Talk:Gegenbauer polynomials

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 * Another important series expansion is given by
 * $$\sum _{n=0}^\infty\frac{C_n^{(\alpha )}(x)}\frac{t^n}{n!}= \Gamma \left(\alpha +\frac 12\right)e^{t x} \frac{ J_{\alpha -\frac{1}{2}}\left(t \sqrt{1-x^2}\right)}{\left( \frac 1 2 t\sqrt{1-x^2}\right)^{\alpha-\frac 1 2}},$$
 * where $$J_\alpha$$ is the Bessel function.

The Askey–Gasper inequality has the generalization
 * $$\sum_{j=0}^n\frac{C_j^\alpha(x)}s^j =\frac{\,_2F_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}}

\ge 0\qquad (-1\le x\le1,\, -1\le s\le 1,\, \alpha\ge 0).$$ for Gegenbauer polynomials.