The symmetric Meixner-Pollaczek polynomials with real parameter

The Symmetric Meixner-Pollaczek polynomials pn(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, pn(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 pn(λ)(x/2,π/2), for λ⩽0, become orthogonal polynomials. It examines the major properties of the polynomials, pn(λ)(x/2,π/2), for λ>0 which are also shared by the polynomials, pn(λ)(x/2,π/2), for λ⩽0.