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.
The Mittag-Leffler polynomials can be defined recursively by M0(x) = 1
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 pn(x) = xn be the power function, then the binomial theorem says
Something like the binomial theorem holds if we replace pn with Mn: