Hypergeometric function Definition and 31 Threads

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 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...

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...

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: $$...

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...

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

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...

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...

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...

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...

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 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...

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...

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

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 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...

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...

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...

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...

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...

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

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...

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...

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

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...