On the von Staudt-Clausen's theorem associated with q-Genocchi numbers

Keywords: Genocchi numbers and polynomials q-Genocchi numbers von Staudt–Clausen's theorem Kummer congruence a b s t r a c t Recently, the von Staudt–Clausen's theorem for q-Euler numbers was introduced by Kim (2013) and q-Genocchi numbers were constructed by Araci et al. (2013). In this paper, we give the corresponding von Staudt–Clausen's theorem for q-Genocchi numbers and also get the Kummer-type congruence for q-Genocchi numbers. Karl von Staudt [5] and Thomas Clausen [3] first introduced a theorem including fractional part of Bernoulli numbers, which are now popularly known as the von Staudt–Clausen's theorem (see [3,5]). Kim [7] proved that q-Euler numbers are p-adic integers and q-Euler numbers that can be shown in terms of the von Staudt–Clausen theorem. It can be seen that these numbers play an important role in the development of several areas of Mathematics such as Number theory, Complex analysis, Mathematical physics and so on. Some special numbers related to Bernoulli numbers, Euler numbers, Genocchi numbers, Frobenius–Euler numbers have been studied by many mathematicians (see [1–25]). Recently, the modified q-Genocchi numbers and polynomials were constructed by Araci et al. [22]. They also established some new identities for the modified q-Genocchi numbers and polynomials. In the complex plane, Genocchi numbers can be expressed as