Orthogonal Study Of Generalized Chebyshev And Gegenbauer Polynomials

For any integer n ? 0,let Up,q,n(x)denote the second kind(p,q)-Chebyshev polynomials.That is,for any postivc integer n>1,Up,q,0(x)= 1,Up,q,1(x)=2px,and Up,q,n+1(x)= 2pxUp,q,n(x)-qUp,q,n-1(x).This paper is using the elementary method,some identities involving power series and(p,q)-Gegenbauer polynomials to study the computational problem of the convolution about the second kind(p,q)-Chebyshev polynomials,and give an computational results for it.Secondly,Gegenbauer polynomials and inner product can be extend-ed.So as to obtain the generalized Gegenbauer polynomials and general-ized inner product space.Thus,we investigate some interesting identities on the Bernoulli,Euler,Hermite and generalized Gegenbauer polynomi-als arising from the orthogonality of generalized Gegenbauer polynomials in the generalized inner product Pn = {p{x)? R[x]|deg p(x)<n}((?)(?q-p2x2)?-1/2P1(x)p2(x)dx).