A kind of orthogonal polynomials and related identities

In this paper we introduce the polynomials {dn(r)(x)} and {Dn(r)(x)} given by dn(r)(x)=∑k=0n(x+r+kk)(x−rn−k)(n≥0), D0(r)(x)=1,D1(r)(x)=x and Dn+1(r)(x)=xDn(r)(x)−n(n+2r)Dn−1(r)(x)(n≥1). We show that {Dn(r)(x)} are orthogonal polynomials for r>−12, and establish many identities for {dn(r)(x)} and {Dn(r)(x)}, especially obtain a formula for dn(r)(x)2 and the linearization formulas for dm(r)(x)dn(r)(x) and Dm(r)(x)Dn(r)(x). As an application we extend recent work of Sun and Guo.