Mahler polynomial

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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

\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 + 3 x^{2};

g_{5} = - x + 10 x^{2};

g_{6} = - x + 25 x^{2} - 15 x^{3};

g_{7} = - x + 56 x^{2} - 105 x^{3};

g_{8} = - x + 119 x^{2} - 490 x^{3} + 105 x^{4};

References

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