Mittag-Leffler polynomials
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 reflection formula  with respect to the main diagonal, e.g. . {| class="wikitable"
! ! ! 1!! 2 ! ! 3 ! ! 4 ! ! 5 ! ! 6 ! ! 7 ! ! 8 ! ! 9 ! ! 10
|-
! 1 
| style="text-align: right;" |1 || style="text-align: right;" |1 || style="text-align: right;" |1 || style="text-align: right;" |1 || style="text-align: right;" |1 || style="text-align: right;" |1 || style="text-align: right;" |1 || style="text-align: right;" |1 || style="text-align: right;" |1 || style="text-align: right;" |1
|-
 ! 2 
| style="text-align: right;" |2 || style="text-align: right;" |4 || style="text-align: right;" |6 || style="text-align: right;" |8 || style="text-align: right;" |10 || style="text-align: right;" |12 || style="text-align: right;" |14 || style="text-align: right;" |16 || style="text-align: right;" |18 || style="text-align: right;" |
|-
 ! 3 
| style="text-align: right;" |3 || style="text-align: right;" |9 || style="text-align: right;" |19 || style="text-align: right;" |33 || style="text-align: right;" |51 || style="text-align: right;" |73 || style="text-align: right;" |99 || style="text-align: right;" |129 || style="text-align: right;" | || style="text-align: right;" |
|-
 ! 4 
| style="text-align: right;" |4 || style="text-align: right;" |16 || style="text-align: right;" |44 || style="text-align: right;" |96 || style="text-align: right;" |180 || style="text-align: right;" |304 || style="text-align: right;" |476 || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" |
|-
 ! 5 
| style="text-align: right;" |5 || style="text-align: right;" |25 || style="text-align: right;" |85 || style="text-align: right;" |225 || style="text-align: right;" |501 || style="text-align: right;" |985 || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" |
|-
 ! 6 
| style="text-align: right;" |6 || style="text-align: right;" |36 || style="text-align: right;" |146 || style="text-align: right;" |456 || style="text-align: right;" |1182 || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" |
|-
 ! 7 
| style="text-align: right;" |7 || style="text-align: right;" |49 || style="text-align: right;" |231 || style="text-align: right;" |833 || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" |
|-
 ! 8 
| style="text-align: right;" |8 || style="text-align: right;" |64 || style="text-align: right;" |344 || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" |
|-
 ! 9 
| style="text-align: right;" |9 || style="text-align: right;" |81 || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" |
|-
 ! 10 
| style="text-align: right;" | 10 || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | ||
|-
 |}

Orthogonality relations
For  the following orthogonality relation holds:

(Note that this is not a complex integral. As each  is an even or an odd polynomial, the imaginary arguments just produce alternating signs for their coefficients. Moreover, if  and  have different parity, the integral vanishes trivially.) Binomial identity 
Being a Sheffer sequence of binomial type, the Mittag-Leffler polynomials  also satisfy the binomial identity 
. Integral representations 
Based on the representation as a hypergeometric function, there are several ways of representing  for  directly as integrals, some of them being even valid for complex , e.g. . Closed forms of integral families
There are several families of integrals with closed-form expressions in terms of  zeta values where the coefficients of the Mittag-Leffler polynomials occur as coefficients. All those integrals can be written in a form containing either a factor  or , and the degree of the Mittag-Leffler polynomial varies with . One way to work out those integrals is to obtain for them the corresponding recursion formulas as for the Mittag-Leffler polynomials using integration by parts. 1. For instance, define for  
 
These integrals have the closed form
  
in umbral notation, meaning that after expanding the polynomial in , each power  has to be replaced by the  zeta value  . E.g. from 
  we get 
 for . 2. Likewise take for 
 

In umbral notation, where after expanding,  has to be replaced by the Dirichlet eta function , those have the closed form
. 3. The following holds for  with the same umbral notation for  and , and completing by continuity . Note that for , this also yields a closed form for the integrals

 
 
4. For , define . If  is even and we define , we have in umbral notation, i.e. replacing   by ,

 
Note that only odd zeta values (odd ) occur here (unless the denominators are cast as even zeta values), e.g. 5. If  is odd, the same integral is much more involved to evaluate, including the initial one . Yet it turns out that the pattern subsists if we define , equivalently . Then  has the following closed form in umbral notation, replacing  by :
, e.g. Note that by virtue of the logarithmic derivative  of  Riemann's functional equation, taken after applying Euler's reflection formula, these expressions in terms of the  can be written in terms of , e.g. 6. For , the same integral  diverges because the integrand behaves like  for . But the difference of two such integrals with corresponding degree differences is well-defined and exhibits very similar patterns, e.g. . See also
 Bernoulli polynomials of the second kind
 Stirling polynomials
 Poly-Bernoulli number

References

 

Polynomials