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.

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  1. T

    I Hypergeometric Limits: Analyzing p(x)

    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...
  2. T

    I Converting Second Order ODE to Hypergeometric Function

    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...
  3. benorin

    I Hypergeometric Functions Identities: n_F_n & (n+1)_F_n

    See attachment for identities and proofs, if you find my proofs are incorrect in some way please post it. Thanks for your time.
  4. rafgger

    A Expectation of a Fraction of Gaussian Hypergeometric Functions

    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...
  5. L

    Legendre polynomials, Hypergeometric function

    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...
  6. cg78ithaca

    A Generalization of hypergeometric type differential equation

    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...
  7. C

    Mathematica Cannot do the integral of the Hyper-geometric function?

    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...
  8. S

    How to plot generalized hypergeometric function in ROOT?

    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
  9. Euler2718

    I Hypergeometric Distribution Calculation in Libreoffice

    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...
  10. Dirickby

    Double Integral: solution with hypergeometric function?

    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...
  11. M

    A Sample size required in hypergeometric test

    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...
  12. E

    I Simplifying integral of Gauss' hypergeometric function

    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...
  13. H

    I Hypergeometric distribution with different distributions

    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...
  14. C

    Principal value of hypergeometric function

    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...
  15. L

    Hypergeometric function. Summation question

    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...
  16. C

    Understanding Probability and Observations in Statistics

    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...
  17. N

    Orthogonal properties of confluent hypergeometric functions

    Hi Can anyone point to me a reference where orthogonal properties of confluent hypergeometric functions are discussed? Navaneeth
  18. C

    Convergence of a hypergeometric

    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...
  19. M

    On the hypergeometric distribution

    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
  20. Soumalya

    TextBooks for Some Topics in Mathematics

    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...
  21. A

    Convert a polynomial to hypergeometric function

    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...
  22. A

    Probability question - hypergeometric distribution?

    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...
  23. D

    Wronskian of the confluent hypergeometric functions

    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]...
  24. J

    Integration with hypergeometric function

    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.
  25. D

    Integral could lead to Hypergeometric function

    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...
  26. B

    How to Reduce a Function to a Hypergeometric Using a Change of Variables?

    I'm having difficulty in solving an exercise. http://imageshack.us/a/img542/484/765z.jpg They ask to reduce it to http://imageshack.us/a/img203/3986/lwqb.jpg making the change of variables x=r^2/(r^2+1) and then to reduce it to a hypergeometric , using...
  27. K

    MHB Verification of poisson approximation to hypergeometric distribution

    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...
  28. J

    How to show integral equal to hypergeometric function?

    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...
  29. alyafey22

    MHB What is the Hypergeometric Challenge #2?

    Prove the following _2F_1 \left(a,1-a;c; \frac{1}{2} \right) = \frac{\Gamma \left(\frac{c}{2} \right)\Gamma \left(\frac{1+c}{2} \right) } {\Gamma \left(\frac{c+a}{2}\right)\Gamma \left(\frac{1+c-a}{2}\right)}.
  30. alyafey22

    MHB Matrix-like hypergeometric function

    How to write the hypergoemtric function in a matrix like form ?
  31. alyafey22

    MHB Solving the Hypergeometric Function Integral Representation

    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 .
  32. R

    Relationship between Legendre polynomials and Hypergeometric functions

    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...
  33. D

    Statistics Hypergeometric Probability Distribution

    Homework Statement Uploaded Homework Equations Uploaded The Attempt at a Solution Is my work correct?
  34. S

    MHB Can the Hypergeometric Equation Prove that tan^-1x = xF(1/2, 1, 3/2, -x^2)?

    Show that, \[\tan^{-1}x = xF\left(\frac{1}{2},\, 1,\, \frac{3}{2},\, -x^2\right)\]
  35. M

    Hypergeometric function problem

    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}...
  36. M

    Understanding Notation and Connections of Hypergeometric Functions

    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...
  37. X

    Probability distributions binomial or hypergeometric

    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...
  38. H

    Binomial and Hypergeometric Distributions

    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...
  39. Z

    Hypergeometric Function D.E. Solution | Near x = -1 | No Quotation Marks

    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?
  40. J

    How Can We Simplify the Hypergeometric Function for Easier Integration?

    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...
  41. U

    Are the sum and the hypergeometric equal?

    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...
  42. M

    Confluent Hypergeometric Function

    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...
  43. F

    Gauss hypergeometric function derivative

    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...
  44. C

    Algebraic Manipulation of Hypergeometric F'n Parameters

    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...
  45. T

    Gauss hypergeometric series

    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}...
  46. T

    Writing this series as a hypergeometric series

    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...
  47. R

    Multivariate Hypergeometric Distribution

    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
  48. X

    Multivariable confluence hypergeometric function

    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...
  49. N

    Hypergeometric Probability Testing: Simple Question

    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...
  50. J

    Hypergeometric equation at z = infinity

    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}} +...
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