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Mittag-Leffler Polynomial

Polynomials M_k(x) which form the associated Sheffer sequence for

f(t)=(e^t-1)/(e^t+1)
(1)

and have the generating function

sum_(k=0)^infty(M_k(x))/(k!)t^k=((1+t)/(1-t))^x.
(2)

An explicit formula is given by

M_n(x)=sum_(k=0)^n(n; k)(n-1)_(n-k)2^k(x)_k,
(3)

where (x)_n 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

M_n(x+y)=sum_(k=0)^n(n; k)M_k(x)M_(n-k)(y).
(4)

The Mittag-Leffler polynomials satisfy the recurrence formula

M_(n+1)(x)=1/2x[M_n(x+1)+2M_n(x)+M_n(x-1)].
(5)

The first few Mittag-Leffler polynomials are

M_0(x)=1
(6)
M_1(x)=2x
(7)
M_2(x)=4x^2
(8)
M_3(x)=8x^3+4x
(9)
M_4(x)=16x^4+32x^2.
(10)

The Mittag-Leffler polynomials M_n(x) are related to the Pidduck polynomials by

P_n(x)=1/2(e^t+1)M_n(x)
(11)

(Roman 1984, p. 127).

See also

Pidduck Polynomial

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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.

Referenced on Wolfram|Alpha

Mittag-Leffler Polynomial

Cite this as:

Weisstein, Eric W. "Mittag-Leffler Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Mittag-LefflerPolynomial.html

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