Extension of Borel Summation
Pp. 193-223 (31)
In order to demonstrate that it is regularisation and not Borel summation which is
responsible for yielding meaningful values to asymptotic series, the gamma function in both types
of generalised terminants is now replaced by the functionf(pk+q)in this chapter. Then the regu-larised values for both types of series are determined by assuming that f(s)is a Mellin transform.
These appear in Propositions 6 and 7. Although both types of series are different from generalised
terminants, the proofs of the propositions are nonetheless based on the exposition in the preceding
chapter. Consequently, the regularised values are referred to as extended Borel-summed forms.
The chapter concludes by considering a complicated example of a Type II series, where the coef-ficients can be expressed as the Mellin transform of the product of the Bessel functionJ
the Macdonald function Kν
(x). As there is no special function equivalent to this asymptotic series,
the MB-regularised forms for the regularised value are derived with the aid of the general theory
in Ch. 7. Then a numerical study of both the extended Borel-summed and MB-regularised forms
is carried out with the index νset equal to 1/3 and -3/5 and for large and small values of|z| over
the principal branch. Once again, the Borel-summed and MB-regularised forms yield identical
regularised values for the series.