Polynomials which form the associated Sheffer sequence for
(1)
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and have the generating function
(2)
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An explicit formula is given by
(3)
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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)
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The Mittag-Leffler polynomials satisfy the recurrence formula
(5)
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The first few Mittag-Leffler polynomials are
(6)
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(7)
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(8)
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(9)
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(10)
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The Mittag-Leffler polynomials are related to the Pidduck polynomials by
(11)
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(Roman 1984, p. 127).