A family of p-adic twisted interpolation functions associated with the modified Bernoulli numbers Yilmaz Simsek a , H.M. Srivastava b, * a Department of Mathematics, Faculty of Art and Science, University of Akdeniz, TR-07058 Antalya, Turkey b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada article info Keywords: q-Bernoulli and twisted q-Bernoulli numbers q-Bernoulli and twisted q-Bernoulli polynomials q-Zeta function Lerch zeta function Hurwitz-Lerch and Lipschitz-Lerch zeta functions L-Function and p-adic L-function Polylogarithm function p-Adic distribuLtions abstract The main purpose of this paper is to construct a family of modified p-adic twisted func- tions, which interpolate the modified twisted q-Bernoulli polynomials and the generalized twisted q-Bernoulli numbers at negative integers. We also give some applications and examples related to these functions and numbers. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction, Definitions and Preliminaries Throughout this paper, the ring of integers, the ring of p-adic integers, the field of p-adic rational numbers and the completion of the algebraic closure of the field of p-adic rational numbers will be denoted by Z; Z p ; Q p and C p , respectively. Let v p be the normalized exponential valuation of C p with jpj p ¼ p v pðpÞ ¼ p 1 : If q 2 C p , then we assume that jq  1j p < p  1 p1 and q x ¼ expðx log qÞ and jxj p 5 1: If q 2 C, then we tacitly assume that jqj < 1. Let the q-number [x] be defined by ½x¼½x; q :¼ 1  q x 1  q ðq – 1Þ; x ðq ¼ 1Þ: 8 < : 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.04.010 * Corresponding author. E-mail addresses: ysimsek@akdeniz.edu.tr (Y. Simsek), harimsri@math.uvic.ca (H.M. Srivastava). Applied Mathematics and Computation 216 (2010) 2976–2987 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc