Summability of formal power series in a direction may be dealt with in
the framework of ultraholomorphic classes associated to strongly regular
sequences (in the sense of V. Thilliez) of positive real numbers. After
commenting on some fundamental aspects of this tool, we will put forward
a concept of multisummability in a direction with respect to a finite,
ordered family of strongly regular sequences. As indicated by Professor
S. Kamimoto, an alternative approach, following the ideas of
B. Malgrange and J.-P. Ramis, is possible. We also discuss the
construction of acceleration kernels and operators in this context, and
show some applications of our technique.
Joint work with J. Jim\'enez-Garrido, A. Lastra and S. Malek.
We give a Hamiltonian system
which is nonintegrable in a domain
containing two singular points
and that is integrable
in some neighborhood of a singular point.
The system is
an arbitrarily small nontrivial perturbation
of an integrable Hamiltonian system
given by confluence of regular singular points
of a generalized hypergeometric system.
Under the nonresonance condition and a certain condition expressed by the Borel transform of the coefficients of the equation we show that the Hamiltonian system is non integrable in the domain containing two irregular singular points as well as locally integrable around an irregular singular point.
REFERENCES
Sasaki, Y. and Yoshino, M.: Nonintegrability of Hamiltonian system
perturbed from integrable system with two singular points {\it submitted.}
Yoshino, M.:
Smooth-integrable and analytic-nonintegrable resonant Hamiltonians.
RIMS K\^oky\^uroku Bessatsu, {\bf B40}, 177-189 (2013)