Mahler polynomial

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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

∑ g n ( x ) t n / n ! = exp ⁡ ( x ( 1 + t − e t ) ) {\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)

g 0 = 1 ; {\displaystyle g_{0}=1;}
g 1 = 0 ; {\displaystyle g_{1}=0;}
g 2 = − x ; {\displaystyle g_{2}=-x;}
g 3 = − x ; {\displaystyle g_{3}=-x;}
g 4 = − x + 3 x 2 ; {\displaystyle g_{4}=-x+3x^{2};}
g 5 = − x + 10 x 2 ; {\displaystyle g_{5}=-x+10x^{2};}
g 6 = − x + 25 x 2 − 15 x 3 ; {\displaystyle g_{6}=-x+25x^{2}-15x^{3};}
g 7 = − x + 56 x 2 − 105 x 3 ; {\displaystyle g_{7}=-x+56x^{2}-105x^{3};}
g 8 = − x + 119 x 2 − 490 x 3 + 105 x 4 ; {\displaystyle g_{8}=-x+119x^{2}-490x^{3}+105x^{4};}

References


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