Mahler polynomial

In mathematics, the Mahler polynomials g_n(x) are polynomials introduced by Mahler^[1] in his work on the zeros of the incomplete gamma function.

Mahler polynomials are given by the generating function

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

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

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

1. Mahler, Kurt (December 1930). "Ueber die nullstellen der unvollstaendigen Gammafunktionen". Rendiconti del Circolo Matematico di Palermo (in Italian). 54 (1): 1–41. doi:10.1007/BF03021175. ISSN 0009-725X.