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 ...

learn more… | top users | synonyms

1
vote
0answers
12 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 $$ ...
2
votes
1answer
28 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
127 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
26 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
votes
1answer
45 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: ...
0
votes
0answers
11 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: ...
1
vote
0answers
20 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 ...
-4
votes
1answer
66 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 ...
1
vote
0answers
46 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
votes
1answer
71 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
vote
2answers
129 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
votes
0answers
44 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 ...
1
vote
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). ...
1
vote
0answers
25 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 = ...
1
vote
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 ...
1
vote
0answers
21 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
votes
0answers
24 views

Indefinite Hypergeometric Integral Transformations

I'm attempting to solve the indefinite integral $$S\left(v\right) = 2a\sqrt{\alpha ...
2
votes
1answer
59 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
votes
0answers
9 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 ...
0
votes
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: ...
0
votes
1answer
69 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
votes
1answer
60 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 ...
1
vote
1answer
69 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: ...
8
votes
1answer
145 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.
1
vote
2answers
35 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
votes
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
votes
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
votes
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
vote
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
vote
1answer
52 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 ...
1
vote
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
votes
0answers
32 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
vote
0answers
21 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
votes
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
21 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
vote
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
23 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
31 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
66 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 & ...
2
votes
0answers
31 views

Limit involving hypergeometric function.

I'm trying to prove a limit involving a Hypergeometric function, which seems to hold numerically, but I'm stuck. I have the following function: $ f(A,B)=\frac{2A\pi ^{5/2} (-1)^B}{{\left(A!\right)^2 ...
1
vote
3answers
76 views

Is there a simple expression for ${}_2F_1(a,a+\tfrac{1}{2};a+1;z)$?

I have been searching through some books and also this but I have not succeed. I wonder if there is a simple equivalent form for ${}_2F_1(a,a+\tfrac{1}{2};a+1;z)$, in terms of elementary functions ...
0
votes
0answers
28 views

Find rule generating seven 6-tuples (k=0,…,6) composed of two sets of consecutive relatively prime fractions

I have the seven 6-tuples listed below, functions of k (0,…,6), composed of four consecutive relatively prime fractions involving fifths, and two involving sixths. I would like to find a rule that ...
10
votes
3answers
316 views

Closed form of factorial and cascading power sum

Consider the following sum: $$ \sum_{i =0}^{j} \left( \frac{(j-i)^ix^i \ln(x)^{(j-i)}\ln(x)^i}{(j-i)!i!} \right) $$ I can simplify the sum to: $$ \ln(x)^j\sum_{i =0}^{j} \left( ...
0
votes
1answer
14 views

hypergeometric distribution and random sampling

Is there any simple and fast algorithm (to be implemented in Javascript) to obtain a sample from the hypergeometric distribution? My needed sample size is very large (100,000,000).
0
votes
1answer
30 views

Expectation of the number balls are drawn

A box contains 6 white balls and 4 black balls. Balls are drawn from the box without replacement until either a white ball is drawn or 3 balls have been drawn. Find the expected number of balls that ...
3
votes
1answer
42 views

Evaluate $\int^\infty_0 t^{a+b-1}(t+1)^{-b-1} U(a+2,a-b+2,ct)dt$

Evaluate $$ \int^\infty_0 t^{a+b-1}\left(t+1\right)^{-b-1} U\left(a+2,a-b+2,ct\right)dt $$ under the condition $a>0$, $b>0$ and $c>0$, where $U(\cdot,\cdot,\cdot)$ denotes the ...
2
votes
1answer
179 views

Definite integral of polynomial times exponential times hypergeometric function of imaginary argument

How would one deal with such an integral? $$I(k)\equiv \int_0^\infty r^n e^{-r(1+\mu)} e^{-{\mathrm i} kr}\:{}_1F_1({\mathrm i}/k+1;2;2{\mathrm i} kr) \, \mathrm{d} r$$ Here $n\in\{0,1\}$, $\mu\in ...
1
vote
0answers
39 views

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 ...