Appell Sequence

An Appell sequence is a Sheffer sequence for (g(t),t). Roman (1984, pp. 86-106) summarizes properties of Appell sequences and gives a number of specific examples.

The sequence s_n(x) is Appell for g(t) iff


for all y in the field C of field characteristic 0, and iff


(Roman 1984, p. 27). The Appell identity states that the sequence s_n(x) is an Appell sequence iff

 s_n(x+y)=sum_(k=0)^n(n; k)s_k(y)x^(n-k)

(Roman 1984, p. 27).

The Bernoulli polynomials, Euler polynomials, and Hermite polynomials are Appell sequences (in fact, more specifically, they are Appell cross sequences).

See also

Appell Cross Sequence, Sheffer Sequence, Umbral Calculus

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Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Netherlands: Reidel, pp. 209-210, 1988.Roman, S. "Appell Sequences." §2.5 and §2 in The Umbral Calculus. New York: Academic Press, pp. 17 and 26-28 and 86-106, 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.

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Appell Sequence

Cite this as:

Weisstein, Eric W. "Appell Sequence." From MathWorld--A Wolfram Web Resource.

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