Mahler polynomial

In mathematics, the Mahler polynomials g_n(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+t–e^t (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, vol. 111, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-594380-2, MR 0741185 Reprinted by Dover, 2005