A note on the Von Staudt-Clausen’s theorem for the weighted q-Genocchi numbers

Byung Moon Kim 1 Lee-Chae Jang 0 0 General Education Institute, Konkuk University , Chungju, 138-701 , Korea 1 Department of Mechanical System Engineering, Dongguk University , Gyeongju, 780-714 , Korea Recently, the Von Staudt-Clausen theorem for q-Euler numbers was introduced by Kim (Russ. J. Math. Phys. 20(1):33-38, 2013) and Araci et al. have also studied this theorem for q-Genocchi numbers (see Araci et al. in Appl. Math. Comput. 247:780-785, 2014) based on the work of Kim et al. In this paper, we give the corresponding Von Staudt-Clausen theorem for the weighted q-Genocchi numbers and also prove the Kummer-type congruences for the generated weighted q-Genocchi numbers. MSC: 11B68; 11S40 As is well known, a theorem including the fractional part of Bernoulli numbers, which is called the Von Staudt-Clausen theorem, was introduced by Karl Von Staudt and Thomas Clausen (see []). In [], Kim has studied the Von Staudt-Clausen theorem for the q-Euler numbers and Araci et al. have introduced the Von Staudt-Clausen theorem associated with q-Genocchi numbers. - f (x) dq(x) = lim N [pN ]q x= f (x)(q)x (see []). From (), we note that From n N, we have = []q f (l)()nlql (see []). X = a + dpZp By (), we get tnn! = t Thus, by (), we have f (x) dq(x) = Zp Thus, by (), we get (see [, ]). ()k (k)qk lim N [pN ]qd x= ()xqdx From (), we have [d]q k= [d]q k= [d]q k= 2 Von Staudt-Clausen theorems From (), we have Thus, by (), we have lim Gn(+),q = Gn+ = q n + n + In [], Kim introduced the following inequality: () n pk . Ln(k) = []qn q[]qn + + pk q q From (), we note that = []q By (), we get ()a+jpk qa+jpk a + jpk n q pk p a= j= pk p a= j= a= j= l= pk p n a= j= l= a= j= l= a= j= l= [a]qnl()a+jqal jpk l qa+jpk q pk p n a= j= l= Thus, by (), we get mod pk From (), we have ()aqa a ()jqpj [a]q + q [p]q [j]qp p pk a= j= p pk n a= j= l= p pk n a= j= l= () Theorem . Let Ln(k) = pk a= ()a[a]qn. Then we have Furthermore mod pk By Theorem ., we get () mod [p]q . [x]qn dq(x) Gn,q Zp. Therefore, by (), we have the following theorem. () mod [p]q . ()an[a]qn = Gn,q From () and (), we note that ()a+[a]qnqa Zp (n ). ()a+[a]qnqa Zp. max Therefore, we obtain the following theorem. p a= . = []qt = t (x)[x]qn dq(x) = n lim n Thus, we have + n[p]qn (p) lim af pp,(a,p)= () (a)()aqa[a]qn + a[p]qn (p)Gn,qp, . Therefore, by (), we obtain the following theorem. () () (a)()aqa[a]qn = Gn,q, [p]qn (p)Gn,qp, . () the weighted p-adic l-function associated with Gn,q, as follows: lp(,q)(s, ) = lim X For k , = k = k It is easy to show that = + pn log a + []q (p)[p]kq () Gk,qp, . mod pn , mod pn . mod pn . Competing interests The authors declare that they have no competing interests. Authors? contributions All authors contributed equally to this work. All authors read and approved the final manuscript. Acknowledgements This paper was supported by Konkuk University in 2015. (...truncated)