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)....

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summation involving a hypergeometric 2F2 function

im trying to find the closed form for the following \begin{equation} \sum_{n=0}^\infty \frac{c^n}{n!}\frac{(a)_n}{(b)_n}\frac{(\alpha+1/2)_n}{(\alpha+3/2)_n}{_2F_2}(-n,1-b-n;1-a-n,1/2;-\frac{d}{c}) \...
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Nested Hypergeometric series

Is it possible to express the following series as a hypergeometric function: $$\sum_{n=0}^\infty (a)_n \sum_{j_1+j_2+\cdots+j_k=n} \frac{1}{(b)_{j_1} (b)_{j_2}\cdots (b)_{j_k}} z^n $$ where $(a)_n, (...
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Complicated identity relating two 2F2 hypergeometric functions

I would like to relate the following hypergeometric 2F2 function: $\,_2F_2 \left(1, n+1-\alpha;n+2,n+2-\alpha-\beta;-x\right)$, where $\alpha,\beta>0$ to another 2F2 function: $\,_2F_2 \left(n+1,...
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Does anyone know a summation formula for Wilson's Polynomials?

Wilson's polynomials are defined as $W_n(x^2; a, b, c, d) := (a+b)_n (a+c)_n (a+d)_n {\space}_4F_3(-n, n+a+b+c+d-1, a+ix, a-ix;a+b, a+c, a+d; 1) $ Does anyone know a summation formula for Wilson's ...
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33 views

Integrating the lower incomplete gamma $\int_0^\infty x^{a-1}e^{-s x} \gamma(b,x) \mathrm{d}x$

I need to prove that $$\int_0^\infty x^{a-1}e^{-s x} \gamma(b,x) \mathrm{d}x = \frac{\Gamma(a+b)}{a(1+s)^{a+b}}F(1,a+b,1+b, 1/(1+s))$$ where $F(a,b,c;x)$ is the hypergeometric function. To show this,...
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How to define hypergeometric function ${}_1 F_1(-n+1;-n+1;z)$ for $n$ positive integer

Consider a truncated Taylor series of the exponential function to approximate $e$: $$ E(n) = \sum_{k=0}^{n-1} \frac{1}{n!} $$ I thought of computing this using the hypergeometric finite series $_1 F ...
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60 views

Integration of Laguerre polynomial $\int_{0}^{x}u^{p-1}(1-u)^{q-1}e^{-\theta u}L_n^{(m)}(\theta u)\mathrm du$

It's been several days that I'm confronted to this integral, without much success in its resolution. To give you more details, in my case: $n$ is an integer $>1$ $m=n-2$ $p,q \in \{n-1, n\}$ $x ...
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1answer
46 views

Summation involving a hypergeometric 1F1 function

I'm trying to find a closed form for the following: \begin{equation} \sum_{n=0}^\infty \frac{(-1/4)_n}{n!(3/2)_n}\left(\frac{i}{2\tau}\right)^{n} {_1F_1(2n+1;2n+2;i k)} \end{equation} Using the ...
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36 views

Integral representation of a Meijer G-function

How to prove that, the integral $$I_{a,b}:=\int_{1}^{+\infty}e^{-at}(1-t^{-1})^b\,dt ; \, a,b>0$$ is given by $\Gamma(b+1)$ times a Meijer G-function, i.e., $$I_{a,b}:=\Gamma(b+1) \times G^{m,...
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1answer
57 views

Integral of incomplete gamma function and limit of hypergeometric function

Let $a > 0$ and consider the integral $$\int_x^\infty \frac{\Gamma(a,t)}{t}\,dt$$ where $\Gamma(a,t)$ is the upper incomplete gamma function $$\Gamma(a,t) = \int_t^\infty x^{a-1} e^{-x} \, dx.$$ ...
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Simplify Hypergeometric function ${\mbox{$_2$F$_2$}(-i,-i;\,1-2\,i,2-i;\,{{\rm e}^{Y}})}$

Any suggestion how I could simplify this function, where $i$ is the complex unit? $${\mbox{$_2$F$_2$}(-i,-i;\,1-2\,i,2-i;\,{{\rm e}^{Y}})}$$ I allready tried, without success, to simplify this ...
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31 views

The integral $\int_0^{\frac{1}{2}}\frac{x^{k-1}}{1-x^{2^k}}dx$ and how simplify the Pochhmammer symbol in related series

Inspired in the shape of useful integrals to compute $\pi$ (see *), I've consider for each integer $k\geq 1$ $$\int_0^{\frac{1}{2}}\frac{x^{k-1}}{1-x^{2^k}}dx=\int_0^{\frac{1}{2}}x^{k-1}\sum_{n=0}^\...
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153 views

Can you prove the following formula for hypergeometric functions?

I wanna prove the following identity for big values of $N\gg 1$ $$ {}_3F_1\left(-N+1,1,1;2;-\frac{1}{N}\right)\to\frac{1}{2}\bigg({}_2F_1\left(1,1;2;1-\frac{1}{N}\right)+\log 2+\gamma\bigg) $$ where $...
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27 views

Question about an hypergeometric function identity

It is well known that for the following identity holds $$ \lim_{b\to\infty}{}_2F_1(a,b;c;z/b)={}_1F_1(a,c;z) $$ where ${}_pF_q$ is an hypergeometric function. Is there a similar identity for $$ \lim_{...
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55 views

Computing a sum involving binomial coefficients

I am doing some (pretty heavy) computations, and I am stuck at a point that can be rephrased as follows: Let $m>n\ge0$ be two integers. Compute $$\sum_{k=0}^n\binom{n}{k}\frac{1}{m-k}x^{n-...
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50 views

Integrating lower incomplete gamma function raised to the power $k$

I'm trying to solve the following integral: $$\int_0^\infty \gamma(t,x)^k x^t e^{-x} \mathrm{d} x$$ I'm fighting with it for quiet a while and didn't get any result. Though, I do have the ...
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86 views

How is the following integral related to confluent hypergeometric functions?

I am solving an integral that appears in a physics paper. $$ -\int_0^{\infty}dt\,\frac{e^{-t}}{t}\bigg[\bigg(1+\frac{3}{N}t\bigg)^N-1\bigg] $$ The paper does not give the full solution, it only gives ...
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27 views

Finite sum of $_{1}F_{2}$ hypergeometric functions

Could you help me with this finite sum? $$ \sum_{k=0}^{n}\binom{n}{k}\,_{1}F_{2}\left(\frac{n+1}{2},\frac{1}{2}+n-k,\frac{1}{2}+k,z\right), $$ where $_{1}F_{2}$ is a hypergeometric function? Thanks....
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n-th order dervative of the hypergeometric function f[x_] := s^(2 x) HypergeometricPFQ[{x}, {1/2 + x, 1 + x}, -(s^2/4)]/Gamma[2*x + 1]

I have recently enountered the following function f[x_] := s^(2 x) * HypergeometricPFQ[{x}, {1/2 + x, 1 + x}, -(s^2/4)]/Gamma[2*x + 1], that is a 1F2 hypergeometric function. I am trying to evaluate ...
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36 views

Finite sum of hypergeometric $_{2}F_{3}$

I need to find an expression for this finite sum $$ \sum_{k=0}^{n}\binom{n}{k}\,_{2}F_{3}\left(\frac{n+1}{2},\frac{n}{2};\frac{1}{2}+n-k,\frac{1}{2}+k,n;z\right), $$ where $_{2}F_{3}$ is a ...
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1answer
63 views

Function inversion (analytical)

Can $t(x)$ be found from: $$A \, t + B\ln\frac{1-t}{t}=x \; ?$$ Here, $A>0, \; B < 0$ and $0 \lt t \lt 1$. The $t(x)$ should be given in analytical form (even if you use, say, Lambert's W - ...
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308 views

Closed form for $ S(m) = \sum_{n=1}^\infty \frac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$?

What is the (simple) closed form for $\large \displaystyle S(m) = \sum_{n=1}^\infty \dfrac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$? Notation: $ \dbinom{2n}n $ denotes the central binomial ...
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45 views

Interesting behavior of the expansion of $_1F_2(\alpha/2;3/2,\alpha/2+1;y^2/4)$ near $y=\infty$

When we use Mathematica 10.0 to expand generalized hypergeometric function $_1F_2(\alpha/2;3/2,1+\alpha/2;y^2/4)$ near $y=\infty$ with $\alpha$ a complex number, we obtain: $${_1F_2}(\alpha/2;3/2,1+\...
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How to proof that $_2 F_1(a,b,c,x)=(1-x)^{-a}{_2F_1(a,c-b,c,\frac{x}{x-1})}$?

My Teacher wrote the following two remarks $$_2 F_1(a,b,c,x)=(1-x)^{-a}{_2F_1(a,c-b,c,\frac{x}{x-1})}$$ and $$_2 F_1(a,b,c,x)=(1-x)^{c-a-b}{_2F_1(c-a,c-b,c,x)}$$ without proof. $_2 F_1$ is ...
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Near $z=\infty$ solutions for generalized hypergeometric functions $_pF_{p+1}(z)$

For differential equation that is satisfied by the hypergeometric function $_2F_1(a_1,a_2;b_1;z)$, around $z=\infty$, if $a_1-a_2$ is not an integer, one has two independent solutions $$(z^{-a_1})...
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Euler/Pfaff transformations for generalized hypergeometric functions $_pF_{p+1}$ functions

For hypergeometric function $_2F_1(a_1,a_2;b_1;z)$ there exists Euler/Pfaff transformations: $$_2F_1(a_1,a_2;b_1;z)=((1-z)^{b_1-a_1-a_2})_2F_1(b_1-a_1,b_1-a_2;b_1;z),\quad \text{Euler transformation}$$...
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Derivative of Kummer's confluent hypergeometric with respect to parameter?

Kummer's confluent hypergeometric function is: $$M(a,b;z)= {_1}F_1(a,b;z)$$ There is an easy recurrence for the derivative of $M$ with respect to $z$. I am interested in the derivative with respect ...
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209 views

a continued fraction related to pythagoras theorem $a^2+b^2=c^2$

For our purpose,let $a,b,c$ and $x\gt2$ be natural numbers such that the positive integers $a,b$ and $c$ form a special pythagorean triple $(a,b,c)$,then it is conjectured that the following is true $...
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How to isolate and solve for k in a Sigma notation probability mass function equation?

"isolate and solve for k:" $$P(X = k) = \sum_{k=0}^n {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}$$ If the above equation is a function of P, how would the equation be stated as a ...
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1answer
24 views

turning a certain chebychev polynomial-like expression into a hypergeometric form

Can the following expression be represented in terms of hypergeometric function $$\sqrt{3}\sin(\arcsin(7/25)/3)-\cos(\arcsin(7/25)/3)$$ It looks similar to the one presented on [this site][1] [1] :...
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How to proof this about Hypergeometric function?

I am trying to understand why the following four properties are true $$_{2}F_{1}(a,b,c,1)=\frac{\Gamma(c)\Gamma(-a-b)}{\Gamma(c-a)\Gamma(c-b)}$$ $$_{2}F_{1}(-n,b,c,1)=\frac{(c-b)_{n}}{(c)_{n}}$$ $$...
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$25 w(w-1)y''+(14-15w)y'+y=0$ - Gauss's Hypergeometric equation

I would like to solve the equation $(x^2-x-6)y''+(5+3x)y'+y=0$ near the singular point $x=3$. I think we have to solve this problem in considering the Gauss's hypergeometric equation on the form $$x(1-...
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Differential equations - Hypergeometric function [duplicate]

I would like to solve the equation $(x^2-x-6)y''+(5+3x)y'+y=0$ at $x=3$. I think we have to solve this problem in considering the Gauss's hypergeometric equation on the form (*) $$x(1-x)y''+[c-(a+b+1)...
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Reconstruct a family of probability distributions having certain generalized hypergeometric moments

Reconstruct and/or otherwise characterize any/or all members of a certain one-parameter ($\alpha =\frac{1}{2}, 1, \frac{3}{2}, 2,\ldots$) family of univariate probability distributions (of quantum-...
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81 views

How to evaluate $\int_{0}^{\infty }\frac{e^{-x^{2}}}{\sqrt{t^{2}+x}}\mathrm{d}x$

How to evaluate the integral below $$\int_{0}^{\infty }\frac{e^{-x^{2}}}{\sqrt{t^{2}+x}}\mathrm{d}x~~~~~~(t>0)$$ The WolframAlpha gave me a horrible answer $$\frac{t}{2}e^{-\frac{t^{4}}{2}}\left \{ \...
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39 views

Hypergeometric 2F3

Please could somebody tell me if there is a simpler form or known function to express the hypergeometric function next: $_2F_3\left(\frac{1}{2},\frac{1}{2};1,1,\frac{3}{2};-4 \pi ^2 a^2\right)$ I ...
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40 views

What are hypergeometric functions in layman terms?

Could someone please explain what are these in layman terms? Someone here told me that and I still can't figure out what they mean on my own after giving Google a number of hits. Wikipedia says this: ...
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47 views

Asymptotics of this HyperGeometric Function

I have a function $$f(x)=x^{2m}\text{ }_2F_1\left(\frac{1}{2},-m;\frac{3}{2};-\frac{1}{x^2}\right)$$ where $x>0$. I am interested in asymptotics in the two extreme limits: $$\lim_{x\rightarrow 0} ...
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1answer
40 views

Probability Question [Hypergeometric Distribution]

I was solving the below problem, and I had a few questions: An urn contains five red marbles and three blue marbles. Four marbles are chosen without replacement from the urn and their colors are ...
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38 views

Repeated Indefinite Integration of Gaussian Integral

I have an integral that can be solved via recursive integration by parts. In my case, $\mathrm{d}v=e^{-ax^{2}}$. Question: Is there a solution or special function defined as the n-th indefinite ...
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31 views

Database of hypergeometric functions

Knuth wrote in Concrete Math that hypergeometric functions are useful because they allow the construction of a "database" of identities, since any sum with the property that the ratio between ...
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20 views

Hypergeometric function asymptotics

When calculating the number of possible states of a spin 1 system in a magnetic field, one obtains the following expression $$\#\text{ of states} \propto \,_2 F_1 \left(-\frac{N-P}{2}, - \frac{N-P}{2} ...
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1answer
81 views

How to evaluate the series $\sum_{i,j,k=0}^{\infty}\left(\frac{(i+j+k)!}{i!j!k!}\right)^2x^{-i-j-k} $?

Suppose the series $$ \Gamma (x) =\sum_{i,j,k=0}^{\infty}\frac{((i+j+k)!)^2}{(i!)^2(j!)^2(k!)^2}x^{-i-j-k} $$ How to evaluate it? It is claimed that for $x <3$ this function converges to elliptic ...
2
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55 views

How to evaluate following integral?

Suppose the integral $$ \tag 1 I = \int \limits_{-\pi}^{\pi}dx \int \limits_{-\pi}^{\pi}\frac{dy}{\tau - \cos (2x) -2\cos(x)\cos(y)}, \quad t > 3 $$ How to evaluate it in terms of elliptic integral?...
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1answer
19 views

multivariate hyper geometric Vs. Indep. drawing without replacement

I think my problem can be seen as a basic urn setup, with another layer tacked on after the fact. I'm looking at drawing from an urn with 3 biological specimens. Each has genotype AA, Aa, or aa. ...
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42 views

Kummer equation, solution to find optimal value

Suppose V follows the mean reverting process $$dV=η( ̅V-V)Vdt+σVdz$$ I want to find the optimal investment rule, and using Itos's lemma I get that the differential equation that F(V) must satisfy $$ ...
2
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1answer
51 views

combinatorial identity involving fraction and product of bionomial coefficients

How can I prove the following identity for $i\geq 1$: $$ \sum_{t=i}^{s-1} \frac{i}{t}\binom{2(s-t-1)}{s-t-1}\binom{2t-i-1}{t-1}= \binom{2s-i-2}{s-1}. $$ Perhaps I'll need to go to hypergeometric ...
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30 views

Weighted Q-binomial Coefficients

A possible identity popped up in a project for college, and if features q-binomial coefficient, which can be interpreted as the generating function for the number of Ferrer's boards fitting into a $k\...
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1answer
57 views

Closed form for $\int^{\pi}_0 \frac{\sin^2 (y)}{a+\cos(y)} \cos(ny) dy$ for integer $n$

I encountered this integral when trying to obtain a Fourier series for the function inside (in connection to this question). Mathematica gives the following general solution (only valid for $|a|>...
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21 views

Approximation of a hypergeometirc-like distribution

Fix $0<\varepsilon<1$. For $m\in\mathbb{N}$, let $$c_m=\max\left\{\frac{{m-1\choose s-1}{m\choose k-s}}{{2m\choose k}}:\;k=1,2,\dots,2m(1-\varepsilon)\;\mbox{and}\;s=1,2,\dots,k\right\}$$ Prove ...