How to represent this quotient of gamma functions as a continued fraction

Asked Oct 20 '17 at 18:40

Active Oct 20 '17 at 19:46

Viewed 197 times

I want to ask how to represent the following quotient as a continued fraction. I think we may use Gauss continued fraction but I am not too confident about it.

$\dfrac{\Gamma(x + \frac{1}{2})}{\Gamma(x + 1)}\tag{1}=\cfrac{a_{1}}{b_{1}+}\cfrac{a_{2}}{b_{2}+}\cfrac{a_{3}}{b_{3}+}\cdots$

Thanks in advance.

edited Oct 20 '17 at 19:46

Jack D'Aurizio

353,855

asked Oct 20 '17 at 18:40

Juan Alfaro

Up to a $\Gamma\left(\frac{1}{2}\right)$ term, the LHS is a Beta function. Have a look here: http://functions.wolfram.com/GammaBetaErf/Beta3/10/ – Jack D'Aurizio Oct 20 '17 at 19:48
1

See Conjectured general continued fraction for the ratio of gamma functions and also Ramanujan's continued fraction for ratio of gamma values which may shed some light – Nicco Oct 21 '17 at 14:51

How to represent this quotient of gamma functions as a continued fraction

0 Answers0