The methods of Borel, Abel and Cesaro, Voronoi & Norlund part I

As promised, we shall consider some classic resummation procedures in concise detail.

One can recall that the sum of an infinite series in the Cauchy sense was the limit of partial sums.

We consider a very general and powerful method which relies on an alternative in taking the limit.

Suppose s_n is a partial sum of the series a_0 + a_1 + a_2 + ... which is under consideration and that S(t) = (p_0)(s_0) + (p_1)(s_1).t + (p_2)(s_2).t^2 + .... is entire for non-negative coefficients p_n and P(t) = (p_0) + (p_1).t + (p_2).t^2 + ...

If lim_{t->infinity} S(t) / P(t) = s

then we say,

a_0 + a_1 + a_2 + ... = s , (P)

(while we have used P to refer to this summation method, it ought not be considered with Abel summation for which P is sometimes used for Poisson who also employed it in his investigations into Fourier series. Hardy in Divergent Series calls this the J method pp79-80)

We put p_n = 1/ (n!) and thus P(t) = e^t to define the method of Borel summation. That is if,

lim_{t->infinity} e^(-t) S(t)

= lim_{t->infinity} e^(-t) [(s_0) + (s_1).t + (s_2).(t^2)/2! + ... + (s_n).(t^n)/n! + ...] = s

then,

a_0 + a_1 + a_2 + ... = s , (B)

Since we have,

n! = integral_{t = 0}^{infinity} (t^n).e^(-t) dt

, by employing a property of the Gamma function we have a similar summation,

a_0 + a_1 + a_2 + ...

= sum_{n = 0}^{infinity} (a_n).[integral_{t = 0}^{infinity} (t^n).e^(-t) dt] / n!

= integral_{t = 0}^{infinity} [sum_{n = 0}^{infinity} (a_n).(t^n)/n!] e^(-t) dt = s

(the interchange of summation and integration is allowed given the convergence of n!)

Should this last integral converge and the series within has a non-zero radius of convergence we say,

a_0 + a_1 + a_2 + ... = s , (B')

Should e^(-t) [sum_{n = 0}^{infinity} (a_n).(t^n)/n!]  -> 0 as t -> infinity the methods B and B' may be easily shown to be equivalent.

See Hardy, Divergent Series pp182-183

The Borel methods certainly allow us to sum more types of series than we had been able to thus far, for instance, it is easily shown that the B sum of 1 + z + z^2 + z^3 + ... = 1 / (1 - z) throughout the half plane Re(z) < 1