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# Appell Sequence

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

The sequence is Appell for iff

 (1)

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

 (2)

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

 (3)

(Roman 1984, p. 27).

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

Appell Cross Sequence, Sheffer Sequence, Umbral Calculus

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## References

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.

Appell Sequence

## Cite this as:

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