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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Definite integral of a hypergeometric function of an imaginary argument

How would one deal with such an integral? $$\int_0^\infty\frac{e^{-n r}}{r}{}_1F_1(i/k+1;2;2i kr) \, \mathrm{d} r$$ Here $F$ is the confluent hypergeometric function, $n\in\mathbb{N}$ and $k>0$ ...
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strange hypergeometric coefficients in Concrete Mathematics book

I don't understand how in this famous book they obtained hypergeometric coefficients for $$ \sum_{k\leq n} z^k \binom{n-k}{k}.\tag{5.74} $$ They say it is $\displaystyle F{-n,\ 1+2\lceil ...
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3answers
45 views

Hypergeometric 2F1 with negative c

I've got this hypergeometric series $_2F_1 \left[ \begin{array}{ll} a &-n \\ -a-n+1 & \end{array} ; 1\right]$ where $a,n>0$ and $a,n\in \mathbb{N}$ The problem is that $-a-n+1$ is ...
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Evaluate $ \int^\infty_0\int^\infty_0 x^a y^{1-a} (1+x)^{-b-1}(1+y)^{-b-1} \exp(-c\frac{x}{y})dxdy $

Evaluate $$ \int^\infty_0\int^\infty_0 x^a y^{1-a} (1+x)^{-b-1}(1+y)^{-b-1} \exp(-c\frac{x}{y})dxdy $$ under the condition $a>1$, $b>0$, $c>0$. Note that none of $a$, $b$ and $c$ is ...
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How to write the integral form of a Fox H function when n=p?

I have a questions regarding Fox H functions For a general Fox H function if $n=p$, when we write the integral form, does the term with the product from $p+1$ to $n$ disappear ?
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34 views

Proof that $\int_0^\infty t^\gamma e^{t/2} \operatorname{erfc}(\sqrt{t}\,) \,dt$ leads to hypergeometric function

I'm looking for the proof of $$\DeclareMathOperator\erfc{erfc} \int_0^\infty t^\gamma e^{t/2} \erfc(\sqrt{t}\,) \, dt = \frac{2^{-2 \gamma -1} \Gamma (2 \gamma +2) \, _2F_1\left(\gamma +1,\gamma ...
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35 views

Can this series be expressed as a Hyper Geometric function

I am trying find a Hyper Geometric function representation of the following series. $$\sum\limits_{k=0}^{\infty} \frac{a^k}{k!}\frac{\Gamma \left(\frac{b+k}{c}\right)}{\Gamma ...
4
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2answers
82 views

Closed form for integral of inverse hyperbolic function in terms of ${_4F_3}$

While attempting to evaluate the integral $\int_{0}^{\frac{\pi}{2}}\sinh^{-1}{\left(\sqrt{\sin{x}}\right)}\,\mathrm{d}x$, I stumbled upon the following representation for a related integral in terms ...
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1answer
27 views

radius of convergence of hypergeometric function

Looking up information on the Bessel function there is a formula as $|z| \to \infty$: $$ I_0(z) \approx \frac{e^z}{\sqrt{2\pi z}} {}_2F_0( \tfrac{1}{2}, \tfrac{1}{2}, \tfrac{1}{2z}) = ...
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1answer
46 views

Problems with a Definite Integral Resulting in the Hypergeometric Function

I am attempting to find $$\int_{a}^{b}(\cos^n{x} )dx$$ by using $$\int(\cos^n{x}) dx=-\frac{\cos^{n+1}x}{n+1}{_2}F_1(\frac{n+1}{2},\frac{1}{2};\frac{n+3}{2};\cos^{2}x)+c.$$ However, when $a$ and $b$ ...
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1answer
29 views

asymptotic expansion for Bessel function $I_0(z)$ in terms of Gauss hypergeometric functions ${}_2F_1$

On the Wikipedia page one can asympotoic formula of the Bessel function $$ I_0(z) \propto \frac{e^z}{\sqrt{2\pi z}} $$ On the Wolfram page there is a more detailed asymptotic formula for the Bessel ...
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56 views

Any way to simplify integral of Confluent Hypergeometric Function of the First Kind?

The integral is this: $$\int_{-\log n}^{0}e^{t(1-s)} \cdot z \cdot {}_1F_1(1-z, 2, t) dt $$ Is there a way to write this in terms of special functions that eliminates the integral and doesn't use ...
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1answer
27 views

Is $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} {(\delta+1)^n}$ for any $n$ ?(in this specific case)

Let $\alpha=(K-1)a$, $\beta=K$ and $\delta=Ka$, where $K>a\ge 1$ ($\delta>\alpha>\beta$). Can we claim that $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} ...
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Analyze the variation of $f(x)=(1+\frac{1}{x})^{-K}$ $_2F_1((K-1)a,K,Ka,\frac{1}{1+x})$ w.r.t. $x$

Is there any way to analyze the variation(w.r.t. $x$) of the following function: $f(x)=(1+\frac{1}{x})^{-K}$$ _2F_1((K-1)a,K,Ka,\frac{1}{1+x})$, where $ _2F_1$ is the Gauss' Hypergeometric Function, ...
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0answers
39 views

Identities for hypergeometric functions ${}_2F_1$ with z=1/2

Is there a closed form (or approximation) for a hypergeometric function of form: $_2F_1(1,b+c;c;\frac{1}{2}) \quad \text{where} \; b,c \in \mathbb{N}$ ? I researched all identities in ...
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21 views

an integral involves mutiply of two Kummer Confluent Hypergeometric function

i met an integral involving the mutiply of two Kummer Confluent Hypergeometric functions as follows: $$\int_{ - \infty }^\infty {\frac{1}{{{x^\alpha }{{\left( {x + k} \right)}^\beta ...
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1answer
15 views

Identity relating hypergeometric function and Legendre polynomial

In my notes I have written down the following relation: $_2F_1(a,a+\frac{1}{2};c;z)=2^{c-1}z^{(1-c)/2}(1-z)^{-a+(c-1)/2}L_{2a-c}^{1-c}\big(\frac{1}{\sqrt{1-z}}\big)\ ,$ where $_2F_1(a,b;c;z)$ is the ...
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1answer
99 views

Evaluate $\int^\pi_{-\pi}(1-a\cos\theta)^{-b-2}\log(1-a\cos\theta)d\theta$

$$ \int^\pi_{-\pi} \left(1-a\cos\theta\right)^{-b-2} \log\left(1-a\cos\theta\right)d\theta$$ Under the condition $0<a<1$ and $b>0$. Mathematica found the following form. $$ ...
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44 views

Show that this Hypergeometric Function equal to this gamma function

I have a question related to hypergeometric functions: Show that ...
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44 views

How do I go with proving that the coefficient of each terms of $\prod^{k=n}_{1}{1-x^k}$ is either 1,-1 or 0?

How do I go with proving that the coefficient of each terms of $\prod^{n}_{k=1}({1-x^k})$ is either 1,-1 or 0 for n that is sufficiently large? Also, is there any pattern in terms of the 1,-1 and 0s?
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43 views

Asymptotics of Hypergeometric series

Suppose we have a $_pF_q$ hypergeometric series that terminates for all $r>t$ for some positive integer $t$, and consider the expression \begin{multline*}\lim_{n\to\infty} ...
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24 views

Simplified expression of $ _2F_1((K-1)a,K,Ka,x) $

Is there any simplified expression of this Hypergeometric function $ _2F_1((K-1)a,K,Ka,x) $ Thanks!
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1answer
41 views

Simple explanation for Hypergeometric distribution probability

I am following through the Hypergeometric distribution: The probability that we select a sample of size $n$ containing $r$ defective items from a population of $N$ items known to contain $M$ ...
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2answers
98 views

Hypergeometric function integral representation

How to prove the following relation? $$ \, _2{F}_1(K,K;K+1;1-m) = \frac{\Gamma (K+1)}{\Gamma (K)} \int_0^{\infty } \frac{1}{(1+x) (m+x)^K} \, dx $$ where $_2{F}_1(.,.;.;.)$ is the hypergeometric ...
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1answer
86 views

how to integrate $\mathrm{arcsin}\left(x^{15}\right)$?

Integral by parts: $$ I = x\sin^{-1}\left(x^{15}\right) - \int\frac{15x^{15}}{\sqrt{1-x^{30}}}dx $$ then what? The answer by wolfram gives an answer contains hypergeometric ${}_2F_1$ function,because ...
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1answer
92 views

Proving that $\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = {1\over\pi}$

While trying to prove that $$(1)\qquad x\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = 1 \implies x=\pi$$ I got to a point, using W|A, where I have to prove that $$\color{red}{(2)\qquad ...
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1answer
145 views

Prove this closed-form of sum of ${_4F_3}$ hypergeometric functions

I think the following identity is true. How could we prove it? $${_4F_3}\left(\begin{array}c 1,1,1,1 \\\tfrac54,2,2\end{array}\middle|\,1\right) + ...
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1answer
226 views
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What more can i do to this infinite sum?

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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2answers
133 views

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[ ...
3
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2answers
106 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 ...
1
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1answer
32 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
53 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: $$ \frac{d}{dx}[x\sin^n(x)] = \sin^n(x) + nx\sin^{n-1}(x)\cos(x) $$ $$ ...
7
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2answers
160 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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1answer
215 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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60 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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1answer
28 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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1answer
37 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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0answers
18 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 ...
0
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1answer
35 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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54 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
342 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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19 views

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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37 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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0answers
19 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 ...
27
votes
4answers
997 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 ...
0
votes
0answers
27 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 ...
0
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1answer
56 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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1answer
22 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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0answers
24 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 ...