Mahler polynomial - Alchetron, The Free Social Encyclopedia

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

∑ 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+t–e^t (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 + 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

Mahler polynomial Wikipedia

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