Analytic foliations in the 2-dimensional complex projective space with algebraic invariant curves are studied when the holonomy groups of these curves are solvable. It is shown that such a condition leads to the existence of a Liouville type first integral, and, under “generic” extra conditions, it is proven that these foliations can be defined by Bernoulli equations.

In this paper, we give some further properties of p-adic q-L-function of two variables, which is recently constructed by Kim 2005 and Cenkci 2006 . One of the applications of these properties yields general classes of congruences for generalized q-Bernoulli polynomials, which are qextensions of the classes for generalized Bernoulli numbers and polynomials given by Fox 2000 , Gunaratne 1995 , an...

By using q-integration, we will give some integral equation which are related to the Barnes’ multiple Bernoulli numbers. The object of this paper is to give explicit q-integral’s formulae which are related to Barnes’ multiple q-Bernoulli polynomials. §

Using the integral expression of Mathieu's series and some integral and analytic inequalities involving periodic functions and the generating function of Bernoulli numbers, we present several new inequalities and estimates for Mathieu's series and generalize Math-ieu's series. Two open problems are proposed.

Scan statistics have been widely applied to test for unusual cluster of events in many scientific areas. It has been of practical interest on how to select the window size of a scan statistic. An adaptive procedure for multiple window scan statistics is proposed and the distributions are studied for independent identically distributed Bernoulli trials and uniform observations on (0, 1) in one-d...

Using Szenes formula for multiple Bernoulli series, we explain how to compute Witten series associated to classical Lie algebras. Particular instances of these series compute volumes of moduli spaces of flat bundles over surfaces, and also multiple zeta values.

We discuss the asymptotic expansions of certain products of Bernoulli numbers and factorials, e.g., n ∏

In [7] we further proved that H1(p−1) ≡ 0 (mod p ) if and only if p divides the numerator of Bp−3, which never happens for primes less than 12 million [2]. There is another important equivalent statement of Wolstenholme’s Theorem by using combinatorics. D.F. Bailey [1] generalizes it to the following form. Theorem 1.1. ([1, Theorem 4]) Let n and r be non-negative integers and p ≥ 5 be a prime. ...