Generalization of power polynomials

A while back I wrote about the Mittag-Leffler function which is a sort of generalization of the exponential function. There are also Mittag-Leffler polynomials that are a sort of generalization of the powers of x; more on that shortly.

Recursive definition

The Mittag-Leffler polynomials can be defined recursively by M0(x) = 1
and

M_{n+1}(x) = \frac{x}{2}\left(M_n(x+1) + 2M_n(x) + M_n(x-1) \right )

for n > 0.

Contrast with orthogonal polynomials

This is an unusual recurrence if you’re used to orthogonal polynomials, which come up more often in application. For example, Chebyshev polynomials satisfy

T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x)

and Hermite polynomials satisfy

 H_{n+1}(x) = x H_n(x) - n H_{n-1}(x)

as I used as an example here.

All orthogonal polynomials satisfy a two-term recurrence like this where the value of each polynomial can be found from the value of the previous two polynomials.

Notice that with orthogonal polynomial recurrences the argument x doesn’t change, but the degrees of polynomials do. But with Mittag-Leffler polynomials the opposite is true: there’s only one polynomial on the right side, evaluated at three different points: x+1, x, and x-1.

Generalized binomial theorem

Here’s the sense in which the Mittag-Leffler polynomials generalize the power function. If we let pn(x) = xn be the power function, then the binomial theorem says

p_n(x+y) = \sum_{k=0}^n {n\choose k}\, p_{k}(x)\, p_{n-k}(y).

Something like the binomial theorem holds if we replace pn with Mn:

M_n(x+y) = \sum_{k=0}^n {n\choose k}\, M_{k}(x)\, M_{n-k}(y).

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