p-adic Invariant Integral on Z p Associated with the Changhee s q-bernoulli Polynomials

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1 It. Joual of Math. Aalysis, Vol. 7, 2013, o. 43, HIKARI Ltd, htt://dx.doi.og/ /ima Adic Ivaiat Itegal o Z Associated with the Chaghee s -Beoulli Polyomials J. J. Seo Deatmet of Alied Mathematics Puyog Natioal Uivesity Busa , Reublic of Koea seo2011@u.ac. T. Kim Deatmet of Mathematics Kwagwoo Natioal Uivesity Seoul , Reublic of Koea tim@w.ac. Coyight c 2013 J. J. Seo ad T. Kim. This is a oe access aticle distibuted ude the Ceative Commos Attibutio Licese, which emits uesticted use, distibutio, ad eoductio i ay medium, ovided the oigial wo is oely cited. Abstact. I this ae, we study some oeties of Chaghee s -Beou lli olyomials which ae deived fom -adic ivaiat itegal o Z. By usig these oeties, we give some iteestig idetities elated to higheode -Beoulli olyomials. Keywods: Chaghee s -Beoulli olyomials, -adic ivaiat itegal, highe-ode -Beoulli olyomials 1. Itoductio Thoughout this ae, Z, Q, C ad C will esectively deote the ig of -adic iteges, the field of -adic umbes, the comlex umbe field ad the

2 2118 J. J. Seo ad T. Kim comletio of algebaic closue of Q. Let ν be the omalized exoetial valuatio of C with ν( 1/. Whe oe tals of -extesio, is vaiously cosideed as a idetemiate, a comlex umbe C, oa -adic umbe C.If C, oe omally assumes < 1. If C, oe omally assumes 1 < 1/ 1, so that g x ex(xlog fo x Z.We use the followig otatio i this ae: [x] [x : ] 1 x, (see [8] [16]. 1 Note that lim 1 [x] x fo ay x with x 1 i the eset -adic case. Let d be a fixed itege ad let be a fixed ime umbe. We set X d lim N Z d N Z, X (a + dz, 0<a<d (a,1 a + d N Z { x X x a(mod d N }, whee a Z lies i 0 a<d N, (see [1]-[11]. Let UD(Z be the sace of uifomly diffeetiable fuctios o Z.Fof UD(Z, the -adic ivaiat itegal o Z is defied by I 0 (f Thus, by(1.1, we see Z f(xdμ 0 (x lim N 1 N N 1 x0 f(x X f(xdμ 0 (x. (1.1 whee f 1 (x f(x + 1. Fom (1.2, we ca deive I 0 (f 1 I 0 (f+f (0, (1.2 1 I 0 (f I 0 (f+ f (i, (see [9], [10], [14], [15] (1.3 i0 df (x dx xi. whee f (x f(x +, f (i As is well ow, the Beoulli olyomials of ode ae defied by the geeatig fuctio to be ( t e xt B e t ( 1 (xt!, ( Z 0 (1.4 0

3 -Adic ivaiat itegal 2119 I the secial case, x 0,B ( (0 B ( ae called the th Beoulli umbes of ode (see [1]-[16]. By (1.2, we easily set e (x 1+ +x t dμ 0 (x 1 dμ 0 (x Z Z times ( t B ( t e t 1!. 0 (1.5 Fom (1.5, we have ad (x x dμ 0 (x 1 dμ 0 (x B (, ( 0, Z Z (1.6 times (x x + x dμ 0 (x 1 dμ 0 (x B ( (x, ( 0. Z Z (1.7 times I this ae, we coside Chaghee -Beoulli umbe ad olyomials which ae deived fom multivaiate adic ivaiat itegal o Z. ad ivestigate some oeties of Chaghee -Beoulli olyomials. 2. Chaghee -Beoulli olyomials Let a 1,a 2,,a,b 1,b 2,,b be ositive iteges. Fo w Z, we coside the modified Chaghee -Beoulli olyomials which ae deived fom multivaiate -adic ivaiat itegal o Z as follows: B (, (w a 1,a 2,,a ; b 1,b 2,,b b 1x 1 + +b x [w + a 1 x a x ] dμ 0(x 1 dμ 0 (x, (2.1 Z Z

4 2120 J. J. Seo ad T. Kim ad B (, (a 1,a 2,,a ; b 1,b 2,,b b 1x 1 + +b x [a 1 x a x ] dμ 0(x 1 dμ 0 (x. (2.2 Z Z Fom (1.1 ad(2.1, we have B (, (w a 1,a 2,,a ; b 1,b 2,,b b 1x 1 + +b x [w + a 1 x a x ] dμ 0(x 1 dμ 0 (x Z Z 1 (1 1 (1 0 0 ( ( w P N 1 x 1,x 2,,x 0 ( 1 lim N P N ( ( log ( w 1 a 1x 1 + +a x +b 1 x 1 + +b x 1 ( a + b [a + b ] Theefoe, by (2.3, we obtai the followig theoem. Theoem 2.1. Fo a 1,a 2,,a,b 1,b 2,,b Z 0 ad w Z, we have B ( whee 0, 1.,(w a 1,a 2,,a ; b 1,b 2,,b ( log 1 ( ( w 1 (1 0 ( a + b 1 [a + b ], (2.3 Let us tae b 1 1,b 2 2,,b. The, by theoem 2.1, we get B (, (w 1, 1,, 1;1, 2,, 1 ( ( log 1 (1 ( log 1 1 (1 0 0 ( + ( w [ + ] 1 ( + ( w! []!, ( ( + (2.4

5 -Adic ivaiat itegal 2121 whee ( [] [ +1] []!, []![] [ 1] [2] [1]. Theefoe, by (2.4, we obtai the followig coollay. Coollay 2.2. Fo 0, 1, we have B (, (w 1, 1,, 1;1, 2,, ( log 1 1 (1 0 ( ( + ( w ( +! []!. The multile Baes Beoulli olyomials ae defied by the geeatig fuctio to be 1 ( w t e xt e w t 1 0 B ( (x w 1,,w t!, (2.5 whee w i > 0, 0 <t<1 (see [8],[9],[15]. The umbes B ( (0 w 1,w 2,,w B ( (w 1,,w ae called the Baes Beoulli umbes of ode. If we tae x x, w + ad t log i (2.5, the we have ( +1 ( + ( +1 1 ( + 1 x s0 B s ( (log s (x +1,,+. s! Note that lim 1 B(, (w a 1,a 2,,a ; b 1,b 2,,b B ( (a 1,,a Fo 1 ad w 0, let B, B (1,(0 1. The we have ( log B, B, (1 (0 1 1 m0 m [m] 1 log m [m]. m0

6 2122 J. J. Seo ad T. Kim Fom (2.1, we ca deive b 1x 1 + +b x [a 1 x a x ] dμ 0(x 1 dμ 0 (x Z Z ( 1 Z Z (b 1 a 1 x 1 + +(b a x [a 1 x a x ] +1 dμ 0 (x 1 dμ 0 (x (2.6 P + 1 (b a x [a 1 x a x ] dμ 0 (x 1 dμ 0 (x Z Z ( 1B +1,(a ( 1,,a ; b 1 a 1,,b a + B, ( (a 1,,a ; b 1 a 1,,b a. Theefoe, by (2.6, we obtai the followig theoem. Theoem 2.3. Fo 0, 1, we have B,(a ( 1,,a ; b 1,,b ( 1B +1,(a ( 1,,a ; b 1 a 1,,b a + B, ( (a 1,,a ; b 1 a 1,,b a. It is easy to show that i ( i ( 1 [a 1 x 1 + a x ] i+ 0 Z Z b 1x 1 + +b x dμ 0 (x 1 dμ 0 (x i 1 ( i 1 ( 1 [a 1 x 1 + a x ] i+ 0 Z Z (b 1+a 1 x 1 + +(b +a x dμ 0 (x 1 dμ 0 (x i 1 ( i 1 ( 1 B ( i+, (a 1,,a ; b 1 + a 1,,b + a. 0 (2.7 Theefoe, by (2.7, we obtai the followig theoem

7 Theoem 2.4. Fo i 1, 0, 1, we have -Adic ivaiat itegal 2123 i 0 ( i ( 1 B ( i+, (a 1,,a ; b 1,,b i 1 ( i 1 ( 1 B ( i+, (a 1,,a ; b 1 + a 1,,b + a. 0 I the secial case, 1, we have ( ( ( 1 B (1, (a 1,b 1 ( 1 [a 1 x] b 1x dμ 0 (x 0 Z (a 1+b 1 x dμ 0 (x Z ( 1 a1 + b 1 log [a 1 + b 1 ]. 0 Fom (2.1 ad (2.3, we have B (, (w a 1,,a ; b 1,,b ( log 1 ( ( ( w a + b 1 (1 [a b ] ( [w] i wi B ( i, i (a 1,,a ; b 1,,b i0 (2.8

8 2124 J. J. Seo ad T. Kim ad B (, (w a 1,,a ; b 1,,b b 1x 1 + +b x [a 1 x a x ] dμ 0(x 1 dμ 0 (x X d X d [l] l 1 i 1,,i 0 b 1i 1 + +b i [ ] w + a1 i a i + a 1 x a x Z Z l l [l] l 1 i 1,,i 0 b 1i 1 + +b i B (, l ( w + a1 i a i l lb 1x 1 + +lb x dμ 0 (x 1 dμ 0 (x a 1,,a ; b 1,,b (2.9 (Distibutio elatio fo Chaghee -Beoulli olyomials A obvious geeatig fuctio F (w, tofb,(w a ( 1,,a ; b 1,,b is obtaied fom Theoem 2.1 as follows; ( log F (w, t 1 0 ( ( log e t ia + b 1 [ia + b ] ( 1 B, ( (w a 1,,a ; b 1,,b t! i0 1 witi i!. (2.10 Note that lim 1 F (w, t a 1 a 2 a t (e a 1t 1 (e a t 1. Diffeetiatig both sides with esect to t i (2.10 ad comaig coefficiets, we have w B, ( (w a 1,,a ; b 1,,b B, ( (w a 1,a 2,,a ; b 1 a 1,,b a ( 1B +1,(w a ( 1,,a ; b 1 a 1,,b a. (2.11

9 -Adic ivaiat itegal 2125 It is easy to show that ( ( ( ( 1 i B ( log i, i (a a + b 1,,a ; b 1,,b. 1 [a i0 1 + b ] (2.12 Theefoe, by (2.11 ad (2.12, we obtai the followig theoem. Theoem 2.5. Fo 0, 1, we have w B, ( (w a 1,,a ; b 1,,b B, ( (w a 1,,a ; b 1 a 1,,b a ( 1B ( +1, (w a 1,,a ; b 1 a 1,,b a Moeove, ( ( ( log ( 1 i B ( i, i (a a + b 1,,a ; b 1,,b 1 [a + b ] i0 1 Let us assume that a 1 a 1,b 1 h, b 2 h 1,,b h +1, whee h Z. The we have B (, (w 1,, 1; h, h 1,,h +1 ( log 1 ( ( + h +1 ( w 1 (1 [ + h +1] 0 1 ( log 1 ( ( +h ( w! ( 1 (1 +h []!. 0 Theefoe, by (2.13, we obtai the followig theoem. Theoem 2.6. Fo Z, 1, 0, we have B (,(w 1, 1,, 1; h, h 1,,h +1 ( log 1 ( ( +h ( w! ( 1 (1 +h []!. 0 (2.13 Let us use the otatio of -oduct as follows : 1 (a; (1 a(1 a (1 a 1 (1 a i. (2.14 i0

10 2126 J. J. Seo ad T. Kim Fom (2.13, we have B (, (w 1, 1,, 1; h, h 1,,h +1 (log 1 (1 (log 1 (1!(log 1 (1 ( w ( 1 + ( + h ( +h +1 : ( ( + h w ( 1 +! ( m + 1 m m0 ( m m0 m 0 m(h +1 ( ( 1 + ( + h m(+h +1 (m+w. (2.15 By (2.15, we get B (,(w 1, 1,, 1; h, h 1,,h +1 (log (1! m0 ( m + 1 m m(h +1 0 ( ( + h ( 1 + (m+w. (2.16 Theefoe, by (2.16, we obtai the followig theoem. Theoem 2.7. Fo 0, 1, we have (w 1, 1,, 1; h, h 1,,h +1 (log (1! ( m + 1 m m B (, m0 m(h +1 0 ( ( + h ( 1 +h (m+w. (2.17

11 -Adic ivaiat itegal 2127 Rema. Fom (2.15, we ca deive B,(w ( +1 1, 1,, 1; h, h 1,,h +1 + (log (1 h (log (1 0 ( (w+1 ( 1 + ( + h ( +h ; 1 ( ( 1 + w ([ + h] ( ( 1(log (1 (log (1 0 0 ( + h ( +h ; 1 ( ( 1 + w ( + h 1 1 ( + h ( +h 1 ; 1 1 ( ( 1 + w ( + h ( +h ; 1 h(1 log B, ( 1 (w 1, 1,, 1; h 1,h 2,,h log B ( 1 1, (w 1, 1,, 1; h, h 1,,h +1 + B,(w ( 1, 1,, 1; h, h 1,,h +1. (2.18 Refeeces 1. S. Aaci, M. Acigoz, A ote o the Fobeius-Eule umbes ad olyomials associated with Bestei olyomials. Adv. Stud. Cotem. Math. 22 (2012, o. 3, A. Bayad, Modula oeties of ellitic Beoulli ad Eule fuctios, Adv. Stud. Cotem. Math. 20 (2010, o. 3, L. Calitz, -Beoulli umbes ad olyomials, Due Math. J. 15(1948, K.-W. Hwag, D. V. Dolgy, T. Kim, S.-H. Lee, O the highe-ode -Eule umbes ad olyomials with weight α, Discete Dy. Nat. Soc. 2011, At. ID , G. Kim, B. Kim, J. Choi, The DC algoithm fo comutig sums of owes of cosecutive iteges ad Beoulli umbes, Adv. Stud. Cotem. Math. 17 (2008, o. 2, D. S. Kim, T. Kim, Y.-H. Kim, D. V. Dolgy, A ote o Euleia olyomials associated with Beoulli ad Eule umbes ad olyomials, Adv. Stud. Cotem. Math. 22 (2012, o. 3, T. Kim, Y.-H. Kim, K.-W. Hwag, O the -extesios of the Beoulli ad Eule umbes, elated idetities ad Lech zeta fuctio, Poc. Jageo Math. Soc. 12 (2009, o. 1,

12 2128 J. J. Seo ad T. Kim 8. T. Kim, O the weighted -Beoulli umbes ad olyomials Adv. Stud. Cotem. Math. 21 (2011, o. 2, T. Kim, S. H. Rim, New Chaghee -Eule umbes ad olyomials associated with -adic -itegals, Comut. Math. Al. 54 (2007, o. 4, T. Kim, S.-H. Rim, O Chaghee-Baes -Eule umbes ad olyomials, Adv. Stud. Cotem. Math. 9 (2004, o. 2, J. V. Leyedees, A. G. Shao, C. K. Wog, Itege stuctue aalysis of the oduct of adacet iteges ad Eule s extesio of Femat s last theoem, Adv. Stud. Cotem. Math. 17 (2008, o. 2, H. Ozde, I. N. Cagul, Y. Simse, Remas o -Beoulli umbes associated with Daehee umbes, Adv. Stud. Cotem. Math. 18 (2009, o. 1, C. S. Ryoo, T.Kim, R. P. Agawal, Exloig the multile Chaghee -Beoulli olyomials, It. J. Comut. Math. 82 (2005, o. 4, C. S. Ryoo, H. Sog, O the eal oots of the Chaghee-Baes -Beoulli olyomials, Poceedigs of the 15th Iteatioal Cofeece of the Jageo Mathematical Society, 63-85, Jageo Math. Soc., Hacheo, Y. Simse, Geeatig fuctios of the twisted Beoulli umbes ad olyomials associated with thei iteolatio fuctios, Adv. Stud. Cotem. Math. 16 (2008, o. 2, Y. Simse, T. Kim, I.-S. Pyug, Baes tye multile Chaghee -zeta fuctios, Adv. Stud. Cotem. Math. 10 (2005, o. 2, Received: Jue 25, 2013

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