ON THE MULTIPLE q-GENOCCHI AND EULER NUMBERS Taekyun Kim 8 0 0 2 Abstract. Thepurposeofthispaperistopresentasystemicstudyofsomefamiliesof multipleq-GenocchiandEulernumbersbyusingmultivariateq-Volkenbornintegral(= n a p-adicq-integral)onZp. Fromthestudiesofthoseq-Genocchinumbersandpolynomials J of higherorderwe derivesome interestingidentitiesrelatedto q-Genocchinumbersand 7 polynomials of higher order. ] T N §1. Introduction . h t a m Let p be a fixed odd prime. Throughout this paper Z , Q , C, and C will, p p p [ respectively, denote the ring of p-adic rational integers, the field of p-adic rational 1 numbers, the complex number field, and the completion of algebraic closure of Qp. v Let v be the normalized exponential valuation of C with |p| = p−vp(p) = p−1 p p p 8 and let q be regarded as either a complex number q ∈ C or a p-adic number q ∈ 7 9 C . If q ∈ C, then we always assume |q| < 1. If q ∈ C , we normally assume p p 0 1. |1−q|p < p−p−11, which implies that qx = exp(xlogq) for |x|p ≤ 1. Here, |·|p is the 0 p-adic absolute value in Cp with |p|p = p1. The q-basic natural number are defined 8 by [n] = 1−qn = 1 + q + ··· + qn−1, ( n ∈ N), and q-factorial are also defined as 0 q 1−q : [n] ! = [n] ·[n−1] ···[2] ·[1] .In thispaperweuse the notationofGaussian binomial v q q q q q i coefficient as follows: X r n [n] ! [n] ·[n−1] ···[n−k +1] a (1) = q = q q q. (cid:18)k(cid:19) [n−k] ![k] ! [k] ! q q q q Note that lim n = n = n·(n−1)···(n−k+1). The Gaussian coefficient satisfies the q→1 k q k n! following recursion(cid:0) f(cid:1)ormu(cid:0)la(cid:1): n+1 n n n n (2) = +qk = qn−k + , cf. [12]. (cid:18) k (cid:19) (cid:18)k −1(cid:19) (cid:18)k(cid:19) (cid:18)k −1(cid:19) (cid:18)k(cid:19) q q q q q Key wordsand phrases. q-Bernoullinumbers,q-Volkenbornintegrals,q-Eulernumbers,q-Stirling numbers. 2000 AMS Subject Classification: 11B68, 11S80 ThispaperissupportedbyJangjeonResearchInstituteforMathematicalScience(JRIMS-10R-2001) Typeset by AMS-TEX 1 From thus recursion formula we derive n = qd1+2d2+···+kdk, cf.[1, 2, 12, 13, 14]. (cid:18)k(cid:19) q d0+···+dkX=n−k,di∈N The q-binomial formulae are known as n n n (b;q)n = 1−bqi−1 = q(k2)(−1)kbk, (cid:18)k(cid:19) iY=1(cid:0) (cid:1) Xk=0 q and n ∞ (3) 1 = 1−bqi−1 −1 = n+k −1 bk, cf.[12]. (b;q) (cid:18) k (cid:19) n iY=1(cid:0) (cid:1) kX=0 q In this paper we use the notation 1−qx 1−(−q)x [x] = , and [x] = . q −q 1−q 1+q Hence, lim [x] = 1, for any x with |x| ≤ 1 in the present p-adic case, cf.[1-18]. q→1 q p For d a fixed positive integer with (p,d) = 1, let X = X = limZ/dpNZ, X = Z , d ←− 1 p N X∗ = ∪ (a+dpZ ), p 0

On the multiple q-Genocchi and Euler numbers

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