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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29 views

Derivation of mean and variance of Hypergeometric Distribution

I need clarified and detailed derivation of mean and variance of a hyper-geometric distribution. If a box contains $N$ balls, $a$ of them are black and $N-a$ are white, and $n$ number of balls are ...
4
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1answer
30 views

Numerically evaluate Gauss' hypergeometric function ${}_{2}F_{1}(a,b;c;x) $ for large $|a|$ or $|b|$ and $x\ll 0$ or $ x \approx 1$?

I need to compute Gauss' hypergeometric function $${}_{2}F_{1}(a,b;c;x)$$ for the case where one of $|a|$ or $|b|$ is large and $x\ll 0$ or $ x \approx 1$. By employing some linear transformations, I ...
11
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2answers
160 views

Integral $\int_0^1\frac{\log(x)\log(1+x)}{\sqrt{1-x}}\,dx$

I'm trying to evaluate this definite integral: $$\int_0^1\frac{\log(x) \log(1+x)}{\sqrt{1-x}} dx$$ It's clear that the result can be expressed in terms of derivatives of a hypergeometric function with ...
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1answer
17 views

Which operations is the set of generalized hypergeometric functions closed under?

Consider the set of all generalized hypergeometric functions. I am trying to figure out which operations this set is closed under. For example, is the sum of two generalized hypergeometric functions ...
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1answer
24 views

How do I write a terminating series representation of $_2 F_1(p, n+1, n+2, x)$

How do I find a terminating series representation of the hypergeometric function $_2 F_1(p, n+1, n+2, x)$, for real $p \in \mathbb{R}$ but $n \in \mathbb{Z}$, $n\geq0$? Mathematica gives (...
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0answers
33 views

How to derive Such infinite sum representation for Hypergeometric function?

I was reading a paper $[1]$ in which authors claimed that we can simplify below Gauss function to finite series if $m $ and $v$ are positive integers. $$ _2F_{1}(v,m+v;m+1;x)=\psi\sum_{c=0}^{v-1} ...
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0answers
27 views

Convergence of a hypergeometric function

The hypergeometric function, ${}_{2}F_1(a,b,c;z)$ can be written in terms of a power series in $z$ as follows, $${}_{2}F_1(a,b,c;z) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} ...
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1answer
31 views

I have question about incomplete gamma functions?

the question is about definition of upper and lower incomplete gamma functions in [1] we can see : lower case : $ \gamma(a,x)=\int_{0}^{x}e^{-t} \ t^{a-1}dt \ \ \ \ , \ \ \ Re(a)>0 $ Upper case: ...
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0answers
20 views

Recognizing and using hypergeometric function

Some expressions that interest me end up having something to do with hyper geometric function. I want to be able to derive such results myself. Where do I begin? For example, the equation $$ ...
1
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0answers
39 views

Integral gaussian hypergeometric function

How can we define integral with interval $[b,\infty)$ $$ \begin{align} C(b,\alpha) & = \int_b^\infty \frac{1}{1+w^{\alpha/2}}\,\mathrm{d}w \\[8pt] & = 2\pi/\alpha \csc(2\pi/\alpha)-b_2 F_1 ...
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1answer
29 views

Max value of hypergeometric distribution?

I'm reading the book Probability Theory: The Logic of Science by Jaynes. While I'm reading chapter 3, on page 56, it says: Although the hypergeometric distribution $h(r)$ appears complicated, it ...
5
votes
2answers
128 views

Limit involving binomial coefficient

I was trying to find the below limit. The sum can be written in a hypergeometric function but it doesn't seem to help me to find the limit. Any help will be appreciated. $$ \lim_{n \rightarrow ...
2
votes
3answers
28 views

Confluent hypergeometric function for positive integers

Do any of you know a simple form for the confluent hypergeometric function with positive integers that involve simple functions? What I actually need to compute is $_1F_1(n,n + m,z)$. I know for ...
4
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1answer
52 views

Simplifying real part of hypergeometric function with complex parameters

I am looking for a simpler representation of the following hypergeometric function with complex parameters in terms of more basic functions and manifestly real parameters: ...
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0answers
14 views

Algebraic hypergeometric functions (?)

After a series of calculations following this question of mine (Constant term of noncommutative $(X+Y+(XY)^{-1})^n$), I've cooked up the following function: ...
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0answers
25 views

Hypergeometric differential equation with nonlinearity

I have come across a problem involving a hypergeometric differential equation (http://mathworld.wolfram.com/HypergeometricDifferentialEquation.html) with a nonlinear term added as in ...
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1answer
72 views

Defective Light bulbs [closed]

Light bulbs come in packages, each containing $N$ light bulbs. Let $p_k$ denote the probability that a package contains $k$ defective light bulbs $(0 ≤ k ≤ m)$. A sample of $n$ light bulbs is taken ...
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0answers
48 views

Deriving towards a hypergeometric function

Can someone explain the details to the equality: $$ x^n \sum_{k=0}^{\lfloor \frac{n}2\rfloor} \binom{n+1}{2k+1}(1-x^{-2})^k = \sum_{k=0}^{\lfloor \frac{n}2 \rfloor} \binom{2k-(n+1)}{k}(2x)^{n-2k}? ...
2
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1answer
73 views

Sum with binomial coefficients and integer powers

I would like to have an analytic expression for the following sum $$ G_{n,a} = \sum_{p=1}^n \frac{(-1)^p p^{2(a+n)}}{(n-p)! (n+p)!} \;. $$ I am not sure it has a closed form, but I would at least ...
1
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2answers
131 views

Closed form of partial hypergeometric sum

Can we get closed form for $$\sum_{k=0}^m \left(-\frac12\right)^k \binom{2m}{m-k}k^p,\quad p\in\mathbb{N}\,?$$ In Concrete Mathematics Knuth describes Gosper's algorithm and its Zeilberger's ...
3
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0answers
56 views

Integral involving Whittaker function

Consider the following integral: $$ \int_1^{\infty} \frac{e^{u/2}}{u}[-\mathrm{Ei}(-u)]\,W_{1,\imath p}(u)\,du, $$ where $\imath=\sqrt{-1}$ and $p>0$ selected so that $W_{1,\imath p}(1)=0$; here ...
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1answer
19 views

Hypergeometric function variance

In a fishing event, a small lake is populated with $75$ trout, among which $25$ are tagged. Each participant is allowed to capture $5$ fish during the day (the fish are not put back into the lake). ...
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26 views

Singularities in the Gauss Hypergeometric Function

I am evaluating the following term in a series: $$I_k = \int\!x^{-3(2k+1)}(1+\lambda x^4)^{-1/2}\,\mathrm dx$$ When I plug this into WolframAlpha, I get the following result: $$I_k = ...
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0answers
26 views

Decomposing a series

When I insert the following function \begin{equation} F(X,Y)=-\frac{1}{Y^{2/3}}\sum _{m=0}^{\infty } \frac{\Gamma \left(\frac{m+2}{3}\right)}{m! \Gamma (m+1)}\left(-\frac{X^2}{2^2 ...
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0answers
23 views

Remove Multiplicative Constant from Hypergeometric Function

I have a function of the form $$f(x;\lambda) = {}_2F_1\left(a,b;c;-\frac{e^{2x}}{\lambda}\right)$$ I need to invert this function to solve for the constant $\lambda = f\left(x\right)$. I could do ...
2
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0answers
24 views

Indefinite Hypergeometric Integral Transformations

I'm attempting to solve the indefinite integral $$S\left(v\right) = 2a\sqrt{\alpha ...
2
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1answer
63 views

Closed-form expression for $\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$?

As per the title, I am looking for a closed-form expression for the integral $$\frac{1}{B(\alpha,\beta)}\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$$ where $a,\alpha,\beta>0$ and ...
0
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0answers
10 views

How can I find the derivative of Meixner_Pollaczek polynomials in general?

$\frac{d^n}{dx^n} \left(P_{n}^{(\lambda)}(x;\phi) \right)= \frac{d^n}{dx^n} \left(\frac{(2\lambda)_n}{n!} e^{i n \phi} \sum_{k=0}^{\infty} \frac{(-n)_k(\lambda+ix)_k}{(2\lambda)_k} \frac{(1-e^{-2 i ...
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1answer
20 views

Some properties about the Kampé de Fériet function

Prove that according to http://www.iosrjournals.org/iosr-jm/papers/Vol8-issue6/L0866770.pdf?id=7287, this special case of Kampé de Fériet function can reduce to Generalized hypergeometric function: ...
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1answer
70 views

How to show this equality holds?

I need to show that the following equality holds for any integer i,j,m,n and p where p is probability (0<=p<=1) Could you please help me?I think I should use hyper-geometric function but I could ...
2
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1answer
69 views

Asymptotic series of Confluent Hypergeometric function $U(a,1,z) $ as $z \to 0$

Consider the Confluent hypergeometric function $U(a,b,z)$, which is a solution of the Kummer's Equation : $$zw''+(b-z)w'-aw=0$$ it has the following integral representation when $- \pi/2 < \arg ...
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1answer
70 views

how to compute the integral $\int_0^1 (1-x^p)^n dx$?

For constants $n$ and $p$, how to compute the integral $\int_0^1 (1-x^p)^n dx$ ? I saw a solution using hypergeometric function and another using incomplete beta function here: ...
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1answer
146 views

Expressing ${}_2F_1(a, b; c; z)^2$ as a single series

Is there a way to express $${}_2F_1\bigg(\frac{1}{12}, \frac{5}{12}; \frac{1}{2}; z\bigg)^2$$ as a single series a la Clausen? Note that Clausen's identity is not applicable here.
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2answers
38 views

What is the probability of getting 2 new balls out of 3 at the third time of picking 3 out of 12?

Here is the question: (don't bother with the title if you don't get the question) You are going to pick 3 balls out of 12 and put it back every time. The ball that gets picked is considered an old ...
2
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1answer
20 views

Simplifying hyperbolic complex function

I am trying to simplify the following fraction: $$\frac{2\tanh(z)}{1-\tanh(z))}$$ I know it equals: $e^{2z}-1$ from wolframalpha, but I have no idea why. I imagine I should be using some trig ...
0
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0answers
16 views

Wilf-Zeilberger context with an extra parameter

Define two sequences $(A_p(X)),(B_p(X))$ of polynomials by $A_p(X)=(-2p-8)X^2+(3p^2+22p+40)X-(p^3+11p^2+40p+48)$ and $B_p(X)=(4p+12)X^2-(3p^2+21p+34)$. Let $(g_p)_{p\geq 1}$ be the sequence of ...
0
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1answer
49 views

Convergence of a series $\sum_{n=0}^{+\infty} {2n\choose n} {\sum_{k=0}^n}{n\choose k} {n\choose {n-k}}({1\over3})^{2k}({1\over6})^{2n-2k}$

I have trouble showing the series $$\sum_{n=0}^{+\infty} {2n\choose n} {\sum_{k=0}^n}{n\choose k} {n\choose {n-k}}\bigg({1\over3}\bigg)^{2k}\bigg({1\over6}\bigg)^{2n-2k}$$ converges or not. I tried to ...
1
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1answer
42 views

Series representation of hypergeometric function reciprocal?

Basically, can you represent $\dfrac{1}{_2F_1(a,b;c;z)}$ as some kind of power series? EDIT: This question came from something I was doing with generating functions were ...
1
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1answer
53 views

Evaluating $\int^{x_2} _{x_1} \sqrt{a - b x^m} ~dx $

Is there any way to evaluate $$\int^{x_2} _{x_1} \sqrt{(a - b x^m)}~ dx $$ where $x_{12} = \pm (a/b)^{1/m}$ without elliptic functions or hypergeometry? Or just any way to solve it. My attempt is to ...
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1answer
48 views

Integration of this using a multi-dimensional hypergeometric function

I want to try and potentially use a Dirichlet - Hypergeometric Function in order to compute the following integral. I would appreciate some help as I'm stuck on how to go about this is a ...
0
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0answers
33 views

Definite integral with Bessel functions

Show that for $Re(\lambda)>0,Re(\mu)>0$ it holds the following identity $\int_0^a x J_\lambda(2a)I_\lambda(2x) J_\mu(2 \sqrt{a^2-x^2}) I_\mu(2 \sqrt{a^2-x^2}) dx = \frac{a^{2 \lambda + 2 \mu + ...
3
votes
0answers
51 views

Closed form expression for a sum

I want to calculate a sum of the form $$\sum_{k=0}^m \frac{\Gamma[m+1+\alpha-k]^2}{\Gamma[m+1-k]^2}\frac{\Gamma[x+k]}{\Gamma[x]k!}$$ where $m>0$ and belongs to integers and $\alpha$ takes half ...
1
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0answers
22 views

Is there a way to express the reciprocal of the hypergeometric function 2F1(a,b;c;z) in terms of a b and c?

I'm trying to use generating functions to get the value of some coefficients, namely $\displaystyle\sum_{m\geq 0} f_{2m}x^m = 1 - \Big(\displaystyle\sum_{m\geq 0} u_{2m}x^m\Big)^{-1}\\$ and $u_{2m} ...
2
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1answer
33 views

Proving that $\lim \limits_{b \rightarrow \infty} F(a,b,\frac{1}{2};\frac{z^2}{4ab})=\cosh z$

I am trying to prove that $\lim \limits_{a,b \rightarrow \infty} F(a,b,\frac{1}{2};\frac{z^2}{4ab})=\cosh z$ . Here $F$ is the hypergeometric function. Here because of two limits I am unable to ...
2
votes
1answer
22 views

Proving that $\lim \limits_{b \rightarrow \infty} F(1,b,1;\frac{z}{b})=e^z$

I am trying to prove that $\lim \limits_{b \rightarrow \infty} F(1,b,1;\frac{z}{b})=e^z$ without using dominated convergence theorem. Here $F$ is the hypergeometric function. I have been able to ...
1
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0answers
34 views

Integral over a product

How the following integral can be computed: $I = \int_0^1 (x-a_1)^{b_1}(x-a_2)^{b_2}...(x-a_n)^{b_n} dx$? Here, $a_i,b_i$ are real numbers and $n$ is a natural number. Are there any techniques for ...
0
votes
0answers
42 views

Closed form expression for 3F2 with positive unit argument

Is there any closed form expression for the Hypergeometric function ${}_3F_2(-n,-n,c;-d/2-n,-d/2-n;1)$ for $n>0$ and $d>0$. The parameter $c$ can be both positive and negative.
0
votes
1answer
24 views

${}_2F_1$ as a FINITE series: How is this result obtained?

I am using the following result: which I have found in this link: http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/03/06/07/10/0001 I am trying to find out how this result ...
0
votes
0answers
34 views

About the extensions from Confluent Hypergeometric Function of the Second Kind

I know that $\int_0^\infty t^{a-1}(1+t)^{c-a-1}e^{-yt}~dt=\Gamma(a)U(a,c,y)$ , where $\text{Re}(a),\text{Re}(y)>0$ . How about $\int_0^\infty t^{a-1}(1+t)^{c-a-1}(1+xt)^{-b}e^{-yt}~dt$ and ...
2
votes
0answers
70 views

How many solutions does Riemann's P-symbol describe?

The Papperitz-Riemann P-symbol $$ \tag 1 y(z) = P \left\{ \begin{matrix} z_1 & z_2 & z_3 & \; \\ \alpha_1 & \alpha_2 & \alpha_3 & z \\ \beta_1 & \beta_2 & ...