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

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

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

The first few examples are (sequence A008299 in the 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, doi:10.1007/BF03021175, JFM 56.0310.01