TY - JOUR
AU - Gerhold, Stefan
AU - Hubalek, Friedrich
AU - Tomovski, Živorad
PY - 2020/09/03
Y2 - 2021/05/13
TI - Asymptotics of some generalized Mathieu series
JF - MATHEMATICA SCANDINAVICA
JA - Math. Scand.
VL - 126
IS - 3
SE - Articles
DO - 10.7146/math.scand.a-121106
UR - https://www.mscand.dk/article/view/121106
SP - 424-450
AB - <p>We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular behavior of certain Dirichlet series, which is then translated into asymptotics for the original series. In the case of power-logarithmic sequences, we obtain precise first order asymptotics. For factorial sequences, a natural boundary of the Mellin transform makes the problem more challenging, but a direct elementary estimate gives reasonably precise asymptotics. As a byproduct, we prove an expansion of the functional inverse of the gamma function at infinity.</p>
ER -