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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Different representations of Appell hypergeometric series

The (first) Appell series: $$F(a; b_1, b_2; c \, | \, z_1, z_2) = \sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n} \sum_{n_1+n_2=n} (b_1)_{n_1} (b_2)_{n_2} \, \frac{z^{n_1}}{n_1!} \frac{z^{n_2}}{n_2!}$$ can ...
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Veiled Appell Integral

It is well known that: $$\frac{\Gamma(c)}{\Gamma(a) \Gamma(c-a)} \int_0^1 du \, \frac{u^{a-1}(1-u)^{c-a-1}}{(1-ux)^{b_1} (1-uy)^{b_2}} = \mathfrak{F} \, (a; b_1, b_2; c \, | \, x, y) = \\ =\sum_{n = ...
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15 views

GKZ systems of differential equations

Roughly defined, the GKZ (Gelfand-Kapranov-Zelevinsky) systems are classes of differential equations that can be solved in terms of generalised hypergeometric functions - for more details on the ...
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1answer
25 views

Question Regarding a Second Order Ordinary Differential Equation

I was wondering if the solution to the following differential equation belongs to a class of special functions. If not, is it exactly solvable? \begin{equation} ...
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1answer
105 views

Hypergeometric function values and the Baxter constant

While I was working on this question by @Vladimir Reshetnikov, I've found the following relations between Gaussian hypergeometric function values and the Baxter constant: ...
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1answer
142 views

Closed-form of the hypergeometric function ${_4F_3}\left(\begin{array}c1,1,\tfrac54,\tfrac74\\\tfrac32,2,2\end{array}\middle|\,-t\right)$

Inspired by this question and by using Mathematica the following conjecture seems to be true for all nonzero complex $t$ number: ...
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37 views

How to solve the general sextic equation with Kampé de Fériet functions?

It is frequently stated, for example on Wolfram Mathworld, that the general sextic equation $$x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x^1 + a_0 = 0$$ can be solved in terms of Kampé de ...
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67 views

Integral of $x e^{cx^3}$

How to evaluate the indefinite integral $\int x e^{cx^3}$. Is there any general form of solution for this integral? some function in terms of hypergeometric function or similar kind of functions? ...
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30 views

Complicated recurrence relation

I would like to know if the following recurrence relation is solvable \begin{equation} (\alpha_{1}n^{2}+\beta_{1}n+\gamma_{1})\ c(n+1)+(\alpha_{0}n^{2}+\beta_{0}n+\gamma_{0})\ ...
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32 views

Binomial Coefficient Probability Question

completely stuck in this probability question. I know to use Hypergeometric probability but im not sure about what numbers i should be using. Any help would be great. A regular deck of 52 playing ...
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56 views

Hypergeometric distribution in probability?

I am struggling with the following question: There are $20$ eggs in the box and three of them are rotten, a) if I pick eggs and replace them, how many eggs do I have to pick on average until ...
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2answers
152 views

Help with C is Euler's constant and $\Gamma(0)=\infty$ in paper

I am referring to a paper by S. Nadarajah & S. Kotz. The notation is simple enough to understand, however i having trouble with $C$ is Euler’s constant and $\Gamma(0)=\infty$ by equation (2.3) I ...
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2answers
38 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 ...
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1answer
42 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 ...
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2answers
205 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
30 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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36 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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31 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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33 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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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 $$ ...
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43 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
31 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 ...
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2answers
135 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 ...
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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 ...
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1answer
56 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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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
29 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
75 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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49 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}? ...
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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 ...
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2answers
139 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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60 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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27 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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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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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 ...
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0answers
25 views

Indefinite Hypergeometric Integral Transformations

I'm attempting to solve the indefinite integral $$S\left(v\right) = 2a\sqrt{\alpha ...
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1answer
64 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 ...
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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
71 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
77 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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...