Some identities involving generalized Gegenbauer polynomials

Abstract

Abstract In this paper, we investigate some interesting identities on the Bernoulli, Euler, Hermite and generalized Gegenbauer polynomials arising from the orthogonality of generalized Gegenbauer polynomials in the generalized inner product 〈 p 1 ( x ) , p 2 ( x ) 〉 = ∫ − α q p α q p ( α q − p 2 x 2 ) λ − 1 2 p 1 ( x ) p 2 ( x ) d x . $$\bigl\langle {{p_{1}}(x),{p_{2}}(x)} \bigr\rangle = \int_{ - \frac{{\sqrt{\alpha q}}}{p}}^{\frac{{\sqrt{ \alpha q} }}{p}} {{\bigl(\alpha q - p^{2}{x^{2}}\bigr)}^{\lambda - \frac{1}{2}}} {p_{1}}(x){p_{2}}(x)\,dx. $$

Keywords

• generalized Gegenbauer polynomials
• orthogonality
• generalized inner product space