# Mahler polynomial

In mathematics, the Mahler polynomials gn(x) are polynomials introduced by Mahler (1930) in his work on the zeros of the incomplete gamma function.

Mahler polynomials are given by the generating function

${\displaystyle \displaystyle \sum g_{n}(x)t^{n}/n!=\exp(x(1+t-e^{t}))}$

Mahler polynomials can be given as the Sheffer sequence for the functional inverse of 1+tet (Roman 1984, 4.9).

The first few examples are (sequence A008299 in the OEIS)

${\displaystyle g_{0}=1;}$
${\displaystyle g_{1}=0;}$
${\displaystyle g_{2}=-x;}$
${\displaystyle g_{3}=-x;}$
${\displaystyle g_{4}=-x+3x^{2};}$
${\displaystyle g_{5}=-x+10x^{2};}$
${\displaystyle g_{6}=-x+25x^{2}-15x^{3};}$
${\displaystyle g_{7}=-x+56x^{2}-105x^{3};}$
${\displaystyle g_{8}=-x+119x^{2}-490x^{3}+105x^{4};}$

## References

• Mahler, Kurt (1930), "Über die Nullstellen der unvollständigen Gammafunktionen.", Rendiconti Palermo (in German), 54: 1–41, JFM 56.0310.01
• Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics, 111, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-594380-2, MR 741185 Reprinted by Dover, 2005