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Pidduck Polynomial

Polynomials P_k(x) which form the Sheffer sequence for

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

and have generating function

sum_(k=0)^infty(P_k(x))/(k!)t^k=t/(1-t)((1+t)/(1-t))^x.
(3)

The first few are

P_0(x)=1
(4)
P_1(x)=2x+1
(5)
P_2(x)=4x^2+4x+2
(6)
P_3(x)=8x^3+12x^2+16x+6.
(7)

The Pidduck polynomials are related to the Mittag-Leffler polynomials M_n(x) by

P_n(x)=1/2(e^t+1)M_n(x)
(8)

(Roman 1984, p. 127).

See also

Mittag-Leffler Polynomial, Sheffer Sequence

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References

Bateman, H. "The Polynomial of Mittag-Leffler." Proc. Nat. Acad. Sci. USA 26, 491-496, 1940.Boas, R. P. and Buck, R. C. Polynomial Expansions of Analytic Functions, 2nd print., corr. New York: Academic Press, p. 38, 1964.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 3. New York: Krieger, p. 248, 1981.Roman, S. The Umbral Calculus. New York: Academic Press, 1984.

Referenced on Wolfram|Alpha

Pidduck Polynomial

Cite this as:

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

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