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
| [画像: s_n(x)=(x^n)/(g(t)) ] |
(2)
|
(Roman 1984, p. 27). The Appell identity states that the sequence s_n(x) is an Appell sequence iff
(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 CalculusExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Appell SequenceCite this as:
Weisstein, Eric W. "Appell Sequence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AppellSequence.html