A Combinatorial Formula for the Linearization Coefficients of General Sheffer Polynomials

We prove a formula for the linearization coefficients of the general Sheffer polynomials, which unifies all the special known results for Hermite, Charlier, Laguerre, Meixner and Meixner–Pollaczek polynomials. Furthermore, we give a new and explicit real version of the corresponding formula for Meixner–Pollaczek polynomials. Our proof is based on some explicit bijections and sign-reversing weight-preserving involutions.