Questions tagged [hypergeometric-function]
In mathematics, the Gaussian or ordinary hypergeometric function ${}_2F_1(a,b;c;z)$ is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.
6
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1answer
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An identity on $\small{}_pF_q\left(\left.\begin{array}{c} a_1+1,a_2+1,\dots ,a_p+1\\ b_1+1,b_2+1,\dots ,b_q+1\end{array}\right| z\right)$
I stumbled upon this relation while trying to answer this post. I was trying to find a relation between the two generalized hypergeometric functions,
$$A=\,_3F_2\left(\color{blue}{\tfrac12,\tfrac12},\...
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votes
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On $\sum_{n=0}^{\infty}{2n+3\choose n+1} \left(\frac{1}{2^n}\cdot\frac{3}{2n+1}\right)^4$ and Gieseking's constant
I. Intro
While trying to solve this post about the function,
$$F(k)=\sum_{n=0}^{\infty}{2n+3\choose n+1} \left(\frac{1}{2^n}\cdot\frac{3}{2n+1}\right)^k$$
for $k=3$, I found out Mathematica can ...
1
vote
0answers
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Rewriting Appell's Hypergeometric Function $F_1$ in terms of Gauss' Hypergeometric Function $_2F_1$
While going through David H. answer on What is $\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx$? I have encountered a step in between I do not really understand. Within the second ...
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Correction terms in the asymptotics of hypergeometric function
I am interested in obtaining the asymptotic expansion of $r(\rho)$ $($which is the inverse of $\rho$ below$)$,
$$\rho=\frac{2b}{1-q}\left(1-\left(\frac br\right)^{1-q}\right)^{1/2}\left(_2F_1\left(\...
0
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0answers
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What hypergeometric transformation rules might I apply to try to simplify a certain expression?
I have (https://mathematica.stackexchange.com/questions/189538/sum-a-certain-hypergeometric-function-based-expression-pertaining-to-an-integrat) a Mathematica expression involving the following six (...
2
votes
0answers
39 views
An identity for generalized hypergeometric function
I think the following identity is true,
$$
\frac{4 (4 s+9)}{3 \Gamma \left(s+\frac{5}{2}\right) \Gamma \left(s+\frac{7}{2}\right)}-\frac{16 (s+2)}{3 \Gamma (s+3)^2}=\frac{\, _3F_2\left(2,s+\frac{5}{2}...
4
votes
0answers
68 views
Hypergeometric identity
I was trying to solve this integral problem and I noticed something that may be true
$$
\int((1-x^r)^{1/r}-x)^{2 n} \mathrm dx
=
\frac{1}{2 n+1}\sum _{j=1}^{2 n+1} (-1)^{j+1} x^j \binom{2 n+1}{j} \, ...
2
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0answers
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Integral involving a Gaussian hypergeometric function and a rational function
Let $x_0,x \in (0,1/10)$ and define:
\begin{equation}
g(x):= F_{2,1}\left[\frac{1}{13},\frac{1}{17},\frac{1}{5}; 100 x^2 \right]
\end{equation}
Then the following identity holds true:
\begin{eqnarray}
...
2
votes
1answer
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Integral involving hypergeometric function
I've worked out the projection of a spherically symmetric power law volume density profile $\rho(r)=br^a$, i.e. its surface density $\sigma(R)$, and am now trying to integrate this in a series of ...
0
votes
0answers
13 views
Identities involving the Gaussian hypergeometric function
By applying the algorithm from Solving linear ordinary, 2nd order differential equations via global integral bases. to the five parameter family of ODEs defined in my first answer to Gauge ...
0
votes
1answer
41 views
How to find integration of function, in form of hypergeometric function, given below?
I would like to prove the left side to right hand side which is in form of hypergeometeric function. Looking for your hints, suggestions and solultions.
$$ \alpha_{1} \int_{0}^{1} (1-z)^{\alpha_{1}+\...
-1
votes
1answer
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which one is worse in terms of probability
There are 2N white balls and N red balls (all balls are same except for the color), to put into K different boxes, such that every box contains 3N/k balls. We say event A happens, if any box has more ...
1
vote
0answers
59 views
Functions whose input is the same as the output?
Given the Dedekind eta function $\eta(\tau)$ and complex number $\tau$. I came across these family of functions,
$${f_2(\tau)= \frac{i}{\sqrt{2}}\frac{\,_2F_1\left(\tfrac14,\tfrac34,1,\,1-\alpha_2\...
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votes
3answers
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Convergent or divergent series? Asymptotic of the series?
\begin{align}
A&=\sum_{n\ge 0} \frac{(-1)^n}{n!} \frac{b^n}{(n+1)^3}\\
&= \sum_{n=0}^{\infty}\frac{(-1)^{n}b^{n}}{n!}\frac{1}{(1+n)^{3}} = 1\,{}_{3}F_{3}(1,1,1;2,2,2;-b)
\end{align}
The series ...
0
votes
0answers
23 views
Hypegeometric differential equation
0
I was wondering if someone, a smart person, can tell me how to transform the following equations (see below) into the well-known Hypergeometric equations. I know that this can be done provided The ...
5
votes
1answer
133 views
Solving an integral equation with inverse Laplace transform
Let $\alpha,\beta,\mu>0$. I am looking for a solution, i.e. a function $g(x)$, that satisfies
$$
\frac{\beta^{\alpha}}{\Gamma(\alpha)}\int_0^\infty g(x)x^{\alpha-1}e^{-\beta x}\,\mathrm dx=\left(\...
1
vote
0answers
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bounds for hypergeometric 2F2 function
Looking for lower- and upper- bounds for the following case of the hypergeometric ${}_pF_q$ function:
$$
{}_2F_{2}(1,k;\;k+1/2+b,k+1/2-b;\;z),
$$
where $k\in\mathbb{N}$, $b\in[0,1/2)$, and $z\ge0$. ...
0
votes
0answers
19 views
Confused on deciding whether its Negative Binomial or Binomial and Negative Hypergeometric or Geometric
I am currently struggling on deciding whether a question is a Negative binomial or a Binomial, and Negative Hypergeometric or a Hypergeometric SRV (Special Random Variable), As I seem to always ...
2
votes
1answer
46 views
Is the hypergeometric function $_2F_1\left[\frac{1}{2},n+\frac{1}{2};n+1;z\right]$ expressible in terms of more elementary functions?
Is this special case of hypergeometric function expressible in terms of more elementary functions :
$$_2F_1\left[\frac{1}{2},n+\frac{1}{2};n+1;z\right]$$
It will also be helpful for me to know, if ...
0
votes
0answers
23 views
Integral involving power, exponential and Confluent Hypergeometric function from 1 to Infinity
I am trying to integrate the following
$\int_{1}^\infty e^{-a\,y}\,y^{b-2}\,W_{\kappa,\nu}(c\,y)\,dy$
where $a,b,\nu\,\in\Bbb R$, with $a,b,\nu>0$ and $c,\kappa\in\Bbb C$, with $Re(c)=0$, $Im(c)&...
1
vote
0answers
42 views
Sum of hypergeometric function
I am trying to evaluate the following sum
$$\sum_{n=1}^N {}_2F_1(-n,n-N,1,x) y^n $$
I notice that according to wolfram alpha,
$$\sum_{n=1}^\infty {}_2F_1(-n,b,c,x) y^n = \frac{_2F_1(1,b,c,\frac{...
2
votes
2answers
73 views
$a_n=(n-1)a_{n-1}+a_{n-2}$
In an answer to a recent question of mine, I was introduced to the recurrence relation $$a_n=(n-1)a_{n-1}+a_{n-2}$$
Where $a_1=0,a_2=1$. I have computational evidence that $a_n=a_{-n}$, as I have ...
0
votes
0answers
23 views
Laplace transform of generalized hypergeometric distribution
What is please the Laplace transform (moment generating function $M(t)$) of a generalised hypergeometric distribution shown below
$$p_X(x)=K\cdot\frac{(a_1)_x\dots(a_p)_x}{(b_1)_x\dots(b_q)_x}\cdot\...
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votes
0answers
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The accuracy of relations related to hypergeometric function
Kummer differential equation is:
$$xy^{''}+(b-x)y^{'}-ay=0$$
It is written in the handbook of mathematical functions(Abramowitz and Stegun 1972) that, this equation has 8 answers, the most general of ...
0
votes
1answer
98 views
Inverse of the asymptotic expansion of Gauss Hypergeometric function
I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of $\rho$ below). Basically I want to series expand $\rho$ for large $r$ (i.e. as $r\to \infty$) and then ...
0
votes
0answers
59 views
Evaluation of $\int_{e}^{\pi} \frac{dx}{4(x^2-1)- (x^2-1)^{\frac14}2^{-\frac14 }}$
I tried many ways to evaluate the below integral but i didn't succeed , I have used Hypergeometric functions but it fails , now My question is the below integral has known closed form ? and how i can ...
5
votes
0answers
62 views
What is this generalization of the Chebyshev polynomials?
For $\varepsilon>0$ consider the tridiagonal matrix
$$L_{\varepsilon}=\begin{bmatrix}
0 & 1 & \ & \ & \ & \ & \ & \ \\
1 & \varepsilon & 1 & \ & \ &...
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votes
0answers
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Is the hypergeometric function a simple resurgent function
In the study of Airy functions you obtain, by Borel transforming components of a transseries, a hypergeometric function $_2F_1(a,b,c|z)$ where the constant are positive. Now I would like to know if ...
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votes
0answers
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Tricomi confluent function with the negative argument
What is the relation between the Tricomi hypergeometric function with the negative argument and the Tricomi hypergeometric function with the positive argument or Kummer hypergeometric function? That ...
0
votes
1answer
59 views
hypergeometric function with special Pochhammer symbol
What is the relationship between these two hypergeometric functions? Can the following function be written as another function of some hypergeometric functions ?
$$1F1(a+b,2a,x)$$ and $$1F1(a+b,a,x)$$ ...
0
votes
1answer
38 views
Solve the equation similar to the Kummer equation
In my calculations, I arrived at the differential equation below. What should I do to solve this equation, which is similar to the Kummer equation? Can the Kummer equation be obtained by changing the ...
0
votes
0answers
48 views
Integral of squared Hypergeometric Function
I am trying to integrate the following
$\int_{0}^{1} {_2{F}_1}\big(-n,1+2m+n,1+m,1-z\big)^2 dz$,
where $m\in\Bbb Z$ and $n\in\Bbb Z$ with $m>0$, $n\geq 0$.
(Basically I want to normalise the ...
1
vote
0answers
21 views
an approximation to the generalized hypergeometric function
Relating to the article An approximation to the generalized hypergeometric function, I would like to calculated example of the Poisson distribution $Po(10)$. If you have the possibility, please see ...
1
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0answers
34 views
Tricomi confluent hypergeometric function
Let $$U(\alpha,\delta,z)=\frac{\Gamma(1-\delta)}{\Gamma(\alpha-\delta+1} {}_1\!F_1(\alpha,\delta,z) + z^{1-\delta}\frac{\Gamma(\delta-1)}{\Gamma(\delta)}\!F_1(\alpha-\delta+1, 2-\delta,z)$$
be the ...
0
votes
1answer
32 views
The convergence and reality of Gauss Hypergeometric Function
The Gauss Hypergeometric function is defined via a power series:
$$F(a,b,c;z)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}\frac{z^n}{n!}$$
In ...
1
vote
1answer
65 views
Closed form for $c_m = \sum_{n=|m|}^{\infty} \left(\dfrac{1}{2}\right)^{2n} (-1)^{m+n}{2n \choose m+n}$, $m$ integer
What is a closed form for
$c_m = \sum_{n=|m|}^{\infty} \left(\dfrac{1}{2}\right)^{2n} (-1)^{m+n}{2n \choose m+n}$
with $m$ an integer? (N.B. the starting index is an absolute value.)
I already ...
2
votes
1answer
53 views
How to get a formula for the definite integral mentioned below
I came upon the problem to get a formula depending on $k\ge 1$ for the definite integral $$\int_0^1 \frac{t^{2k}}{(1-t^2)^k} dt$$
Please note that 1 is a singularity of the integrand.
I tried on ...
2
votes
1answer
58 views
A connection between the confluent hypergeometric function and Bessel functions.
Let $b$ and $w$ be real parameters subject to $b\neq 1$. Let $x \in {\mathbb R}$.
Define:
\begin{equation}
{\mathfrak N}(w,b):= \frac{2^{\frac{1}{2 (1-b)}+1} \left(\frac{1}{b-1}\right)^{\frac{1}{2 (b-...
0
votes
0answers
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Do we have a formula to express $_3F_2(L+r-1+2q,1,r+2;L+r+2,r+3;1)$ in closed form?
Note: This is not a homework.
I am interested to calculate $_3F_2(L+r-1+2q,1,r+2;L+r+2,r+3;1)$ where $L,r\geq 1$ are integers and $q\in [0,1]$ is a real number.
In particular,
Question: Do we ...
6
votes
2answers
350 views
How to calculate the Kampé de Fériet function?
This is a continuation of this post.
The following is my original question in that post.
Question: Is it possible to express
$$\sum_{l=0}^\infty \sum_{r=0}^\infty\frac{\Gamma(L+r-2q)}{\Gamma(L+...
1
vote
0answers
21 views
Simplifying specific $\,_4F_3$ hypergeometric function?
Consider the hypergeometric function
$$\,_4F_3\left({{a,b+n,c-n,d}\atop{a+1,b,c}};1\right)$$
with $a,b,c,d\in\mathbb{C}$ and $n\in\mathbb{N}$. Is there any way to simplify it to bits and pieces that ...
1
vote
1answer
37 views
multiplying a hypergeometric series
We are able to calculate the value of the sum
$\sum_{k=0}^\infty \frac{(a_1)_k(a_2)_k\dots(a_p)_k}{(b_1)_k(b_2)_k\dots(b_{p-1})_k}\cdot\frac{x^k}{k!}$, which equals the generalised hypergeometric ...
6
votes
2answers
444 views
Simplify $\sum_{l=0}^\infty \sum_{r=0}^\infty\frac{\Gamma(L+r-2q)}{\Gamma(L+r-1+2q)} \frac{\Gamma(L+r+l-1+2q)}{\Gamma(L+r+l+2)}\frac{r+1}{r+l+2}$
This question is a continuation of this post.
Let $r,l,L\geq 1$ be integers.
Assume that $q\in [0,1]$ is a real number.
The authors obtained the following equation $36$ in their paper (I express in ...
2
votes
2answers
38 views
If $a,b,c$ are positive integers with $c\leq a+b,$ can I conclude that $_2F_1(a,b;c;1)$ diverges?
Recall that the hypegeometric series is defined by
$$_2F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}$$
where $z\in \mathbb{C}$ with $|z|<1$ and $(a)_n = a(a+1)...(a+n-1)...
0
votes
2answers
18 views
Probability Function Hypergeometric distribution
The question:
A shipment of 2500 car headlights contains 200 defective. You choose from this shipment without replacement until you have 18 which are not defective. Let X be the number of defective ...
1
vote
1answer
57 views
Understanding Mathematica's formula for $ \int_0^{\infty } x^a \exp \left(-\frac{c x^2+f x}{b}\right) \, dx $
My goal is to integrate the following function:
$$
\int_0^{\infty } x^a \exp \left(-\frac{c x^2+f x}{b}\right) \, dx
$$
where, $a, b, c > 0$ and $a, b, c, f \in \mathbb{R}$.
Mathematica gives me ...
0
votes
0answers
19 views
Is a hypergeometric sum the minimum of a “potential” function?
I'm wondering if values of a generalized hypergeometric function can be written as solutions to an optimization problem, like this:
$$_q F_p (a_1, \dots, a_p;b_1,\dots,b_q;x)=\min_{t}\psi(a_1, \dots, ...
2
votes
0answers
57 views
My attempt at $f''(x)+\sin(f'(x))=0$ does it work?
Here's an attempt at a fairly innocent-looking separable differential equation.
$$f''(x)+\sin f'(x)=0$$
$$\int\frac{f''(x)}{\sin f'(x)}dx=-(x+c)$$
let $y=f'(x)$ then $dy=f''(x)dx$
$$\int\csc y\,dy=-(x+...
0
votes
1answer
34 views
Computation of a sum involving gamma functions
Let $l$ denote a positive integer and $m$ be an integer $-l \leq m \leq l$. I would like to prove the following identity:
$$\sum_{0 \leq j \leq \left\lfloor\frac{l - m}{2}\right\rfloor}\sum_{0 \leq k ...
0
votes
0answers
19 views
hypergeometric function - express in terms of hypergeometric function of another variable
I am very new at this, so please be happy to correct my mistakes.
Suppose I have a Hypergeometric function $ {}_2F_1(a,b;c;z) $, where $z$ is the form
$ z = Ax^2 + Bx, \quad where \space A,B > 0 ...
