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Peters Polynomial

Polynomials s_k(x;lambda,mu) which are a generalization of the Boole polynomials, form the Sheffer sequence for

g(t)=(1+e^(lambdat))^mu
(1)
f(t)=e^t-1
(2)

and have generating function

sum_(k=0)^infty(s_k(x;lambda,mu))/(k!)t^k=[1+(1+t)^lambda]^(-mu)(1+t)^x.
(3)

The first few are

s_0(x;lambda,mu)=2^(-mu)
(4)
s_1(x;lambda,mu)=2^(-(mu+1))(2x-lambdamu)
(5)

and

s_2(x;lambda,mu)=2^(-(mu+2))[4x(x-1)+(2-4x)lambdamu+mu(mu-1)lambda^2].
(6)

See also

Boole Polynomial

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References

Boas, R. P. and Buck, R. C. Polynomial Expansions of Analytic Functions, 2nd print., corr. New York: Academic Press, p. 37, 1964.Roman, S. "The Peters Polynomial." §4.6 in The Umbral Calculus. New York: Academic Press, p. 128, 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

Peters Polynomial

Cite this as:

Weisstein, Eric W. "Peters Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PetersPolynomial.html

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