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 ...

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Prove that the relationship exists

This question sprung out from another post of mine that was in part by Semiclassical, he Proved the Following: $$ \sum_{n=0}^{\infty} {}_2F_1(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1)/n! = 2\pi ...
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Evaluating a series of hypergeometric functions

I would like to prove (or disprove) the following statement: $$ \sum_{n=0}^\infty \left[\frac{{}_2{\rm F}_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1\right)}{n!}\right] = \frac{\pi}{2} \left[ ...
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61 views

Closed form of the sequence ${_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right)$

Is there a closed-form of the following sequence? $$a_n={_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right),$$ where $_2F_1$ is the hypergeometric function and $n ...
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22 views

Multiplying the results of a hypergeometric distribution to get a total probability

For a (trading) card game I would like to determine the probability of a specific hand from a deck of cards. I can determine the probability of a single card occurring any number of times in an ...
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1answer
44 views

Integral of a function which yields a hyper-geometric function

Note that $n$ is an arbitrary constant. $$ \int(\sin^n(x))dx $$ I start by using the obvious integrating by parts and get: $$ \frac{d}{dx}[x\sin^n(x)] = \sin^n(x) + nx\sin^{n-1}(x)\cos(x) $$ $$ ...
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141 views

A Sine integral: problem I

Is it possible to demonstrate a solution for the integral \begin{align} \int_{0}^{\infty} x^{n} \, \sin\left( a x^{2} + \frac{b}{x^{2}} \right) \, dx \end{align}
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206 views

Prove ${_2F_1}\left(\begin{array}c\tfrac16,\tfrac23\\\tfrac56\end{array}\middle|\,\frac{80}{81}\right)=\frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$

I've found the following hypergeometric function value by numerical observation. The identity matches at least for $100$ digits. ...
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23 views

Gauss hypergeometric function with complex conjugate parameters

The Wolfram function site provides a wealth of information about special values of the Gauss hypergeometric function ${}_2F_1(a_1,a_2;b;z)$. It seems, however, that there aren't any information about ...
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59 views

What is the closed form of a certain sum

The following sum appears in a problem of Mathematical Epidemiology: $$P(m)=\sum _{p=0}^{\infty } \left( \sum _{q=0}^{\infty }2\,{\frac {\min \{ p,q \} {{\rm e}^{-m}} \left( \frac{m}{2} \right) ...
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21 views

Expression of the Runge function's derivative

I am trying to get the nth derivative of the Runge function i.e. i want : $$\dfrac{d^n}{dx^n} \dfrac{1}{1+25x^2}.$$ Mathematica gives me the answer : $$\dfrac{d^n}{dx^n} \dfrac{1}{1+25x^2}=\dfrac{n! ...
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27 views

Integral versus hypergeometric series: how to solve this?

How can I resolve the following indefinite integral using hypergeometric series? $$ \int (x^3 + 1)^\frac{1}{3} \,dx $$ Wolfram Alpha indicates that the series of Appell are used, but how to get to ...
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13 views

Monotonicity of Gauss Hypergeometry function

I have had some Gauss Hypergeometry functions (2F1(a,b;c;x)) involved in my work. For intuition, I wish to know the monotonicity (how does it change with x) of these functions. Specifically, is ...
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1answer
28 views

Show that this Markov chain is recurrent

So I have a Markov chain on the nonnegative integers such that, starting from $x$, the chain goes to $x+1$ with probability $p$, $0<p<1$, and goes to state $0$ with probability $1-p$. I'm ...
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42 views

Moment Generating Function of a Beta random variable.

After getting some excellent help on this problem in the statistics SE, I am reformuluating my question. Let me know if I should just delete it and ask a new one. Let $V$ be a $Beta(\alpha,1)$ ...
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3answers
276 views

The closed form of $\lim_{x\to\frac{4}{3}}\frac{\partial}{\partial x}\left[\,_2{\rm{F}}_1\left(\frac{1}{3},1;x;-1\right)\right]$

Do you think the following limit might have a closed form? Some hints or clues? $$\lim_{x\to\frac{4}{3}}\frac{\partial}{\partial x}\left[\,_2{\rm{F}}_1\left(\frac{1}{3},1;x;-1\right)\right]$$
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Integral of Hypergeometric Function with polynomial, power, exponential and logarithm function

I am trying to work out this integral. If there is no closed form, can you think of any approximations to it?
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34 views

A generalization of Clausen's formula

Clausen's formula, $${}_{2}F_{1}(a, b; c; x)^2 = {}_{3}F_{2}(2a, 2b, a + b; 2a + 2b, c; x), \quad c = a + b + \frac{1}{2},$$ is well known. Does anyone know if this formula has been generalized for an ...
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17 views

Evaluation of hypergeometric function when the argument is sum of two matrices

If $_pF_q\left(a_1,\cdots,a_p;b_1,\cdots,b_q;A+B\right)$ is a hypergeometric function whose third argument is a sum of two positive definite symmetric matrices, $A$ and $B$, then is there any way to ...
26
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806 views

Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$

I want to find a closed form for this integral: $$I=\int_0^1\frac{\ln^2x}{\sqrt{x^2-x+1}}dx\tag1$$ Mathematica and Maple cannot evaluate it directly, and I was not able to find it in tables. A numeric ...
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21 views

Representing a binomial expansion as a hypergeometric series

I just started learning about hypergeometric series and to start off with an example problem, I wanted to try to express the binomial expansion as a hypergeometric series. In specific, I wanted to ...
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1answer
38 views

MLE of Urn Without Replacement Problem

An urn has W white balls where W is unknown. Suppose that R=5 Red balls are added to the urn and then a random sample of N=10 balls is selected. But this time, Suppose that we continue to draw balls ...
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20 views

Hypergeometric Distrbution issue

So I am trying to figure out some probabilities using hypergeometric distribution, and I am a little confused on one part I am trying to figure out. I know how to figure out what the probability is ...
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17 views

How to express a hypergeometric confluent function in the pFq form?

I would like to implement the following function due to Erdeliy and need it as $_pF_q^{(\alpha)}(a,b;c,d)$ $\Phi_3=\Sigma_{m} \Sigma_{n} \frac{(\beta)_m}{(\gamma)_{m+n}m!n!}x^{m}y^{n}$, where the ...
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1answer
107 views

Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

I'm trying to find a closed form of this sum: $$S=\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}.\tag{1}$$ WolframAlpha gives a large expressions containing multiple ...
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53 views

Relations between the Eisenstein series and the hypergeometric series

It is known that $$E_4(\tau) = {}_{2}F_{1}\left(\frac{1}{12}, \frac{5}{12}; 1; \frac{1}{J(\tau)}\right)^4$$ and $$E_6(\tau) = {}_{2}F_{1}\left(\frac{1}{12}, \frac{7}{12}; 1; \frac{1}{1 - ...
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108 views

Prove: $\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3$

I'd like to evaluate the following definite integral: $$\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3.$$ ...
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19 views

Approximate $_2F_1(a,b;c;x)$ for large (maybe negative) values of $a, b, c$?

I need asymptotic approximations of the Hypergeometric function $_2F_1(a,b;c;x)$ for large positive values of $a, b, c$. Specifically, I need approximations for all the possible regimes, in which one ...
5
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63 views

Integral's Closed-form expression in terms of hypergeometric function

I want to solve the following integral: $$I = 2\left[\int_{0}^{1}\dfrac{y^m}{(1 - ay)^{m + 1}\sqrt{1 - y^2}}\mathrm{d}y+\int_{0}^{1}\dfrac{y^m}{(1 + ay)^{m + 1}\sqrt{1 - y^2}}\mathrm{d}y\right]$$ ...
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1answer
130 views

How to evaluate the integral $\int_{1}^{\infty} x^{-5/3} \cos\left((x-1) \tau\right) dx$

I would like to evaluate the following integral: $$\int_{1}^{\infty} x^{-5/3} \cos\left((x-1) \tau\right) dx$$ I get the Integral by Maple and it gives the Lommel function. After that, I will search ...
2
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46 views

Continued Fraction Expansion

While reading "Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, π, and the Ladies Diary " [p.602] from $F\left( -\dfrac {1}{2},-\dfrac {1}{2};1;\lambda^{2}\right)=1+\dfrac {\lambda ...
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Hypergeometric function ratios: $\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$?

I need a numerically stable way to compute the following ratio: $$\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)},$$ All the parameters are real numbers, with $a< 0$,$\ $ $b,c > 0$ and ...
13
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368 views
8
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1answer
100 views

Prove $_2F_1\!\left(\frac76,\frac12;\,\frac13;\,-\phi^2\right)=0$

Please help me to prove the identity $$_2F_1\!\left(\frac76,\frac12;\,\frac13;\,-\phi^2\right)=0,$$ where $\phi=\frac{1+\sqrt5}2$ is the golden ratio.
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1answer
59 views

Symbolic Integration involving hypergeometric functions

What's the best way to symbolically evaluate this integral? $$\frac{1}{\hbar}\int_{-\infty}^\infty e^{iux/\hbar}\Psi^{*}_n(p-u/2)\Psi_n(p+u/2)\,du$$ where: $$\Psi_n(p)=\frac{1}{(1+\alpha ...
20
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478 views
21
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413 views

Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$

I discovered the following conjecture numerically, but have not been able to prove it yet: $$_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47.\tag1$$ The equality holds ...
17
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1answer
297 views

Prove ${\large\int}_0^\infty\left({_2F_1}\left(\frac16,\frac12;\frac13;-x\right)\right)^{12}dx\stackrel{\color{#808080}?}=\frac{80663}{153090}$

I discovered the following conjectured identity numerically (it holds with at least $1000$ digits of precision). How can I prove it? ...
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29 views

a question on sum of q_binomials

I was trying to calculate something and at some point I get the following sum: \begin{equation} \sum_{t=0,t \text{ even}}^{s}{s+3n \brack s-t}\sum_{i = 0}^{t/2}q^{2i^2}{t/2+2n-i \brack t/2-i}{n ...
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388 views

Prove ${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$

I discovered the following conjecture by evaluating the integral numerically and then using some inverse symbolic calculation methods to find a possible closed form: $$\int_0^\infty\frac{\ln ...
7
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1answer
102 views

Derivative of a generalized hypergeometric function

Let $$f(a)={_2F_3}\left(\begin{array}c1,\ 1\\\tfrac32,\ 1-a,\ 2+a\end{array}\middle|-\pi^2\right).$$ How to find $f'(0)$ in a closed form?
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2answers
50 views

Evaluation of Indefinite Integral resulting in Hypergeometric Function

I am attempting to derive the result: $$ \int \left(1+x^n\right)^{-1/m}dx= x\,_2F_1\left(\frac 1m,\frac 1n;1+\frac 1n;-x^n\right)$$ First, I start off with the binomial expansion of the integrand to ...
3
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1answer
73 views

Does $\int { y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$ have a closed form

I am trying to solve the following indefinite integral $$F_Y(y) = \int {y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$$ Where $J_0$ is the Bessel function of the first kind. I tried ...
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1answer
59 views

Hypergeometric function representation

Is it possible to express the following sum in terms of the hypergeometric function $_2F_1$: $$ f(x) = \sum_{n=0}^\infty\frac{(-ax)^n}{n!~\Gamma(b-n)} $$ with $a$ and $b$ constant values ($x>0$ ...
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1answer
38 views

$\displaystyle k^{th}$ derivative of a Gaussian function with zero mean

The gaussian function is: $$f(x,\mu,\sigma)=\dfrac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\dfrac{(x-\mu)^2}{\sigma^2}\right)$$ Putting $\mu=0$, we can get the $\displaystyle k^{th}$ derivative of this ...
2
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2answers
80 views

Approximate formula for the series: $\sum_{k=1}^{+\infty}\dfrac{k^x}{(k!)^x}$

I found that this series: $$S(x)=\sum_{k=1}^{+\infty}\dfrac{k^x}{(k!)^x}$$ can be very well approximated in this way: $$S(x)=\dfrac{1}{\left(a+b\exp(cx)\right)^d}$$ with: $a=0.1876$, $b=-0.1895$, ...
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2answers
25 views

Can b=0 in the confluent hypergeometric function U(a,b,z)?

I am confused about the possible values of b in the confluent hypergeometric function of the second kind U(a,b,z). Specifically can b=0? I know that the U function can be expressed as $$U(a,b,z)=\pi ...
2
votes
0answers
38 views

Motivating Hypergeometric Series

Are there a few 'nice' or 'natural' ways to motivate the existence of the Hypergeometric series $$F(a,b;c:x) = 1 + \frac{ab}{c}x+\frac{1}{2!} \frac{a(a+1)b(b+1)}{c(c+1)}x^2+...?$$ Are there ...
0
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0answers
62 views

Integral $\int^\infty_{-\infty}\int^\infty_{-\infty}(\frac{(x-x_1)^2+(y-y_1)^2}{s_1^2}+1)^{-a_1-1}(\frac{(x-x_2)^2+(y-y_2)^2}{s_2^2}+1)^{-a_2-1}dxdy$

Under $x_i,y_i\in\mathbb R$, $s_i>0$ and $a_i>0$ for $i=1,2$, is there any good function to express the following integral? $$\int^\infty_{-\infty}\int^\infty_{-\infty} ...
2
votes
2answers
134 views

Compute the indefinite integral $I=\int y^{-a}(1−y)^{b-1} dy$ or $I=\int_{d}^1 y^{-a}(1−y)^{b-1} dy$

I need to calculate the indefinite integral $I=\int y^{-a}(1−y)^{b-1} dy$, where $a$, $b$ are REAL NUMBERS and $b>0$. (my goal is to determine the definite integral $I=\int_{d}^1 y^{-a}(1−y)^{b-1} ...
1
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0answers
37 views

About the integral forms of Appell Hypergeometric Function and Lauricella Functions

According to http://en.wikipedia.org/wiki/Appell_series#Integral_representations and http://en.wikipedia.org/wiki/Lauricella_hypergeometric_series#Integral_representation_of_FD, it is known that ...