Sheffer Sequence

A sequence s_n(x) is called a Sheffer sequence iff its generating function has the form

where

with A_0,B_1!=0. Sheffer sequences are sometimes also called poweroids (Steffensen 1941, Shiu 1982, Di Bucchianico and Loeb 2000).

If f(t) is a delta series and g(t) is an invertible series, then there exists a unique sequence s_n(x) of Sheffer polynomials s_n(x) satisfying the orthogonality condition

where delta_(nk) 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 (1,f(t)) is called the associated sequence for f(t), 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 (g(t),t) is called the Appell sequence of g(t), and Roman (1984, pp. 86-106) summarizes properties of Appell sequences and gives a number of specific examples.

If s_n(x) is a Sheffer sequence for (g(t),f(t)), then for any polynomial p(x),

The sequence s_n(x) is Sheffer for (g(t),f(t)) iff

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

A sequence is Sheffer for (g(t),f(t)) for some invertible g(t) iff

for all n>=0 (Roman 1984, p. 20). The Sheffer identity states that a sequence s_n(x) is Sheffer for (g(t),f(t)) for some invertible f(t) iff it satisfies some binomial-type sequence

for all y in C, where p_n(x) is associated to f(t) (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 s_(-1)(x)=0, s_0(x)=1, and n>=0 (Meixner 1934; Sheffer 1939; Chihara 1978; Roman 1984, pp. 156-160).

The connection coefficients c_(nk) in the expression

are given by

where s_n(x) is Sheffer for (g(t),f(t)) and r_n(x) is Sheffer for (h(t),l(t)). 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 r_n(x) is Sheffer for (h(t),l(t)) (Roman 1984, pp. 132-138).

See also

Appell Cross Sequence, Appell Sequence, Binomial-Type Sequence, Cross Sequence, Steffensen Sequence, Umbral Calculus

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References

Chihara, T. S. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach, 1978.Di Bucchianico, A. and Loeb, D. "A Selected Survey of Umbral Calculus." Electronic J. Combinatorics Dynamical Survey DS3, 1-34, April 2000. http://www.combinatorics.org/Surveys/#DS3.Meixner, J. "Orthogonale Polynomsystem mit linern besonderen Gestalt der eryengenden Funktion." J. London Math. Soc. 9, 6-13, 1934.Roman, S. "Sheffer Sequences." Ch. 2 and §4.3 in The Umbral Calculus. New York: Academic Press, pp. 2, 6-31, and 107-130, 1984.Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684-760, 1973.Sheffer, I. M. "Some Properties of Polynomial Sets of Type Zero." Duke Math. J. 5, 590-622, 1939.Shiu, E. S. W. "Steffensen's Poweroids." Scand. Actuar. J. 2, 123-128, 1982.Steffensen, J. F. "The Poweroid, an Extension of the Mathematical Notion of Power." Acta Math. 73, 333-366, 1941.

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Sheffer Sequence

Cite this as:

Weisstein, Eric W. "Sheffer Sequence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ShefferSequence.html

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