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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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 ...
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20 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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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? ...
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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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32 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
61 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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+150

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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1answer
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: ...
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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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9 views

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 ...
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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 ...
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8 views

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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204 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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24 views

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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24 views

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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42 views

$25 w(w-1)y''+(14-15w)y'+y=0$ - 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 ...
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43 views

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 (*) ...
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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 ...
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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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1answer
38 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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43 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
36 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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36 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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29 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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18 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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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 ...
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52 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 ...
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1answer
18 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 ...
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1answer
55 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 ...
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20 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 ...
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14 views

On Hyper-geometric function differential equation

The hypergeometric function $$_2F_1(a,b;c+n:z) = \sum_{m=0}^\infty \frac{(a)_m(b)_m}{(c+n)_m}\frac{z^m}{m!}$$ should satisfy the differential equation $$z(1-z)\frac{d^2u}{dz^2} + ...
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40 views

Does this sum of series have a specific name?

$$ \sum_{n = 0}^\infty \frac{(4n)!}{(2n)!}k^n $$ It looks like a hypergeometric function, but a little bit different. Is there a specific name for this series or any function for this?
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25 views

On Hyper-geometric Functions and its recurrence relation

I research in generating functions of Hyper-geometric functions $_2F_1(a+n,b;c+n;x)$ using Lie group theoretic method and so the recurrence relation is important in this method. I want recurrence ...
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1answer
41 views

How do I use the hypergeometric function?

I was given a long list of integrals involving sines and cosines as homework. Being the slothful person that I am, I tried to find a general formula for these integrals as a function of the sine's and ...
4
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85 views

How to prove sum related to hyperbolic tangents $\sum_{k=0}^{n-1}\frac{\tanh(…)}{1+\frac{\tanh^2x}{\tan^2(…)}}=\tanh(2nx)$

I have no Idea how to start I think to switch it to definite integral, use complex analysis, or some real analysis tricks and at the end I failed to make any progress. $$ \displaystyle ...
5
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1answer
101 views

May I know how this integral was evaluated by using the theory of elliptic integrals?

I can not solve the following integral using the theory of elliptic integrals: $$\int_a^b \frac{\sin(x)}{\sqrt{c-\sin(x)}}dx$$ Where $a\geq 0, b>0, c>0$. Wolfram$|$Alpha showed the following ...
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1answer
167 views

May I know how this integral was evaluated using hypergeometric function?

I can not solve the following integral using the hypergeometric function: $$\int_a^b (\sin x)^{(1/n)}dx$$ Wolframalpha showed the following result. but I do not understand how Wolframalpha came ...
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27 views

How can this summation be written in terms of hypergeometric function?

I came across an expression as \begin{equation} \sum_{k=0}^n(-1)^k\binom{2n}{k}(n-k)^{2n-1}. \end{equation} It seems similar to the definition of the hypergeometric function, is it possible to ...
5
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1answer
138 views

Integral involving power of trigonometric functions

I'm having a technical problem evaluating the following integral: $$\int_{r=0}^1\int_{\theta=0}^{\pi \over2} \cos^{2\epsilon -1}\theta \sin^{\epsilon-1}\theta e^{-ikr\sin^\epsilon\theta}d\theta dr$$ ...
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1answer
17 views

Limits of Kummer Confluent Hypergeometric function for fixed z

I have the following: \begin{equation} \sum_{j=0}^{\alpha_{2}}C\, e^{az}\, M(-\alpha_{2}+j,-\alpha_{3}+j,-\lambda z)\Bigg|_{z=-\infty}^{0} \end{equation} $C$ is a constant, $a,\alpha_{2},\alpha_{3}, ...
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1answer
89 views

On a “coincidence” of two sequences involving $a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$

This was inspired by this post. Define, $$a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$$ $$b_n = \sum_{k=0}^n \binom{-\tfrac{1}{2}}{k}\big(-\tfrac{1}{2}\big)^k$$ where ...
1
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0answers
31 views

Solving summation with incomplete gamma function

I solved an indefinite integral that gave me the following result: \begin{equation} f(u) = -\sum_{k=0}^{n}\binom{n}{k}\frac{\mathrm{sgn}(u)^{k+1}\, b^{n-k}}{2\, a^{\frac{k+1}{2}}}\, ...
2
votes
0answers
40 views

Integral of combination of power, exponential, and kummer hypergeometric function

I am trying to solve a couple integrals of the form: \begin{equation} \int_{0}^{\infty} x \, e^{-a(x-b)^{2}}\, M(-\alpha,-\beta,\lambda x) \end{equation} $\alpha > 0$ and $\beta > 0$ are ...
1
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1answer
57 views

Integrals involving whittaker functions.

I want to compute the following integrals: $$ \int y^{a} e^{\frac{1}{2}y}M_{k,m}(y)dy $$ where a is an arbitrary constant and $M_{k,m}$ is a whittaker function of the first kind. I already know that ...
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2answers
87 views

Expected number of successes before first failure (Hypergeometric distribution)

Think of the following scenario: We are a group of $42$ people. I tag you, and you tag another person. This other person tags another person, etc. The "chain" of tagging stops when a person has been ...