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Appell Cross Sequence

A sequence

s_n^((lambda))(x)=[h(t)]^lambdas_n(x),

where s_n(x) is a Sheffer sequence, h(t) is invertible, and lambda ranges over the real numbers is called a Steffensen sequence. If s_n(x) is an associated Sheffer sequence, then s_n^((lambda)) is called a cross sequence. If s_n(x)=x^n, then

s_n^((lambda))(x)=[h(t)]^lambdax^n

is called an Appell cross sequence.

Examples include the Bernoulli polynomial, Euler polynomial, and Hermite polynomial.

See also

Appell Sequence, Cross Sequence, Sheffer Sequence, Steffensen Sequence

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References

Roman, S. "Cross Sequences and Steffensen Sequences." §5.3 in The Umbral Calculus. New York: Academic Press, pp. 140-143, 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 Cross Sequence

Cite this as:

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

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