// Workers AI · dad joke modeWhat did the Gould polynomial say? "I'm a degree above the rest.

In mathematics, the Gould polynomials $G_{n}$, $n\in \mathbb {N}$, are polynomials introduced by Henry W. Gould. They are defined in terms of the falling factorial by

$G_{n}(x;a,b)={\frac {x}{x-an}}{\Big (}{\frac {x-an}{b}}{\Big )}_{n}$.

They form the Sheffer sequence for $f(t)=e^{at}(e^{bt}-1)$, $b\neq 0$, which has generating function

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

The inverse function is given by

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

See also

• Umbral calculus

External links

• Weisstein, Eric W. "Gould polynomials". MathWorld.