In mathematics, the Mittag-Leffler polynomials are the polynomials gn(x) or Mn(x) studied by Mittag-Leffler (1891). Mn(x) is a special case of the Meixner polynomial Mn(x;b,c) at b = 0, c = -1.
Definition and examples
Generating functions The Mittag-Leffler polynomials are defined respectively by the generating functions
∑ n = 0 ∞ g n ( x ) t n := 1 2 ( 1 + t 1 − t ) x {\displaystyle \displaystyle \sum _{n=0}^{\infty }g_{n}(x)t^{n}:={\frac {1}{2}}{\Bigl (}{\frac {1+t}{1-t}}{\Bigr )}^{x}} and
∑ n = 0 ∞ M n ( x ) t n n ! := ( 1 + t 1 − t ) x = ( 1 + t ) x ( 1 − t ) − x = exp ( 2 x artanh t ) . {\displaystyle \displaystyle \sum _{n=0}^{\infty }M_{n}(x){\frac {t^{n}}{n!}}:={\Bigl (}{\frac {1+t}{1-t}}{\Bigr )}^{x}=(1+t)^{x}(1-t)^{-x}=\exp(2x{\text{ artanh }}t).}
They also have the bivariate generating function
∑ n = 1 ∞ ∑ m = 1 ∞ g n ( m ) x m y n = x y ( 1 − x ) ( 1 − x − y − x y ) . {\displaystyle \displaystyle \sum _{n=1}^{\infty }\sum _{m=1}^{\infty }g_{n}(m)x^{m}y^{n}={\frac {xy}{(1-x)(1-x-y-xy)}}.}
Examples The first few polynomials are given in the following table. The coefficients of the numerators of the g n ( x ) {\displaystyle g_{n}(x)} can be found in the OEIS, though without any references, and the coefficients of the M n ( x ) {\displaystyle M_{n}(x)} are in the OEIS as well.
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