What is Hypergeometric: Definition and 77 Discussions
In mathematics, the Gaussian or ordinary hypergeometric function 2F1(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). Every second-order linear ODE with three regular singular points can be transformed into this equation.
For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Erdélyi et al. (1953) and Olde Daalhuis (2010). There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic.
I have been working with some Hypergeometric functions whose behavior I am not quite familiar with. Suppose the equation I wish to analyze is
##p(x) = (e^{x}-1)^{2i}\left({}_{2}F_{1}(a,b;c;e^{x}) + {}_{2}F_{1}(a+1,b+1;c+1;e^{x})\right)## where ##a,b,c## are all complex valued and we have...
I believe it is the case that any linear second order ode with at most 3 regular singular points can be transformed into a hypergeometric function. I am trying to solve the following equation for a(x):
where E, m, v, k_{y} are all constants and I believe turning it into hypergeometric form will...
I am looking for the expectation of a fraction of Gauss hypergeometric functions.
$$E_X\left[\frac{{}_2F_1\left(\begin{matrix}x+a+1\\x+a+1\end{matrix},a+1,c\right)}{{}_2F_1\left(\begin{matrix}x+a\\x+a\end{matrix},a,c\right)}\right]=?$$
Are there any identities that could be used to simplify or...
Homework Statement
_2F_1(a,b;c;x)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{(c)_nn!}x^n
Show that Legendre polynomial of degree ##n## is defined by
P_n(x)=\,_2F_1(-n,n+1;1;\frac{1-x}{2})
Homework Equations
Definition of Pochamer symbol[/B]
(a)_n=\frac{\Gamma(a+n)}{\Gamma(a)}
The Attempt at a...
I am aware that hypergeometric type differential equations of the type:
can be solved e.g. by means of Mellin transforms when σ(s) is at most a 2nd-degree polynomial and τ(s) is at most 1st-degree, and λ is a constant. I'm trying to reproduce the method for the case where λ is not constant...
Dear friends:
It's strange that Mathematica can do the integral of ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x^2)##, however, fails when it's changed to ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x-x^2)##.
Are there any major differences between this two types? Is it possible to do the second kind of integral...
Hello everyone
I am trying to write code in ROOT.I want to plot generalized hypergeometric function pFq with p=0 and q=3 i.e I want to plot 0F3(;4/3,5/3,2;x) as a function of x using TF1 class.I am not getting how to plot this function in ROOT.Kindly help me out.
Thanks in Advance
Given this libreoffice command:
HYPGEOM.DIST(X; NSample; Successes; NPopulation; Cumulative)
>X is the number of results achieved in the random sample.
>NSample is the size of the random sample.
>Successes is the number of possible results in the total population.
>NPopulation is the size...
Homework Statement
Hello, I've recently encountered this double integral
$$\int_0^1 dv \int_0^1 dw \frac{(vw)^n(1-v)^m}{(1-vw)^\alpha} $$
with ## \Re(n),\Re(m) \geq 0## and ##\alpha = 1,2,3##.
Homework Equations
I use Table of Integrals, Series and Products by Gradshteyn & Ryzhik as a...
I have a hypergeometric distribution with:
N=total population of red and green balls, I now this
K=total number of red balls, I don't know this
n=sample size (number of investigated balls), I can choose this
k=number of investigated balls that are red, I don't know this
Red balls are a problem...
Hello all,
I have this integral, and currently I'm evaluating it using Mathematica numerically, which takes time to be evaluated. Can I write it in a way that the integral has a formula in the Table of Integrals?
\int_0^{\infty} F\left(a_1,a_2;a_3;a_4-a_5x\right) e^{-x}\,dx
where...
Hello,
For this type of question:
There are 5 green and 45 red marbles in the urn. Standing next to the urn, you close your eyes and draw 10 marbles without replacement. What is the probability that exactly 4 of the 10 are green?
I understand that I can use Hypergeometric distribution, which...
I am looking to write the hypergeometric function $${}_2F_1\left(1,1,2+\epsilon, -\frac{\alpha}{\beta}\right) = \int_0^1\,dt\,\frac{(1-t)^{\epsilon}}{1-tz + i\delta},$$ where ##z=-\alpha/\beta## and ##0< \beta < - \alpha##, in terms of its real and imaginary part. The ##i\delta## prescription...
Homework Statement
It is very well known that ## \sum^{\infty}_{n=0}x^n=\frac{1}{1-x}##. How to show that
## \sum^{\infty}_{n=0}\frac{(a)_n}{n!}x^n=\frac{1}{(1-x)^a}##
Where ##(a)_n=\frac{\Gamma(a+n)}{\Gamma(a)}##
[/B]Homework Equations
## \Gamma(x)=\int^{\infty}_0 e^{-t}t^{x-1}dt##
The...
The assignment was already turned in a while ago, but I am currently reviewing all the past homework and trying to resolve the problems I couldn't understand. The website software gives the correct multiple choice or numerical answer, but not the steps. They gave me a weird answer and I didn't...
The hypergeometric function, ##{}_{2}F_1(a,b,c;z)## can be written in terms of a power series in ##z## as follows, $${}_{2}F_1(a,b,c;z) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}\,\,\,\,\,\text{provided}\,\,\,\,|z|<1$$
So we may reexpress any hypergeometric function as a...
While I do understand the story of the hypergeometric distribution, I was wondering if there's anything "geometric" about it, or if there's any connection between the distribution and "geometry". Can anyone throw some light on it?
Thanks,
Madhav
Hi,
I need suggestions for picking up some standard textbooks for the following set of topics as given below:
Ordinary and singular points of linear differential equations
Series solutions of linear homogenous differential equations about ordinary and regular singular points...
i want to write a hypergeometric function (2F1(a,b;c,x)) as function of n that generate polynomials below
n=0 → 1
n=1 → y
n=2 → 4(ω+1)y^2-1
n=3 → y(2(2ω+3)y^2-3)
n=4 → 8(ω+2)(2ω+3)y^4-6(6+4ω)y^2+3
... → ...
2F1(a,b;c,x)=1+(ab)/(c)x+(a(a+1)b(b+1))/(c(c+1))x^2/2!+...
the...
Hi,
I have never quite worked this type of probability question out, so would like some help please.
Imagine this scenario:
There are 4 people sat around a table, A, B, C and D.
A is sitting opposite C, B is sitting opposite D.
There is a bag with 16 balls numbered 1-16.
The balls are...
According to [Erdely A,1953; Higher Transcendental Functions, Vol I, Ch. VI.] the confluent hypergeometric equation
\frac{d^2}{d x^2} y + \left(c - x \right) \frac{d}{d x} y - a y = 0
has got eight solutions, which are the followings:
y_1=M[a,c,x]
y_2=x^{1-c}M[a-c+1,2-c,x]...
How to integrate:
_{2}F_{1}(B;C;D;Ex^{2})\,Ax
where _{2}F_{1}(...) is the hypergeometric function, x is the independent variable and A, B, C, D, and E are constants.
How can I perform this integral
\begin{equation}
\int^∞_a dq \frac{1}{(q+b)} (q^2-a^2)^n (q-c)^n ?
\end{equation}
all parameters are positive (a, b, and c) and n>0.
I tried using Mathemtica..but it doesn't work!
if we set b to zero, above integral leads to the hypergeometric...
I'm having difficulty in solving an exercise.
http://imageshack.us/a/img542/484/765z.jpg [Broken]
They ask to reduce it to
http://imageshack.us/a/img203/3986/lwqb.jpg [Broken]
making the change of variables x=r^2/(r^2+1)
and then to reduce it to a hypergeometric , using...
How can I verify that
$\lim_{N,M,K \to \infty, \frac{M}{N} \to 0, \frac{KM}{N} \to \lambda} \frac{\binom{M}{x}\binom{N-M}{K-x}}{\binom{N}{K}} = \frac{\lambda^x}{x!}e^{-\lambda}$,
**without** using **Stirling's formula** or the **Poisson approximation to the Binomial**?
I have been stuck on...
Hi,
I would like to show directly,
\int \frac{e^{at}}{e^{it}+e^{-it}}dt=\frac{e^{(i+a) t} \text{Hypergeometric2F1}\left[1,\frac{1}{2}-\frac{i a}{2},\frac{3}{2}-\frac{i a}{2},-e^{2 i t}\right]}{i+a}
I realize I can differentiate the antiderivative to show the relation but was wondering...
Prove the following
{}_2 F_1 \left( a,b; c ; x \right) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int^1_0 t^{b-1}(1-t)^{c-b-1} (1-xt)^{-a} \, dt
Hypergeometric function .
Homework Statement
If we define \xi=\mu+\sqrt{\mu^2-1}, show that
P_{n}(\mu)=\frac{\Gamma(n+\frac{1}{2})}{n!\Gamma(\frac{1}{2})}\xi^{n}\: _2F_1(\frac{1}{2},-n;\frac{1}{2}-n;\xi^{-2}) where P_n is the n-th Legendre polynomial, and _2F_1(a,b;c;x) is the ordinary hypergeometric function...
Homework Statement
Calculate
_2F_1(\frac{1}{2},\frac{1}{2},\frac{3}{2};x)
Homework Equations
_2F_1(a,b,c;x)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{n!(c)_n}x^n
(a)_n=a(a+1)...(a+n-1)
The Attempt at a Solution
(\frac{1}{2})_n=\frac{1}{2}\frac{3}{2}\frac{5}{2}...\frac{2n-1}{2}...
Hypergeometric function is defined by:
_2F_1(a,b,c,x)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{n!(c)_n}x^n
where ##(a)_n=a(a+1)...(a+n-1)##...
I'm confused about this notation in case, for example, ##_2F_1(-n,b,b,1-x)##.
Is that
_2F_1(-n,b,b,1-x)=\sum^{\infty}_{n=0}\frac{(-n)_n}{n!}(1-x)^n
or...
Homework Statement
A committee of 16 persons is selected randomly from a group of 400 people, of whom are 240 are women and 160 are men. Approximate the probability that the committe contains at least 3 women.
I just want to know if it's hyper geometric or binomial. I suspect it's hyper...
Homework Statement
We have an urn with 5 red and 18 blues balls and we pick 4 balls with replacement. We denote the number of red balls in the sample by Y. What is the probability that Y >=3? (Use Binomial Distribution)
Homework Equations
The Attempt at a Solution
Okay, so we...
Homework Statement Find the general solution in terms of Hypergeometric functions near x = -1 :
(1-x2)y'' - (5x2 - 9)/5x y' + 8y = 0 The Attempt at a Solution
Here the coefficient of y' contains 9/5x which causes problem. The general form contains the coefficient of y' as A+Bx
How do I solve this?
Now, i am getting the problem with this type of function. Giving z belongs to C(field of complex numbers), f(z)=hypergeometric(1,n/2,(3+n)/2,1/z).
Do you know how we can obtain a simple performance of f(z) which allows us to take the integral of f(z)/sqrt(1-z) from 1 to Y(an particular...
I was looking at finding a series solution to a 2nd order DE the other day and came up with the following (for one of the solutions, and there was a somewhat similar series for the other solution).
\sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!} \prod_{m=1}^{k-1} (3m+1)
Wolfram said the solutions...
Homework Statement
Hermite differential equation: y"(x) -2xy'(x)+2ny(x)=0Homework Equations: y(x)=C_{n}(x)H_{n}(x) though it won't have to do with my 1st question directly & change of variable: z=x^{2}
The Attempt at a Solution: procedure: dy/dx=2\sqrt{z}dy/dz
1st Question: I want to find now...
Homework Statement
I want to differentiate the Gauss hypergeometric function:
_2F_1[a,b;c;\frac{k-x}{z-x}]
with respect to z
Homework Equations
The derivative of
_2F_1[a,b;c;z]
with respect to z is:
\frac{ab}{c} _2F_1[1+a,1+b;1+c;z]
The Attempt at a Solution
Can I treat this as...
Hi guys, I'm dealing with a function whose integral (via Wolfram integrator) carries a hypergeometric function term: 2F1(\frac{1}{4}, \frac{1}{2}, \frac{3}{2}, z).
I need to evaluate this function twice for every integral, but |z| will often be greater than 1, so I can't use the hypergeometric...
Homework Statement
Express \sum_{n=0}^{\infty} \frac{1}{(\frac{2}{3})_n} \frac{(z^3/9)^n}{n!} in terms of the Gauss hypergeometric series.
Homework Equations
The Gauss hypergeometric series has 3 parameters a,b,c: _2 F_1 (a,b;c;z) = \sum_{n=0}^{\infty}...
Homework Statement
Write \displaystyle \sum_{k=0}^{\infty} \frac{1}{9^k (\frac{2}{3})_k} \frac{w^{3k}}{k!} in terms of the Gauss hypergeometric series of the form _2 F_1(a,b;c;z).
Homework Equations
The Gauss hypergeometric series is http://img200.imageshack.us/img200/5992/gauss.png...
Homework Statement
I worked out A) just fine it seems (given the answer in the book), but B) I'm not sure how to take this out. Below was a try but I'm not sure i was even on the right track. Any ideas?
Homework Equations
The Attempt at a Solution
I'm looking for any kind of reference on a multivariable generalization of a (confluent) hypergeometric function.
In particular, Horns list is a list of 34 two-variable hypergeometric functions, 20 of which are confluent. Then one of these has the following series expansion:
\Phi_2(\beta...
Hi everyone.
So I'm afraid I don't really know much about statistics, but I am trying to learn by working through a book, and taking some examples (I have mathematics experience, but from a biological perspective).
Just now, I am looking at the hypergeometric probability...
Homework Statement
Show that by letting z = \zeta^-1 and u = \zeta^{\alpha}v(\zeta) that the differential equation,
z(1-z)\frac{d^{2}u(z)}{d^{2}z}+{\gamma - (\alpha+\beta+1)z}\frac{du(z)}{dz}-\alpha \beta u(z) = 0
can be reduced to
\zeta(1-\zeta)\frac{d^{2}v(\zeta)}{d\zeta^{2}} +...