Talk:Gegenbauer polynomials

Unreferenced content moved here

Moved unreferenced content that someone added. Given the recent sockpuppetry involving IP addresses in Austria and User:A. Pichler in our articles on special functions, please provide citations before adding such content. Sławomir Biały (talk) 00:48, 20 October 2012 (UTC)

• Another important series expansion is given by

$\sum _{n=0}^{\infty }{\frac {C_{n}^{(\alpha )}(x)}{2\alpha +n-1 \choose n}}{\frac {t^{n}}{n!}}=\Gamma \left(\alpha +{\frac {1}{2}}\right)e^{tx}{\frac {J_{\alpha -{\frac {1}{2}}}\left(t{\sqrt {1-x^{2}}}\right)}{\left({\frac {1}{2}}t{\sqrt {1-x^{2}}}\right)^{\alpha -{\frac {1}{2}}}}},$

where $J_{\alpha }$ is the Bessel function.

The Askey–Gasper inequality has the generalization

$\sum _{j=0}^{n}{\frac {C_{j}^{\alpha }(x)}{2\alpha +j-1 \choose j}}s^{j}={\frac {\,_{2}F_{1}\left(\alpha -{\frac {1}{2}},{\frac {1}{2}};\alpha +{\frac {1}{2}};{\frac {s^{2}(1-x)(1+x)}{1-2sx+s^{2}}}\right)}{\sqrt {1-2xs+s^{2}}}}\geq 0\qquad (-1\leq x\leq 1,\,-1\leq s\leq 1,\,\alpha \geq 0).$

for Gegenbauer polynomials.