Mittag-Leffler transform

I keep running into Mittag-Leffler. A couple days ago I wrote about his polynomials. Today I ran across his regularization method for summing a divergent series. Before that I wrote about his generalization of the exponential function, which is closely related to his summation method.

The exponential function has power series

\exp(x) = \sum_{k=0}^\infty \frac{x^k}{\Gamma(k+1)}

where we’ve written the denominator as Γ(k+1) rather than the customary k! in preparation for defining the Mittag-Leffler function

E_{\alpha, \beta}(x) = \sum_{k=0}^\infty \frac{x^k}{\Gamma(\alpha k+\beta)}

So when α = β = 1, the Mittag-Leffler function reduces to the exponential function.

Now we apply a similar idea to transforming a series. Given a formal (i.e. not necessarily convergent) power series

A(z) = \sum_{k=0}^\infty a_k z^k

the Borel transform of the series is

 B(t) = \sum_{k=0}^\infty \frac{a_k}{k!}\, t^k

If we replace k! with Γ(1 + αk) we get the Mittag-Leffler transform

B_\alpha(t) = \sum_{k=0}^\infty \frac{a_k}{\Gamma(1 + \alpha k)}\, t^k

which reduces to the Borel transform when α = 1.

So the Mittag-Leffler function is to the exponential function as the Mittag-Leffler transform is to the Borel transform.

Also, the Mittag-Leffler function is the Mittag-Leffler transform of the series

1 + z + z² + z³ + … .

Applications

Why would you take a Borel transform or a Mittag-Leffler transform? The transformed series might converge where the original series does not. This could be useful in finding an analytic continuation of a power series beyond its disk of convergence.

If you take the limit as α goes to zero of the Mittag-Leffler transform Bα you get the Mittag-Leffler sum of the formal power series A. If the original series converges, the Mittag-Leffler sum will converge to the same value. But the Mittag-Leffler sum might converge when the original sum does not.

And why would you want to sum a divergent series? This is delicate business. The result could be nonsense, but it could also be meaningful and useful. You might, for example, discover an efficient approximation by applying this to an asymptotic series.

If you’re worried whether your result is correct, and you should be, then you could use Borel summation or Mittag-Leffler summation as a technique that gives you a candidate solution. You could pretend that the result was an inspired guess that came to you in a dream. If you can prove that the candidate solution is correct, then it’s correct, even if you can’t justify the process that led to its discovery.

Related post

Three ways to sum a divergent series

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