Tagged Questions

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 (ODE)....

17 views

summation involving a hypergeometric 2F2 function

im trying to find the closed form for the following \sum_{n=0}^\infty \frac{c^n}{n!}\frac{(a)_n}{(b)_n}\frac{(\alpha+1/2)_n}{(\alpha+3/2)_n}{_2F_2}(-n,1-b-n;1-a-n,1/2;-\frac{d}{c}) \...
27 views

20 views

Does anyone know a summation formula for Wilson's Polynomials?

Wilson's polynomials are defined as $W_n(x^2; a, b, c, d) := (a+b)_n (a+c)_n (a+d)_n {\space}_4F_3(-n, n+a+b+c+d-1, a+ix, a-ix;a+b, a+c, a+d; 1)$ Does anyone know a summation formula for Wilson's ...
33 views

Integrating the lower incomplete gamma $\int_0^\infty x^{a-1}e^{-s x} \gamma(b,x) \mathrm{d}x$

I need to prove that $$\int_0^\infty x^{a-1}e^{-s x} \gamma(b,x) \mathrm{d}x = \frac{\Gamma(a+b)}{a(1+s)^{a+b}}F(1,a+b,1+b, 1/(1+s))$$ where $F(a,b,c;x)$ is the hypergeometric function. To show this,...
36 views

46 views

Summation involving a hypergeometric 1F1 function

I'm trying to find a closed form for the following: $$\sum_{n=0}^\infty \frac{(-1/4)_n}{n!(3/2)_n}\left(\frac{i}{2\tau}\right)^{n} {_1F_1(2n+1;2n+2;i k)}$$ Using the ...
36 views

153 views

24 views

How to isolate and solve for k in a Sigma notation probability mass function equation?

"isolate and solve for k:" $$P(X = k) = \sum_{k=0}^n {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}$$ If the above equation is a function of P, how would the equation be stated as a ...
24 views

turning a certain chebychev polynomial-like expression into a hypergeometric form

Can the following expression be represented in terms of hypergeometric function $$\sqrt{3}\sin(\arcsin(7/25)/3)-\cos(\arcsin(7/25)/3)$$ It looks similar to the one presented on [this site][1] [1] :...
25 views

43 views

39 views

Hypergeometric 2F3

Please could somebody tell me if there is a simpler form or known function to express the hypergeometric function next: $_2F_3\left(\frac{1}{2},\frac{1}{2};1,1,\frac{3}{2};-4 \pi ^2 a^2\right)$ I ...
40 views

What are hypergeometric functions in layman terms?

Could someone please explain what are these in layman terms? Someone here told me that and I still can't figure out what they mean on my own after giving Google a number of hits. Wikipedia says this: ...
47 views

81 views

How to evaluate the series $\sum_{i,j,k=0}^{\infty}\left(\frac{(i+j+k)!}{i!j!k!}\right)^2x^{-i-j-k}$?

Suppose the series $$\Gamma (x) =\sum_{i,j,k=0}^{\infty}\frac{((i+j+k)!)^2}{(i!)^2(j!)^2(k!)^2}x^{-i-j-k}$$ How to evaluate it? It is claimed that for $x <3$ this function converges to elliptic ...
55 views

How to evaluate following integral?

Suppose the integral $$\tag 1 I = \int \limits_{-\pi}^{\pi}dx \int \limits_{-\pi}^{\pi}\frac{dy}{\tau - \cos (2x) -2\cos(x)\cos(y)}, \quad t > 3$$ How to evaluate it in terms of elliptic integral?...