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. This has been one of the reasons to introduce the .)
Binomial identity Being a Sheffer sequence of binomial type, the Mittag-Leffler polynomials also satisfy the binomial identity .
Rodrigues-type formula The classical Mittag–Leffler polynomials satisfy a Rodrigues' formula involving the central difference operator. This formula can be derived using their connection to the Meixner–Pollaczek polynomials.
The classical Mittag–Leffler 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 Meixner–Pollaczek polynomials and the known Rodrigues formula for the latter family.
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 , which is closely linked to the fact that the generating function features . 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,or see formula (14) in https://mathworld.wolfram.com/RiemannZetaFunction.html 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. .
7. For , define
If is even, let
and write
Then
Here denotes the coefficient of in the Mittag-Leffler polynomial . When is odd (), only even coefficients of contribute; when is even (), only odd coefficients contribute. Consequently, the expansion involves only odd Riemann zeta values. Equivalently, it is a linear combination of derivatives of the Riemann zeta function at negative even integers.
Examples with (even, ), using :
Examples with (odd, ), using
8. For odd, , the same integrals
involve derivatives of the Riemann zeta function at negative odd integers. Let again
Then we have the closed form
where denotes the coefficient of in the Mittag-Leffler polynomial , and is given explicitly by
The last term in the sum, corresponding to , vanishes identically whenever , since then ; it thus contributes only when , where it lands exactly at the top degree of .
The coefficient of has the closed form
where the second expression follows by Lagrange inversion from . Writing for the higher-order (Nørlund) Bernoulli numbersSee Norlund polynomials on MathWorld. defined by
this expands into the finite closed-form sum
Examples with and :
For reference,
for .
Reduced Mittag-Leffler polynomials 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 Rajković et al. (2024). Definition 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 . Explicit low-degree terms The first few monic reduced polynomials are:
Orthogonality relation The reduced Mittag-Leffler polynomials are orthogonal on the real line with weight function : . All zeros of (and hence of ) are real. Differential equations 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 .