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In mathematics, the Mahler polynomials gn(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+tet (Roman 1984, 4.9).

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, 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 0741185 Reprinted by Dover, 2005

Mathematics Encyclopedia