In mathematics, the Mahler polynomials ''g''_''n''(''x'') are polynomials introduced by in his work on the zeros of the incomplete gamma function. Mahler polynomials are given by the

generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary ser ...

\displaystyle \sum g_n(x)t^n/n! = \exp(x(1+t-e^t))

Mahler polynomials can be given as the

Sheffer sequence In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are na ...

for the functional inverse of 1+''t''–''e''^''t'' . The first few examples are :

g_0=1;

g_1=0;

g_2=-x;

g_3=-x;

g_4=-x+3x^2;

g_5=-x+10x^2;

g_6=-x+25x^2-15x^3;

g_7=-x+56x^2-105x^3;

g_8=-x+119x^2-490x^3+105x^4;

References

* *{{Citation , last1=Roman , first1=Steven , title=The umbral calculus , url=https://books.google.com/books?id=JpHjkhFLfpgC , publisher=Academic Press Inc. arcourt Brace Jovanovich Publishers, location=London , series=Pure and Applied Mathematics , isbn=978-0-12-594380-2 , mr=741185 , year=1984 , volume=111 Reprinted by Dover, 2005 Polynomials