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 .) Initial values The table of the initial values of (these values are also called the "figurate numbers for the n-dimensional cross polytopes" in the OEIS) may illustrate the recursion formula (1), which can be taken to mean that each entry is the sum of the three neighboring entries: to its left, above and above left, e.g. . It also illustrates the