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

[math]\displaystyle{ \displaystyle \sum g_n(x)t^n/n! = \exp(x(1+t-e^t)) }[/math]

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

The first few examples are (sequence A008299 in the OEIS)

[math]\displaystyle{ g_0=1; }[/math]

[math]\displaystyle{ g_1=0; }[/math]

[math]\displaystyle{ g_2=-x; }[/math]

[math]\displaystyle{ g_3=-x; }[/math]

[math]\displaystyle{ g_4=-x+3x^2; }[/math]

[math]\displaystyle{ g_5=-x+10x^2; }[/math]

[math]\displaystyle{ g_6=-x+25x^2-15x^3; }[/math]

[math]\displaystyle{ g_7=-x+56x^2-105x^3; }[/math]

[math]\displaystyle{ g_8=-x+119x^2-490x^3+105x^4; }[/math]

References

• Mahler, Kurt (1930), "Über die Nullstellen der unvollständigen Gammafunktionen." (in German), Rendiconti Palermo 54: 1–41