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
![画像: sum_(k=0)^infty(s_k(x;lambda,mu))/(k!)t^k=[1+(1+t)^lambda]^(-mu)(1+t)^x.](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/PetersPolynomial/NumberedEquation1.svg){kind=link}
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 PolynomialExplore with Wolfram|Alpha
WolframAlpha
More things to try:
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 PolynomialCite this as:
Weisstein, Eric W. "Peters Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PetersPolynomial.html