The symmetric Meixner–Pollaczek polynomials with real parameter

Abstract

The Symmetric Meixner–Pollaczek polynomials p n ( λ ) ( x / 2 , π / 2 ) , for λ > 0 are well-studied polynomials. These are polynomials orthogonal on the real line with respect to a continuous, positive real measure. For λ ⩽ 0 , p n ( λ ) ( x / 2 , π / 2 ) are also polynomials, however they are not orthogonal on the real line with respect to any real measure. This paper defines a non-standard inner product with respect to which the polynomials p n ( λ ) ( x / 2 , π / 2 ) , for λ ⩽ 0 , become orthogonal polynomials. It examines the major properties of the polynomials, p n ( λ ) ( x / 2 , π / 2 ) , for λ > 0 which are also shared by the polynomials, p n ( λ ) ( x / 2 , π / 2 ) , for λ ⩽ 0 .