Mittag-Leffler polynomials
In mathematics, the Mittag-Leffler polynomials are the polynomials gn or Mn studied by.
Mn is a special case of the Meixner polynomial Mn at b = 0, c = -1.
Definition and examples
Generating functions
The Mittag-Leffler polynomials are defined respectively by the generating functionsThey 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.Properties
The polynomials are related by and we have for. Also.Explicit formulas
Explicit formulas are, and
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 byAnother recursion formula, which produces an odd one from the preceding even ones and vice versa, is
As for the, we have several different recursion formulas:
Concerning recursion formula, the polynomial is the unique polynomial solution of the difference equation, normalized so that. Further note that and 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.
Initial values
The table of the initial values of may illustrate the recursion formula, 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..Orthogonality relations
For the following orthogonality relation holds:Binomial identity
Being a Sheffer sequence of binomial type, the Mittag-Leffler polynomials also satisfy the binomial identityIntegral 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 occur here, 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 :
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.