^2}}\text{d}x,\int_{0}^{1}\frac{xK(x)^2\ln\left(1-x^2\right)}{\sqrt{1-x^2}}\text{d}x$
I am recently interested in integrals containing an elliptic integral $K(x)$, which is defined by $\int_{0}^{1} \frac{1}{\sqrt{1-t^2}\sqrt{1-x^2t^2} }\text{d}t$ for $|x|<1$ and $x$ is the elliptic ...
- 2,979
1
vote
1
answer
48
views
Limit of hypergeometric function when first three parameters are all large
I have encountered an interesting limit involving a particular parameterisation of the hypergeometric function. The function of interest to me uses the parameters $1 \leqslant k \leqslant n \leqslant ...
- 3,930
-1
votes
0
answers
52
views
Why can't I use the binomial distribution to model the probability of choosing k of n winning lottery numbers
I'm currently working through John Rice's Mathematical Statistics and Data Analysis. In Ch. 2 of his book, Rice uses the the hypergeometric distribution to model the probability of selecting $k$ of 6 ...
4
votes
1
answer
284
views
Evaluate a strange integral involving generalized hypergeometric function ${}_1F_2$
$$
\small I=\int_{0}^{\infty}x^2e^{-4x}
\left (x^{-2/3}\Gamma\left ( \frac13 \right )^2\,_1F_2
\left ( \frac{1}{3};\frac{2}{3},\frac43;x^2 \right )^2
-\frac{9}{4}x^{2/3}\Gamma\left ( \frac{2}{3} \...
- 2,979
0
votes
1
answer
41
views
Integration using hypergeometric function
I would like to calculate the definite integral
$\int_0^\infty \frac{dx}{x(x^a + x^{-b})}$.
From Wolfram Alpha, the indefinite integral is
$x^b\ _2F_1(1;b/(a+b);b/(a+b)+1;x^{a+b})/b$ Where $_2F_1$ is ...
- 33
0
votes
0
answers
47
views
Function $f$ subordinate to function $g$
I'm trying to study a research paper, during that i come across this word Subordinate, i searched this a lot on google but didn't get any satisfactory answer. I understand the definition they gave in ...
- 1,052
3
votes
0
answers
67
views
Integral of $\arccos$ function combined with exponontial function
Is it possible to calculate this integral?
$$\int_0^1 x^{j}\ e^{ax^2}\arccos(x)\ dx,\qquad a\in\mathbb{R}_+,\ j\in\mathbb{N}. $$
Although the integral looks neat and fairly simple in form. I tried ...
- 69
0
votes
0
answers
30
views
Solving hypergeometric like differential equations using Integral transforms.
It is known that for 2nd order differential equations with 3 singularities that are regular, can be converted into the hypergeometric differential equation.
As an example of this, we can see the first ...
- 53
0
votes
0
answers
43
views
Using Inverse Laplace Transform to solve a differential equation involving F(s), F(s+a),F(s-a) and others .
I'm trying to get the inverse Laplace transform of the following expression:
\begin{equation}
\begin{split}
2 F(s)=\frac{-\alpha ^2 A}{\left(-B -\alpha ^2 A -s^{2}\right)}F(s+2 \alpha )+\frac{-\...
- 53
1
vote
0
answers
25
views
Hypergeometric differential equation with fractional polynomial coefficient
I have a differential equation of the form
$$
y''(x)+P(x)y'(x)+Q(x)y(x) =0
$$
where $P$ and $Q$ are rational functions
$$
P(x)=\frac{c-2ia-(c+2b-2ia)x}{cx(1-x)}\\
Q(x)=\frac{(-1+(1-x)^{2/c})a^{2}+x(-b^...
- 19
8
votes
2
answers
443
views
Integral: $\int_{1}^{\infty} \operatorname{arctanh} \left(k\,\sqrt{\frac{x^2-1}{x^2}} \right)\, e^{-\alpha \, x \,} dx$
Is there any chance to find a closed form for these integrals?
$$I_1(k,\alpha)=\int_{1}^{\infty} \operatorname{arctanh} \left(k\,\sqrt{\frac{x^2-1}{x^2}} \right)\, e^{-\alpha \, x \,} dx$$
$$I_2(k,\...
- 189
2
votes
2
answers
64
views
Solve $\int_0^1\frac{x(x+1)^b}{\alpha-x}dx$ using hypergeometric functions
How can we solve this integral?
$$\int_0^1\frac{x(x+1)^b}{\alpha-x}dx$$
I used geometric series
$$=\int_0^1(x+1)^b\sum_{n=0}^\infty\left(\frac{x}{\alpha}\right)^{n+1}dx=\sum_{n=0}^\infty\int_0^1\left(\...
- 1,628
2
votes
1
answer
39
views
How to merge odd series and even series of hypergeometric function of Legendre polynomials into one hypergeometric function?
On the Wolfram MathWorld page of Legendre Differential Equation, Legendre polynomials are represented as
$$
P_l(x) = c_n
\begin{cases}\begin{align*}
&_2F_1\left(-\frac{1}{2}(l), \frac{1}{2}(l + 1);...
- 282
2
votes
0
answers
34
views
Integral representaion of confluent hyper-geometric function
How can I prove these two integrals are equivalent ($\alpha,\beta, z>0$):
\begin{equation}
\frac{1}{\Gamma(\beta)}\int_0^\infty t^{\beta-1}(1+t)^{-\alpha}e^{-zt} dt,
\end{equation}
and
\begin{...
- 73
2
votes
0
answers
115
views
Asymptotics of ratio of q-Pochhammer symbols
In (9.23) of https://arxiv.org/abs/1908.08875 , they claim that it is "easy" to show the following limits $$\lim_{y\to 0}\frac{(yq^\frac{2-r+m}{2};q)_\infty}{(y^{-1}q^\frac{r+m}{2};q)_\infty}...
- 81
2
votes
1
answer
77
views
Converting a differential equation to a hypergeometric equation
I would like to transform the following differential equation into a hypergeometric equation:
${z^2}\frac{{{d^2}W}}{{d{z^2}}} + z\frac{{{d^{}}W}}{{d{z^{}}}} + \left( {{A^2} - {B^2}\left( {1 - {C_{}}z ...
2
votes
0
answers
19
views
Which numerical quadrature is the most suitable for an integral that integrates an exponentially increasing function?
I have an integral function of the form
$$
F_n(x) = \int_0^x du \left( K_n(u)L_n(x) - K_n(x)L_n(u) \right)u^p K_n(u)
$$
where
$$
K_n(x) = x^{(m+1)/2}e^{-x/2}M(a_n,m+1,x), \\
L_n(x) = x^{(m+1)/2}e^{-x/...
- 357
1
vote
1
answer
73
views
Integration of Gauss ${}_{2}F_{1}$ hypergeometric function
The indefinite integral representation of Gauss hypergeometric function is
$$\int {{}_2{F_1}\left( {a,b;c;z} \right)dz = \frac{{c - 1}}{{\left( {a - 1} \right)\left( {b - 1} \right)}}} {}_2{F_1}\left( ...
1
vote
1
answer
83
views
The integral of $(1+x^2)^{-a} \ln (1+x^2)$
Consider the integral $$ \int_0^{\infty} (1+x^2)^{-a} \ln (1+x^2) dx$$ where $a > 0$ is big enough. Is it expressible in terms of some known special functions (gamma function, hypergeometric ...
- 1,053
1
vote
0
answers
86
views
Reduction of $_3\text F_2(a,a,1-b;a+1,a+1;x)$ with the hypergeometric function
A derivative of the incomplete beta function $\text B_x(a,b)$ uses hypergeometric $_3\text F_2$
$$\frac{d\text B_x(a,b)}{da}=\ln(x)\text B_x(a,b)-\frac{x^a}{a^2}\,_3\text F_2(a,a,1-b;a+1,a+1;x)$$
Now ...
- 7,279
1
vote
2
answers
124
views
How to evaluate $\int\frac{x^ndx}{\sqrt{ax^2+bx+c}}$ for natural $n$
How do we evaluate the following integral?
$$\int\frac{x^ndx}{\sqrt{ax^2+bx+c}}$$
I am strictly looking for a solution for natural $n$
I wanted to try a trig substitution
$$\int\frac{x^ndx}{\sqrt{ax^2+...
- 1,628
0
votes
0
answers
28
views
What values of $m$ does the elliptic integral of the first kind $\text F(x,m)$ or the Jacobi amplitude relate to a single hypergeometric series?
Similarly to:
Does the Incomplete Beta function have forms of Elliptic E besides $\frac14 \text B_{\sin^2(2x)}\left(\frac12,\frac34\right)=\text E(x,2)$?
we have simpler transformations for the ...
- 7,279
4
votes
1
answer
138
views
Asymptotic of $_3F_2(1, \frac{1}{2}+d+n, 1+c+n; 1+2c, 2+2n;1)$ as $n\to \infty$
Let $c,d$ be in a small neighborhood of $0$; I think the limit
$$\begin{aligned}&\quad \lim_{n\to \infty} 4^{-n} n^{3c-d} {_3F_2}(1, \frac{1}{2}+d+n, 1+c+n; 1+2c, 2+2n;1) \\
&= \lim_{n\to \...
- 17.7k
1
vote
0
answers
17
views
Solving second order ODE - Hermite polynomial and Kummer confluent hypergeometric function
I have the following system of two second order ODEs:
\begin{align}
(L -D)V+Q^{-1}g(x)=0
\end{align}
where
\begin{align}
L = \frac{1}{2} \sigma_{x}^{2} I \partial_{x}^{2}+(A - a_1 x) I \partial_{x}
\...
- 183
0
votes
1
answer
49
views
$\sum_{i=1}^{n} \frac {x^{2i-1}}{\sqrt{2i}}$ as polylogarithm
$$\sum_{i=1}^{n} \frac {x^{2i-1}}{\sqrt{2i}}$$
It is very clear for me that it has to be polylogarithm function but as it is partial sum I tried to split the series as
$$\sum_{i=1}^{\infty} \frac {x^{...
- 3
1
vote
1
answer
86
views
How to find a closed form expression for the integral of an exponential divided by a polynomial over an infinite range?
I am trying to find a general solution of integrals with the following form
$$I= \int_{-\infty}^{\infty}\frac{e^{-(x-\mu)^2/\sigma^2}}{(1+a x^2)^b} dx$$
where $\{a,b,\mu, \sigma\}\in\mathbb{R}$, $a,b,\...
- 107
1
vote
1
answer
64
views
How to simplify RMM limit of integral?
This was posted a while back as an analysis problem from Romanian Mathematical Magazine. It is prove that $\lim _{n\to \infty }\frac{4^n}{\sqrt{n}}\int _0^1\frac{\sqrt{x^2+\frac{1}{e}}}{\left(x^{\frac{...
- 45
0
votes
0
answers
30
views
Proving that a Fuchsian differential equation has solutions in terms of the hypergeometric function.
I am trying to prove that some kind of fuchsian differential equations have solutions on the form of trygonometric functions.
What we know
From the book,"Linear Differential Equations and Group ...
- 53
3
votes
0
answers
154
views
$\sum\limits_{n=1}^\infty\frac{(-1)^\frac{(5+\sqrt3i)n}6x^n}{n!}(\frac ns )^{(n-1)}\,_2\text F_1\left(1-n,-sn;2-sn;s\right)$ closed form or integral.
From:
Completing the explicit $\lim\limits_{c\to0}\text I^{-1}_{cx}(r,c)$/ inverse $\int_0^x\frac{t^{r-1}}{1-t}dt,r\in\Bbb Q$ series expansion
here is a Lagrange inversion expansion with $r=\frac23$ ...
- 7,279
1
vote
0
answers
30
views
Solutions of 2nd order Fuchsian differential equations as Heun or Hypergeometric functions.
I have been trying to find a solution on any of these Fuchsian differential equations...
$v''(z)+\frac{A_1+B_1z }{z (z-1) }v'(z)+\frac{C_1+ D_1 z}{z^2 (z-1)^2 }v(z)=0$
$u''(z)+\frac{A_0+B_0 z^2 }{z(z-...
- 53
3
votes
1
answer
93
views
Laplace transforms of Appell functions F1, F2, F3, F4 relative to one of the two variables
Appell's two-variable functions $F_1, F_2, F_3$ and $F_4$ are known to have numerous uses in applied mathematics, notably mathematical Physics. I am looking for generalized Laplace transforms (if they ...
- 362
2
votes
0
answers
44
views
Generalization of Hermite polynomials for negative, fractional degree.
After solving a differential equation I've gotten a solution given as a linear combination of Hermite polynomials and confluent hypergeometric functions. The caveat is that the Hermite polynomials ...
- 31
0
votes
0
answers
45
views
Does this second order linear ODE admits a solution?
I am considering the following ODE in $x$:
$$x^2v''(x) + \alpha x v'(x) + \beta v(x)=0, $$
where $\alpha,\beta \in\mathbb{R}$ and one of the boundary conditions says $v(0)=\gamma>0$. If I consider ...
- 359
1
vote
0
answers
35
views
Can one express hypergeometric function with 1 as the first and -1 as the last argument by some factorials or gamma functions [closed]
As a result of integration in Mathematica I got:
\begin{equation}
_{2}F_{1}(1,\frac{1}{2}(2+n-r),\frac{1}{2}(2-n-r),-1)\quad n,r\in\mathbf{N}
\end{equation}
I try to express it as some factorials (as ...
- 11
1
vote
0
answers
38
views
Integral representation of ${}_3 F_2$ allowing negative integers
There exist many integral representations of the generalised hypergeometric function ${}_3 F_2$ but these are all assuming that the entries are positive integers. I have a problem involving,
$${}_3F_2 ...
- 1,676
2
votes
1
answer
101
views
Quotient of generalised hypergeometric series ${}_3 F_2$
I have a function, defined for $x>0$ as
$$f(x) = (1+x)^3 \cdot \frac{{}_3F_2\left(\left\{1,\frac{1}{2}+\frac{1}{2x},\frac{1}{2x}\right\},\left\{1+\frac{1}{2x},\frac{3}{2}+\frac{1}{x}\right\},1\...
- 82
1
vote
1
answer
60
views
Evaluate $\sum^{\infty}_{n=0} (-1)^n \frac{z^{4n+1}}{(4n+1)!}$
I want to evaluate $\sum^{\infty}_{n=0} (-1)^n \frac{z^{4n+1}}{(4n+1)!}$ but while doing my research, I noticed that
\begin{align*}
&\sin(z) = \sum^{\infty}_{n=1} (-1)^n \frac{z^{2n+1}}{(2n+1)!} \\...
- 1,596
1
vote
2
answers
69
views
Ideas/Methods for solving an non linear DE with non constant coefficients
I have to solve the following equation.
$2 A x^2 u''(x)-2 A x u''(x)+2 A x u'(x)-A u'(x)+B u(x)+\frac{C u(x)}{x}$
I have been thinking in a way of solving this since it is pretty complicated.
I tried ...
- 53
7
votes
1
answer
167
views
Transformation Identities of the $_2F_1$ function
From Wolfram Functions we have the following identities for the hypergeometric function $_2F_1$:
$$\begin{align}
_2F_1\left(a,c-b;c;\tfrac{z}{z-1}\right)&=(1-z)^a\,_2F_1(a,b;c;z)\tag1\\
_2F_1\left(...
- 15.9k
1
vote
1
answer
50
views
Solution of a hypergeometric equation
I am thinking the differential equation
$(1+e^x)\frac{d^2 y}{dx^2}+(2+e^x-\delta)\frac{dy}{dx}+y=0$.
Here $\delta$ is a parameter I want it to approaches to zero. I think this can be transformed into ...
- 41
3
votes
1
answer
86
views
A method for approximating a hypergeometric function to an elementary function
I found the following hypergeometric function as the antiderivative of some function in the interval where x is between 0 and 1.
${}_2F_1(-\frac{1}{k}, \frac{1}{k}; 1 + \frac{1}{k}; x^k)x$
Or by ...
- 101
3
votes
1
answer
45
views
Closed form for integer moments of Poisson random variable?
I'm interested in a closed-form (special functions included) for the integer moments of the Poisson distribution. Let $X\sim\operatorname{Poisson}(\lambda)$. Then we have for the moment generating ...
- 4,615