Mahler multinomial expression

I call the multinomial expression that there is introduced in 0 points of studies of the incomplete gamma function by Mahler (1930) Mahler multinomial expression (I will burn Mahler and come British: Mahler polynomials) g_n(x) in the mathematics.

The Mahler multinomial expression is given by the next generating function.

${\displaystyle \displaystyle \sum g_{n}(x)t^{n}/n!=\exp(x(1+t-e^{t}))}$

The Mahler multinomial expression is provided as a Schaefer line for the inverse function of 1+t–e^t (Roman 1984, 4.9).

I give below some of the Mahler multinomial expression of beginning (progression A008299 of the online integer row Dictionary).

${\displaystyle g_{0}=1;}$

${\displaystyle g_{1}=0;}$

${\displaystyle g_{2}=-x;}$

${\displaystyle g_{3}=-x;}$

${\displaystyle g_{4}=-x+3x^{2};}$

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

${\displaystyle g_{6}=-x+25x^{-}15x^{3};}$

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

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

References

• Mahler, Kurt (1930), "Über die Nullstellen der unvollständigen Gammafunktionen." (German), Rendiconti Palermo 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 741185, http://books.google.com/books?id=JpHjkhFLfpgC Reprinted by Dover, 2005

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