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

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

and Hermite polynomials satisfy

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 p_n(x) = xⁿ be the power function, then the binomial theorem says

Something like the binomial theorem holds if we replace p_n with M_n:

Related posts

• Sum of all spheres
• Orthogonal polynomials
• Bell polynomials