7 research outputs found
Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials
We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials
in detail. The starting point is their Fourier
series on which, it is shown, remains valid as an asymptotic expansion
over compact subsets of the complex plane. This is used to determine explicit
estimates on the constants in the approximation, and also to analyze
oscillatory phenomena which arise in certain cases.
These results are transferred to the Apostol-Euler polynomials
via a simple relation linking them to the
Apostol-Bernoulli polynomials.Comment: 16 page
Existence and reduction of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials
summary:One can find in the mathematical literature many recent papers studying the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, defined by means of generating functions. In this article we clarify the range of parameters in which these definitions are valid and when they provide essentially different families of polynomials. In particular, we show that, up to multiplicative constants, it is enough to take as the âmain familyâ those given by and as an âexceptional familyâ both of these for
Some functional relations derived from the Lindelöf-Wirtinger expansion of the Lerch transcendent function
The Lindelöf-Wirtinger expansion of the Lerch transcendent function implies, as a limiting case, Hurwitzâs formula for the eponymous zeta function. A generalized form of M šobius inversion applies to the Lindelöf-Wirtinger expansion and also implies an inversion formula for the Hurwitz zeta function as a limiting case. The inverted formulas involve the dynamical system of rotations of the circle and yield an arithmetical functional equation
BernoulliâDunkl and ApostolâEulerâDunkl polynomials with applications to series involving zeros of Bessel functions
We introduce BernoulliâDunkl and ApostolâEulerâDunkl polynomials as generalizations of Bernoulli and ApostolâEuler polynomials, where the role of the derivative is now played by the Dunkl operator on the real line. We use them to find the sum of many different series involving the zeros of Bessel functions