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

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

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 OEIS)

$g_0=1;$

$g_1=0;$

$g_2=-x;$

$g_3=-x;$

$g_4=-x+3x^2;$

$g_5=-x+10x^2;$

$g_6=-x+25x^2-15x^3;$

$g_7=-x+56x^2-105x^3;$

$g_8=-x+119x^2-490x^3+105x^4;$

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