Salvy, Shackell. Asymptotics of inverse functions. JSC 1999(T)(21s).djvu: eKnigu

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Salvy, Shackell. Asymptotics of inverse functions. JSC 1999(T)(21s).djvu

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Date Dec 19, 2004

B.9)
By the induction hypothesis
y'(x + 9G) - u'n(x + 9G) = Gn+1(x + 9G)ipljn(x + 9G) - Gn+I(x)ipltn(x + 9G), B.10)
since G(x + 9G) lies between G(x) and G(x + G) and B.6) applies...
In the case of the Bell numbers, the saddle-point equation reads
ReR -l = n, C.3)
from which our algorithm retrieves the classical expansion
log log n 1 log log n (log log n — 2) ...
Using our algorithm we thus compute the next level of the multiseries,
which yields
where ? is defined as an inverse function by
C + ln(C) = ln(n)...
These
objects and their further generalizations are studied in statistics, where they are used to
model classification hierarchies...