In mathematics, the Mittag-Leffler polynomials are the polynomials gn(x) or Mn(x) studied by Mittag-Leffler (1891).
The Mittag-Leffler polynomials are defined respectively by the generating functions They also have the bivariate generating function[1] The first few polynomials are given in the following table.
can be found in the OEIS,[2] though without any references, and the coefficients of the
Explicit formulas are (the last one immediately shows
, a kind of reflection formula), and In terms of the Gaussian hypergeometric function, we have[4] As stated above, for
can be defined recursively by Another recursion formula, which produces an odd one from the preceding even ones and vice versa, is
is the unique polynomial solution of the difference equation
, we can apply the reflection formula to one of the identities and then swap
are polynomials, the validity extends from natural to all real values of
(these values are also called the "figurate numbers for the n-dimensional cross polytopes" in the OEIS[6]) 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.
the following orthogonality relation holds:[7] (Note that this is not a complex integral.
is an even or an odd polynomial, the imaginary arguments just produce alternating signs for their coefficients.
have different parity, the integral vanishes trivially.)
Being a Sheffer sequence of binomial type, the Mittag-Leffler polynomials
also satisfy the binomial identity[8] Based on the representation as a hypergeometric function, there are several ways of representing
directly as integrals,[9] some of them being even valid for complex
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
, 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.
These integrals have the closed form in umbral notation, meaning that after expanding the polynomial in
has to be replaced by the Dirichlet eta function
, this also yields a closed form for the integrals 4.
) occur here (unless the denominators are cast as even zeta values), e.g. 5.
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[13]
has the following closed form in umbral notation, replacing
: Note that by virtue of the logarithmic derivative
of Riemann's functional equation, taken after applying Euler's reflection formula,[14] these expressions in terms of the
But the difference of two such integrals with corresponding degree differences is well-defined and exhibits very similar patterns, e.g.