Mahler polynomial

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In mathematics, the Mahler polynomials g_n(x) are polynomials introduced by Mahler{{harvtxt|Mahler|1930|authorlink=Kurt Mahler}} in his work on the zeros of the incomplete gamma function.

Mahler polynomials are given by the generating function

:$\backslash displaystyle\; \backslash sum\; g\_n(x)t^n/n!\; =\; \backslash exp(x(1+t-e^t))$

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

The first few examples are {{OEIS|A008299}}

:$g\_0=1;$

:$g\_1=0;$

:$g\_2=-x;$

:$g\_3=-x;$

:$g\_4=-x+3x^2;$

:$g\_5=-x+10x^2;$

:$g\_6=-x+25x^2-15x^3;$

:$g\_7=-x+56x^2-105x^3;$

:$g\_8=-x+119x^2-490x^3+105x^4;$