What is Hypergeometric function: Definition and 31 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 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...
  2. Tapias5000

    Integral ## \int _{ }^{ }\frac{1}{\sqrt{x^3+1}}dx ##

    I also don't understand how to get the descending factorials for this hypergeometric series, I also know that there is another way to write it with gamma functions, but in any case how am I supposed to do this? If I write it as a general term, wolfram will give me the result which leaves me...
  3. R

    I Inverse Laplace transform of a rational function

    I struggle to find an appropriate inverse Laplace transform of the following $$F(p)= 2^n a^n \frac{p^{n-1}}{(p+a)^{2n}}, \quad a>0.$$ WolframAlpha gives as an answer $$f(t)= 2^n a^n t^n \frac{_1F_1 (2n;n+1;-at)}{\Gamma(n+1)}, \quad (_1F_1 - \text{confluent hypergeometric function})$$ which...
  4. Vick

    A Associated Legendre Function of Second Kind

    The associated Legendre Function of Second kind is related to the Legendre Function of Second kind as such: $$ Q_{n}^m(z)= (-1)^m (1-z^2)^{m/2} \frac{d^m}{dz^m}(Q_{n}(z)) $$ The recurrence relations for the former are the same as those of the first kind, for which one of the relations is: $$...
  5. 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...
  6. 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...
  7. 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
  8. 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...
  9. 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...
  10. 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...
  11. 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...
  12. 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...
  13. 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.
  14. 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...
  15. 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...
  16. 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...
  17. alyafey22

    MHB Matrix-like hypergeometric function

    How to write the hypergoemtric function in a matrix like form ?
  18. 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}...
  19. 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...
  20. 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?
  21. 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...
  22. 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...
  23. 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...
  24. 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...
  25. M

    Calculating Hypergeometric Function 2F1 for |z|>1

    I posted this in the Advanced Physics forum as well, but it occurred to me that this might be a more appropriate place. I'd delete it in Advanced Physics, but I can't see where to do that. Homework Statement I'm need to integrate the function \frac{A}{(1+B^2x^2)^{\frac{C+1}{2}}} which...
  26. B

    Difficult integral involving hypergeometric function

    I am trying to calculate the following integral I=\int_0^\infty\frac{x}{(x+ia)^2} {\mbox{$_2$F$_1\!$}}\left(\frac{1}{2},b,\frac{3}{2},-\frac{x^2}{c^2}\right) dx. I tried several different ways but drew a complete blank. Is there a way of converting this nasty hypergeometric function into...
  27. S

    What is the notation for hypergeometric functions and what does it represent?

    Homework Statement I have seen some hypergeometric function in the form: 2F1=(a,b;c;d), Is there such thing as: 2F1=(a,b,c;d) Homework Equations The Attempt at a Solution I don't understand why sometimes we have a comma and sometimes we have a semi-colon. thank you
  28. B

    Hypergeometric Function around z=1/2

    Hello, for some calculation I need the behaviour of the hypergeometric function 2F1 near z=\tfrac{1}{2}. Specifically I need _2 F_1(\mu,1-\mu,k,\tfrac{1}{2}+i x) with x\in \mathbb{R} near 0, and 1/2\leq\mu\leq 2, 1\leq k \in \mathbb{N}. Differentiating around x=0 and writing the...
  29. G

    What is the Hypergeometric Function for the Given Series?

    Homework Statement write the following serie in the form of hypergeometric function: f(x)=\sum\frac{(-1*(x^2))^n}{(2^n)(2n-1)(2n+1)(2n+3)} n changes from 0 to \infty Homework Equations hypergeometric function: The Attempt at a Solution guys i have thought about this for 2...
  30. G

    What is the formula for F(a, a+1/2, 3/2, z^2) and its general form?

    How would i go about showing the special case F(1, b, b; x) of the hypergeometic function is the geometric series and also how the geometric series is = 1/ (1 -x) Cheers, Dave
  31. W

    I'm not sure if I understand the question - please clarify!

    I am having trouble with a problem that asks me to show that if I change the variable of integration of the following equation from t to t-1 the following http://mathworld.wolfram.com/images/equations/EulersHypergeometricTransformations/equation1.gif (disregard that z in the denominator, that...
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