In mathematics, the Mittag-Leffler polynomials are the polynomials gn(x) or Mn(x) studied by .
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 and
They also have the bivariate generating function
Examples
The first few polynomials are given in the following table. The coefficients of the numerators of the can be found in the OEIS, though without any references, and the coefficients of the are in the OEIS as well. { | class="wikitable" !n !! gn(x) !! Mn(x) |- | 0 || || |- | 1 || || |- | 2 || || |- | 3 || || |- | 4 || || |- | 5 || || |- | 6 || || |- | 7 || || |- | 8 || || |- | 9 || || |- | 10 || || |}
Properties
The polynomials are related by and we have for . Also .
Explicit formulas
Explicit formulas are
(the last one immediately shows , a kind of reflection formula), and
, which can be also written as
, where denotes the falling factorial. In terms of the Gaussian hypergeometric function, we have
Reflection formula
As stated above, for , we have the reflection formula .
Recursion formulas
The polynomials can be defined recursively by
, starting with and . Another recursion formula, which produces an odd one from the preceding even ones and vice versa, is , again starting with .
As for the , we have several different recursion formulas:
Concerning recursion formula (3), the polynomial is the unique polynomial solution of the difference equation , normalized so that . Further note that (2) and (3) are dual to each other in the sense that for , we can apply the reflection formula to one of the identities and then swap and to obtain the other one. (As the are polynomials, the validity extends from natural to all real values of .)