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)[reply]

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.

Relation to spheres

I'd like to see a section relating these to spheres. If I surmise correctly, if one is computing functions on the n-sphere $S_{n}$ and one has a function $f(\langle x,y\rangle )$ of the dot product $\langle x,y\rangle$ of two points $x,y\in S_{n}$ then one would use an expansion in $C_{j}^{\alpha }(\langle x,y\rangle )$ with $\alpha =n/2$ . The reason for this is the weighting $(1-t^{2})^{\alpha -1/2}$ with $t=\langle x,y\rangle$ is just the uniform Lesbesgue measure on the sphere after integrating out all the other angles. This sounds about right to me, but being lazy and stupid means I would rather read about it here on WP than to ask some LLM about it.

Oh, I guess it should also link to some treatment of spherical harmonics on these spheres. Hmm. Except after skimming that article, I'm confused, again... Argh. 67.198.37.16 (talk) 03:20, 21 March 2025 (UTC)[reply]

Oh foo. In fact this article does sort-of already kind-of say this, in the "applications" section, but I was skimming so quick I missed it. Its late, I'm tired. Maybe I should go to bed instead of posting on WP. 67.198.37.16 (talk) 03:27, 21 March 2025 (UTC)[reply]