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Gould Polynomial

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The polynomials G_n(x;a,b) given by the associated Sheffer sequence with

f(t)=e^(at)(e^(bt)-1),
(1)

where b!=0. The inverse function (and therefore generating function) cannot be computed algebraically, but the generating function

sum_(k=0)^infty(G_k(x;a,b))/(k!)t^k=e^(xf^(-1)(t))
(2)

can be given in terms of the sum

f^(-1)(t)=sum_(k=1)^infty1/b(-(b+ak)/b; k-1)(t^k)/k.
(3)

This results in

G_n(x;a,b)=x/(x-an)((x-an)/b)_n
(4)

where (x)_n is a falling factorial. The first few are

G_0(x;a,b)=1
(5)
G_1(x;a,b)=x/b
(6)
G_2(x;a,b)=-((2a+b-x)x)/(b^2)
(7)
G_3(x;a,b)=((3a+b-x)(3a+2b-x)x)/(b^3)
(8)
G_4(x;a,b)=-((4a+b-x)(4a+2b-x)(4a+3b-x)x)/(b^4).
(9)

The binomial identity obtained from the Sheffer sequence gives the generalized Chu-Vandermonde identity

(x+y)/(x+y-an)((x+y-an)/b; n) =sum_(k=0)^nx/(x-ak)y/(y-a(n-k))((x-ak)/b; k)((y-a(n-k))/b; n-k)
(10)

(Roman 1984, p. 69; typo corrected).

In the special case a=-b/2, the function f(t) simplifies to

f(t)=e^(bt/2)-e^(-bt/2)=2sinh(1/2bt),
(11)

which gives the generating function

sum_(k=0)^infty(G_k(x;-1/2b,b))/(k!)t^k=exp[(2xsinh^(-1)(1/2t))/b],
(12)

giving the polynomials

G_0(x;-1/2b,b)=1
(13)
G_1(x;-1/2b,b)=x/b
(14)
G_2(x;-1/2b,b)=(x^2)/(b^2)
(15)
G_3(x;-1/2b,b)=-(x(b-2x)(b+2x))/(4b^3)
(16)
G_4(x;-1/2b,b)=-(x^2(b-x)(b+x))/(b^4).
(17)

See also

Central Factorial, Falling Factorial, Sheffer Sequence

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References

Gould, H. W. "Note on a Paper of Sparre-Anderson." Math. Scand. 6, 226-230, 1958.Gould, H. W. "Stirling Number Representation Problems." Proc. Amer. Math. Soc. 11, 447-451, 1960.Gould, H. W. "A Series of Transformation for Finding Convolution Identities." Duke Math. J. 28, 193-202, 1961.Gould, H. W. "Note on a Paper of Klamkin Concerning Stirling Numbers." Amer. Math. Monthly 68, 477-479, 1961.Gould, H. W. "A New Convolution Formula and Some New Orthogonal Relations for the Inversion of Series." Duke Math. J. 29, 393-404, 1962.Gould, H. W. "Congruences Involving Sums of Binomial Coefficients and a Formula of Jensen." Amer. Math. Monthly 69, 400-402, 1962.Roman, S. "The Gould Polynomials and he Central Factorial Polynomials." §4.1.4 in The Umbral Calculus. New York: Academic Press, pp. 67-70, 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.

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Gould Polynomial

Cite this as:

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

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