Gould polynomials

In mathematics the Gould polynomials G_n(x; a,b) are polynomials introduced by H. W. Gould and named by Roman in 1984.^[1] They are given by ^[2]

\displaystyle \exp(xf^{-1}(t))=\sum _{n=0}^{\infty }G_{n}(x;a,b){\frac {t^{n}}{n!}}

where

f(t)=e^{at}(e^{bt}-1)

so

f^{-1}(t)={\frac {1}{b}}\sum _{k=1}^{\infty }{\binom {-(b+ak)/b}{k-1}}{\frac {t^{k}}{k}}

References

^ Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics, vol. 111, London: Academic Press Inc. Harcourt Brace Jovanovich Publishers, ISBN 978-0-12-594380-2, MR 0741185, Reprinted by Dover, 2005
^ Gould, H. W. (1961), "A series transformation for finding convolution identities", Duke Math. J. Volume 28, Number 2, 193-202.

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