In mathematics, the Mittag-Leffler polynomials are the polynomials g<sub>n</sub>(x) or M<sub>n</sub>(x) studied by .
M<sub>n</sub>(x) is a special case of the Meixner polynomial M<sub>n</sub>(x;b,c) at b = 0, c = -1.
The Mittag-Leffler polynomials are defined respectively by the generating functions
They also have the bivariate generating function
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.
The polynomials are related by and we have for . Also .
Explicit formulas are
(the last one immediately shows , a kind of reflection formula), and
In terms of the Gaussian hypergeometric function, we have
As stated above, for , we have the reflection formula .
The polynomials can be defined recursively by
Another 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 (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 .)
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. .
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. This has been one of the reasons to introduce the .)
Being a Sheffer sequence of binomial type, the Mittag-Leffler polynomials also satisfy the binomial identity
The classical MittagâÂÂLeffler polynomials satisfy a Rodrigues' formula involving the central difference operator. This formula can be derived using their connection to the MeixnerâÂÂPollaczek polynomials.
The classical MittagâÂÂLeffler polynomials are given by the Rodrigues-type formula
where is the central difference operator defined by
and higher powers are obtained by successive application. The weight-like function is the Gamma ratio
This representation is obtained from the relation of to the MeixnerâÂÂPollaczek polynomials and the known Rodrigues formula for the latter family.
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.
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 :
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.
The reduced Mittag-Leffler polynomials are a family of real polynomials derived from the classical Mittag-Leffler polynomials by means of an imaginary-argument transformation. This yields real-valued polynomials that are orthogonal on the real line with respect to a hyperbolic weight function, in contrast to the classical version which uses orthogonality on the imaginary axis. The reduced form and its properties (including finite- and infinite-order differential equations) were studied in detail by RajkoviÃÂ et al. (2024).
The reduced Mittag-Leffler polynomials for are defined as
where are the classical Mittag-Leffler polynomials satisfying the generating function
with initial conditions and . They obey the three-term recurrence relation
with initial values
The polynomials satisfy the parity relation
The associated monic reduced Mittag-Leffler polynomials are given by
which satisfy the monic recurrence
with and .
The first few monic reduced polynomials are:
The reduced Mittag-Leffler polynomials are orthogonal on the real line with weight function :
All zeros of (and hence of ) are real.
Every monic reduced polynomial satisfies a finite-order ordinary differential equation of order :
where the coefficients are
For example, for , we have . The coefficients are:
The differential equation becomes:
This can be verified by computing the derivatives: , , , , and substituting into the equation.
Additionally, they satisfy the infinite-order (operator) differential equation
meaning is an eigenfunction of the operator with eigenvalue .
The operators and act on functions via their Taylor series. For example, for :
Thus:
confirming that is an eigenfunction with eigenvalue .
These differential properties arise because form a Sheffer sequence with exponential generating function
together with the theory of Sheffer sequences.