Mittag-Leffler Polynomial

Polynomials  which form the associated Sheffer sequence for

(1)

and have the generating function

(2)

An explicit formula is given by

(3)

where  is a falling factorial, which can be summed in closed form in terms of the hypergeometric function, gamma function, and polygamma function. The binomial identity associated with the Sheffer sequence is

(4)

The Mittag-Leffler polynomials satisfy the recurrence formula

(5)

The first few Mittag-Leffler polynomials are

			(6)
			(7)
			(8)
			(9)
			(10)

The Mittag-Leffler polynomials  are related to the Pidduck polynomials by

(11)

(Roman 1984, p. 127).

REFERENCES:

Bateman, H. "The Polynomial of Mittag-Leffler." Proc. Nat. Acad. Sci. USA 26, 491-496, 1940.

Roman, S. "The Mittag-Leffler Polynomials." §4.1.6 in The Umbral Calculus. New York: Academic Press, pp. 75-78 and 127, 1984.