Mahler polynomial explained

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

Mahler polynomials are given by the generating function

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

Which is close to the generating function of the Touchard polynomials.

The first few examples are

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