Gould Polynomial
The polynomials 
given by the associated Sheffer sequence with
 |
(1)
|
where 
. The inverse function (and therefore generating function) cannot be computed algebraically, but the generating function
 |
(2)
|
can be given in terms of the sum
 |
(3)
|
This results in
 |
(4)
|
where 
is a falling factorial. The first few are
The binomial identity obtained from the Sheffer sequence gives the generalized Chu-Vandermonde identity
 |
(10)
|
(Roman 1984, p. 69; typo corrected).
In the special case 
, the function 
simplifies to
 |
(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],](http://mathworld.wolfram.com/images/equations/GouldPolynomial/NumberedEquation7.gif) |
(12)
|
giving the polynomials
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.
