The polynomials are related by
and we have
for
. Also
.
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[4]

As stated above, for
, we have the reflection formula
.
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[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.
. It also illustrates the reflection formula
with respect to the main diagonal, e.g.
.
More information nm ...
n m | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
1 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 |
2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | |
3 |
3 | 9 | 19 | 33 | 51 | 73 | 99 | 129 | | |
4 |
4 | 16 | 44 | 96 | 180 | 304 | 476 | | | |
5 |
5 | 25 | 85 | 225 | 501 | 985 | | | | |
6 |
6 | 36 | 146 | 456 | 1182 | | | | | |
7 |
7 | 49 | 231 | 833 | | | | | | |
8 |
8 | 64 | 344 | | | | | | | |
9 |
9 | 81 | | | | | | | | |
10 |
10 | | | | | | | | | |
Close
Binomial identity
Being a Sheffer sequence of binomial type, the Mittag-Leffler polynomials
also satisfy the binomial identity[8]
.
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,[10] 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[11] 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[12]
.
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[13]
, 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,[14] 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.
.