Sheffer Sequence

A sequence is called a Sheffer sequence iff its generating function has the form

where

with . Sheffer sequences are sometimes also called poweroids (Steffensen 1941, Shiu 1982, Di Bucchianico and Loeb 2000).

If is a delta series and is an invertible series, then there exists a unique sequence of Sheffer polynomials satisfying the orthogonality condition

where is the Kronecker delta (Roman 1984, p. 17). Examples of general Sheffer sequences include the actuarial polynomials, Bernoulli polynomials of the second kind, Boole polynomials, Laguerre polynomials, Meixner polynomials of the first and second kinds, Poisson-Charlier polynomials, and Stirling polynomials.

The Sheffer sequence for is called the associated sequence for , and Roman (1984, pp. 53-86) summarizes properties of the associated Sheffer sequences and gives a number of specific examples (Abel polynomial, Bell polynomial, central factorial, Bell polynomial, falling factorial, Gould polynomial, Mahler polynomial, Mittag-Leffler polynomial, Mott polynomial, power polynomial). The Sheffer sequence for is called the Appell sequence of , and Roman (1984, pp. 86-106) summarizes properties of Appell sequences and gives a number of specific examples.

If is a Sheffer sequence for , then for any polynomial ,

The sequence is Sheffer for iff

for all in the field of characteristic 0, where is the compositional inverse function of (Roman 1984, p. 18). This formula immediately gives the generating function associated with a given Sheffer sequence.

A sequence is Sheffer for for some invertible iff

for all (Roman 1984, p. 20). The Sheffer identity states that a sequence is Sheffer for for some invertible iff it satisfies some binomial-type sequence

for all in , where is associated to (Roman 1984, p. 21). The recurrence relation for Sheffer sequences is given by

(Roman 1984, p. 50). A nontrivial recurrence relation is given by

for , , and (Meixner 1934; Sheffer 1939; Chihara 1978; Roman 1984, pp. 156-160).

The connection coefficients in the expression

are given by

where is Sheffer for and is Sheffer for . This can also be written in terms of the polynomial of coefficients

which is Sheffer for

(Roman 1984, pp. 132-138).

A duplication formula of the form

is given by

where is Sheffer for (Roman 1984, pp. 132-138).