Some identities involving generalized Gegenbauer polynomials

• Main Author: Zhaoxiang Zhang
• Format: Article
• Language: English
• Published: SpringerOpen 2017-12-01
• Series: Advances in Difference Equations
• Subjects: generalized Gegenbauer polynomials; orthogonality; generalized inner product space
• Online Access: http://link.springer.com/article/10.1186/s13662-017-1445-2

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 . $$\bigl\langle {{p_{1}}(x),{p_{2}}(x)} \bigr\rangle = \int_{ - \frac{{\sqrt{\alpha q}}}{p}}^{\frac{{\sqrt{ \alpha q} }}{p}} {{\bigl(\alpha q - p^{2}{x^{2}}\bigr)}^{\lambda - \frac{1}{2}}} {p_{1}}(x){p_{2}}(x)\,dx. $$