Mittag-Leffler polynomials


In mathematics, the Mittag-Leffler polynomials are the polynomials gn(x) or Mn(x) studied by Mittag-Leffler (1891).

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


They also have the bivariate generating function[1]




The first few polynomials are given in the following table. The coefficients of the numerators of the   can be found in the OEIS,[2] though without any references, and the coefficients of the   are in the OEIS[3] as well.

n gn(x) Mn(x)



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[4]


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  .[5] 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[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.  .

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

Orthogonality relations


For   the following orthogonality relation holds:[7]


(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.)

Binomial identity


Being a Sheffer sequence of binomial type, the Mittag-Leffler polynomials   also satisfy the binomial identity[8]


Integral representations


Based on the representation as a hypergeometric function, there are several ways of representing   for   directly as integrals,[9] 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,[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.


See also



  1. ^ see the formula section of OEIS A142978
  2. ^ see OEIS A064984
  3. ^ see OEIS A137513
  4. ^ Özmen, Nejla & Nihal, Yılmaz (2019). "On The Mittag-Leffler Polynomials and Deformed Mittag-Leffler Polynomials". {{cite journal}}: Cite journal requires |journal= (help)
  5. ^ see the comment section of OEIS A142983
  6. ^ see OEIS A142978
  7. ^ Stankovic, Miomir S.; Marinkovic, Sladjana D. & Rajkovic, Predrag M. (2010). "Deformed Mittag–Leffler Polynomials". arXiv:1007.3612. {{cite journal}}: Cite journal requires |journal= (help)
  8. ^ Mathworld entry "Mittag-Leffler Polynomial"
  9. ^ Bateman, H. (1940). "The polynomial of Mittag-Leffler" (PDF). Proceedings of the National Academy of Sciences of the United States of America. 26 (8): 491–496. Bibcode:1940PNAS...26..491B. doi:10.1073/pnas.26.8.491. ISSN 0027-8424. JSTOR 86958. MR 0002381. PMC 1078216. PMID 16588390.
  10. ^ see at the end of this question on Mathoverflow
  11. ^ answer on math.stackexchange
  12. ^ similar to this question on Mathoverflow
  13. ^ method used in this answer on Mathoverflow
  14. ^ or see formula (14) in