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Boole Polynomial

Polynomials s_k(x;lambda) which form a Sheffer sequence with

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

and have generating function

sum_(k=0)^infty(s_k(x;lambda))/(k!)t^k=((1+t)^x)/(1+(1+t)^lambda).
(3)

The first few are

s_0(x;lambda)=1/2
(4)
s_1(x;lambda)=1/4(2x-lambda)t
(5)
s_2(x;lambda)=1/4[2x(x-lambda-1)+lambda].
(6)

Jordan (1965) considers the related polynomials r_n(x) which form a Sheffer sequence with

g(t)=1/2(1+e^t)
(7)
f(t)=e^t-1.
(8)

These polynomials have generating function

sum_(k=0)^infty(r_n(x))/(k!)t^k=(2(1+t)^x)/(2+t).
(9)

The first few are

r_0(x)=1
(10)
r_1(x)=1/2(2x-1)
(11)
r_2(x)=1/2(2x^2-4x+1)
(12)
r_3(x)=1/4(4x^3-18x^2+20x-3).
(13)

The Peters polynomials are a generalization of the Boole polynomials.

See also

Peters 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.Jordan, C. Calculus of Finite Differences, 3rd ed. New York: Chelsea, 1965.Roman, S. The Umbral Calculus. New York: Academic Press, 1984.

Referenced on Wolfram|Alpha

Boole Polynomial

Cite this as:

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

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