An operator approach to the ω-transfer correspondence between Appell and Central ω-Sheffer Families

Abstract

Let  denote the central difference operator of step . The polynomial sequences that are Sheffer relative to     -Sheffer class have been studied through their generating functions, but their relation to the classical Appell class has not been made explicit. We construct a single linear operator  on the space of polynomials which carries every Appell sequence to its central -Sheffer partner, and which depends on the step alone and not on the invertible series defining the sequence. The operator is characterised as the unique isomorphism conjugating the derivative into , and its matrix is that of the -central factorial numbers. We then solve the connection problem between the two classes. The connection matrix is shown to be a conjugation, , and this is the feature specific to the central setting to be an even polynomial deformation of the identity: every entry lies in  with , and . For the basic sequence the matrix has checkerboard support,  whenever . Exploiting the deformation structure we obtain the first-order expansion s_n^[ω] (x)=A_n (x)-ω^2/24 n(n-1)(n-2) x A_(n-3) (x)+O(ω^4 ), valid for every Appell sequence, together with a determinantal inversion formula. Two families are worked out completely — the -central factorials and the -central Bernoulli polynomials  with full symbolic verification.