A note on the Lebesgue Radon Nikodym theorem with respect to weighted and twisted p-adic invariant integral on Z p

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1 Notes on Number Theory and Discrete Mathematics ISSN Vol. 21, 2015, No. 1, A note on the Lebesgue Radon Nikodym theorem with resect to weighted and twisted -adic invariant integral on Z Joo-Hee Jeong 1, Jin-Woo Park 2, Seog-Hoon Rim 1 and Joung-Hee Jin 1 1 Deartment of Mathematics Education, Kyungook National University Taegu , Reublic of Korea s: jhjeong@knu.ac.kr, shrim@knu.ac.kr, jhjin@knu.ac.kr 2 Deartment of Mathematics Education, Sehan University YoungAm-gun, Chunnam, , Reublic of Korea a @knu.ac.kr Abstract: In this aer we will give the Lebesgue Radon Nikodym theorem with resect to weighted and twisted -adic q-measure on Z. In secial case, if there is no twisted, then we can derive the same result as Jeong and Rim, 2012; If the case weight zero and no twist, then we derive the same result as Kim Keywords: -adic invariant integral, -adic q-measure, Lebesgue Radon Nikodym theorem. AMS Classification: 11B68, 11S80. 1 Introduction Let be a fixed odd rime number. Throughout this aer, the symbols Z, Q, and C denote the ring of -adic integers, the field of -adic rational numbers, and the comletion of the algebraic closure of Q, resectively. The -adic norm. is defined by x = v(x) = r for x = r s, t where s and t are integers with (, s) = (, t) = 1 and r Q (see [1 16]). When one seaks of q-extension, q can be regard as an indeterminate, a comlex q C, or a -adic number q C. In this aer we assume that q C with q 1 < 1 1 and we use the notations of q-numbers as follows: [x] q = [x : q] = 1 qx 1 q, and [x] q = 1 ( q)x. 1 + q 10

2 For n Z +, let C n = {w w n = 1} be the cyclic grou of order n and let T be the sace of locally constant sace, i.e, T = lim C n = n 0 C n. For any ositive integer N, let a + N Z = { x Z x a ( mod N )} (1) where a Z satisfies the condition 0 a < N (see [1, 2, 3, 5-9]). It is known that the fermionic -adic q-measure on Z is given by Kim as follows: µ q (a + N Z ) = ( q)a [ N ] q = 1 + q 1 + q N ( q) a, (2) (see [5, 7, 12 16]). Let C(Z ) be the sace of continuous functions on Z. From (2), the fermionic -adic q- integral on Z is defined by Kim as follows: 1 I q (f) = f(x)dµ q (x) = lim Z N [ N ] q N 1 f(x)( q) x, (3) where f C(Z )(see [1, 5, 7, 12-16]). For w T, we consider the twisted q-euler olynomials ε w n,q(x) as Z q y w y e [x+y]q dµ q (y) = From (4), we can derive the following equation. n=0 ε w n,q(x) tn n! (4) ε w n,q(x) = [2] q (1 q) n n ( 1) l q lx 1 + wq, (5) l (see [16]). From (5), we note that lim q 1 ε w n,q(x) = ε w n (x). In secial case, x = 0, we have ε w n,q(0) = ε w n,q are the n-th twisted q-euler number, we have ε w n,q = q y w y [x] n q dµ q (y) = Z l=0 [2] q (1 q) n n l=0 ( 1) l wq l. (6) In secial, we have ε w 0,q = [2] q 1 + w. (7) The idea for generalizing the fermionic integral is relacing the fermionic Haar measure with weakly (strongly) fermionic measure Z satisfying µ 1 (a + n Z ) µ 1 (a + n+1 Z ) δ n,q, (8) where δ n,q 0, a is a element of Z, and δ n,q is indeendent of a (for strongly fermionic measure, δ n,q is relaced by C n, where C is a ositive constant)(see [5, 7, 8, 13]). 11

3 Let f(x) be a function defined on Z. The fermionic integral of f with resect to a weakly fermionic measure µ 1 is Z f(x)dµ 1 (x) = lim n 1 f(x)µ 1 (x + n Z ), if the limit exists. If µ 1 is a weakly fermionic measure on Z, then the Radon Nikodym derivative of µ 1 with resect to the Haar measure on Z as follows: f µ 1 (x) = lim µ 1 (x + n Z ), (9) (see [5, 7]). Note that f µ 1 is only a continuous function on Z. Let UD(Z ) be the sace of uniformly differentiable functions on Z. For f UD(Z ), let us define µ 1,f as follows: µ 1,f (x + n Z ) = f(x)dµ 1 (x), (10) x+ n Z where the integral is the fermionic -adic invariant integral. From (9), we can easily note that µ 1,f is a strongly fermionic measure on Z (see [5, 7]). Since µ 1,f (x + n Z ) µ 1,f (x + n+1 Z ) = n 1 f(x)( 1) x = n f(n ) n n C n, f(x)( 1) x where C is ositive consatnt. In this aer, we will give the Lebesgue Radon Nikodym theorem with resect to weighted and twisted -adic q-measure on Z. In secial case, if there is no twisted, then we can derive the same result as Jeong and Rim, 2012 (see [5]). In the case of weight zero and no twisted, then we derive the same result as Kim, 2012 (see [7]). 2 The Lebesgue Radon Nikodym theorem with resect to the weighted -adic q-measure For any ositive integer a and n with a < n and f UD(Z ), we define µ w f, q, weighted and twisted ferminonic measure on Z as follows: µ w f, q(a + n Z ) = w x q x f(x)dµ 1 (x) (11) a+ n Z where the integral is the fermionic -adic invariant integral on Z. From (11), we note that 12

4 m 1 µ w f, q(a + n Z ) = lim f(a + n x)( 1) a+nx q a+nx w a+n x m m n 1 = ( 1) a w a q a lim f(a + n x)( 1) x q n x m (12) = ( 1) a Z w a f(a + n x)q a+nx dµ 1 (x). By (12), we get µ w f, q(a + n Z ) = ( 1) a Z w a f(a + n x)q a+nx dµ 1 (x). (13) Thus, by (13), we have where f, g UD(Z ) and α, β are ositive constants. By (11), (12), (13) and (14), we get µ αf+βg, q = α µ f, q + β µ g, q, (14) µ w f, q (a + n Z ) f wq, (15) where f wq = su x Z f(x)q x w x. Let P (x) C [[x] q ] be an arbitrary q-olynomial. Now we show µ w is a strongly weighted and twisted fermionic -adic invariant measure on Z. Without a loss of generality, it is enough to rove the statement for P (x) = [x] k q. For a Z with 0 a < n, we have Note that m n 1 µ w (a + n Z ) = lim m ( q)a w a q ni [a + i n ] k q( 1) i. (16) i=0 [a + i n ] k q = ([a] q + q a [ n ] q [i] q n ) k. (17) By (6), (16) and (17), { } µ w, q = [a] k q ε 0,q n + k[a] q k 1 q a [ n ] q [i] q n + + q ak [ n ] k q[i] k q. n By (7), (12) and (13), we easily get µ w (a + n Z ) ( 1) a q a w a [a] k qε wn 0,q n (mod [n ] q ) ( 1) a 1 + qn 1 + w n P (a)q a w a (mod [ n ] q ). For x Z, let x x n (mod n ) and x x n+1 (mod n+1 ), where x n, x n+1 Z with 0 x n < n and 0 x n+1 < n+1. Then we have (18) µ w (a + n Z ) µ w (a + n+1 Z ) C v(1 qn), (19) 13

5 where C is ositive constant and n 0. Let (a) = lim µ w (a + n Z ) = ( 1) a q a w a. Then, by (18) and (19), we see that (a) = ( 1) a q a w a [a] k q = ( 1) a q a w a P (a). (20) Since (x) is continuous function on Z, for x Z, we have (x) = ( 1) x q x w x [x] k q, (k Z + ). (21) Let g UD(Z ). Then, by (19), (20) and (21), we get Z g(x)d µ w (x) = lim n 1 g(x) µ w (x + n Z ) n 1 = lim g(x)q x w x [x] k q( 1) x = g(x)q x w x [x] k qdµ 1 (x). Z (22) Therefore, by (22), we obtain the following theorem. Theorem 2.1. Let P (x) C [[x] q ] be an arbitrary q-olynomial. Then µ w weighted and twisted fermionic -adic invariant measure on Z. Then we have is a strongly (x) = ( 1) x q x w x P (x) for all x Z. Furthermore, for any g UD(Z ), g(x)d µ w (x) = g(x)p (x)q x w x dµ 1 (x), Z Z where the second integral is weighted and twisted fermionic -adic invariant integral on Z. Then Let f(x) = n=0 a x ) n,q( be the Mahler q-exansion of continuous function on Z n q, where ( ) x n Then we note that lim a n,q = 0. Let q f m (x) = = [x] q[x 1] q [x n + 1] q. [n] q! m ( ) x a i,q i i=0 q C [[x] q ]. (f f m ) qw su a n,q. (23) n m 14

6 The function f(x) can be rewritten as f = f m + f f m. Thus, by (14) and (23), we get µ w f, q(a + n Z ) µ w f, q(a + n+1 Z ) From Theorem 2.1., we note that max{ w µ fm, q(a + n Z ) µ w f m, q(a + n+1 Z ), w µ f fm, q(a + n Z ) µ w f f m, q(a + n+1 Z ) (24) } µ w f f m, q(a + n Z ) f f m C 1 2v(1 qn), (25) where C 1 are ositive constants. For m 0, we have f = f m. So, we see that w µ fm, q(a + n Z ) µ w f m, q(a + n+1 Z ) = f m([ n ] q )q n [ n [ n ] 2 ] 2 q f m q x w x [ n ] 2 q C 2 2v(1 qn), q (26) where C 2 is a ositive constant. By (25), we get ( 1) a q a w a f(a) µ w f, q(a + n Z ) max{ q a w a q a w a f(a) f m (a)q a w a, q a w a f m (a) µ w f m, q(a + n Z ), µ w f fm, q(a + n Z ) } (27) max{ q a w a f(a) fm (a), fm (a) µ w f m, q(a + n Z ), (f f m ) qw } Let us assume that fix ɛ > 0, and fix m such that f f m < ɛ. Then we have Thus, by (28), we have ( q) a w a f(a) µ w f, q(a + n Z ) ɛ for n 0. (28) f, q (a) = lim µ w f, q(a + n Z ) = ( 1) a q a w a f(a). (29) Let m be the sufficiently large number such that f f m n. Then we get For g UD(Z ), we have µ w f, q(a + n Z ) = µ w f m, q(a + n Z ) + µ w f f m, q(a + n Z ) g(x)d µ w f, q(x) = Z = ( 1) a q a w a f(a) (mod [ n ] 2 q). Z f(x)g(x) [2] q 1 + w qx w x dµ 1 (x). Let f be the function from UD(Z ) to Li(Z ). We easily see that w x q x µ 1 (x + n Z ) is a strongly weighted and twisted -adic invariant measure on Z and (fqw ) µ 1 (a) w a q a µ 1 (a + n Z ) C 3 2v(1 qn), 15

7 where f qw (x) = q x w x f(x) and C 3 is ositive constant and n Z +. If µ w 1, q is associated with strongly weighted and twisted fermionic invarinat measure on Z, then we have w µ 1, q (a + n Z ) (f qw ) µ 1 (a) C 4 2v(1 qn), where n > 0 and C 4 is ositive constant. For n 0, we have q a w a µ 1 (a + n Z ) µ w 1, q(a + n Z ) q a w a µ 1 (a + n Z ) (f qw ) µ w 1 (a) + (fqw ) µ 1 (a) µ w 1, q(a + n Z ) K, (30) where K is ositive constant. Hence, wqµ 1 µ w 1, q is a weighted and twisted measure on Z. Therefore, we obtain the following theorem. Theorem 2.2. Let wqµ 1 be a strongly weighted and twisted -adic invariant measure on Z, and assume that the fermionic weighted and twisted Radon Nikodym derivative (f qw ) µ 1 on Z is uniformly differentiable function. Suose that µ w 1, q is the strongly weighted and twisted fermionic -adic invariant measure associated with (f qw ) µ 1. Then there exists a weighted and twisted measure µ w 2, q on Z such that w x q x µ 1 (x + n Z ) = µ w 1, q(x + n Z ) + µ w 2, q(x + n Z ). 3 Conclusion The Theorem 2.2. is the version of the Lebesgue Radon Nikodym theorem with resect to weighted and twisted -adic q-measure on Z. In secial case, if there is no twisted, then we can derive the same result as Jeong and Rim, 2012 (see [5]). In the case of weight zero and no twisted, then we derive the same result as Kim, 2012 (see [7]). References [1] Bayad, A. & T. Kim. (2011) Identities involving values of Bernstein, q-bernoulli, and q- Euler olynomials. Russ. J. Math. Phys., 18(2), [2] Calabuig, J. M., P. Gregori & E. A Sanchez Perez. (2008) Radon Nikodym derivative for vector measures belonging to Kothe function sace. J. Math. Anal. Al. 348, [3] Choi, J., T. Kim & Y. H. Kim. (2011) A note on the q-analogues of Euler numbers and olynomials, Honam Mathematical Journal, 33(4), [4] George, K. (2008) On the Radon Nikodym theorem, Amer. Math. Monthly, 115,

8 [5] Jeong, J. & S. H. Rim. (2012) A note on the Lebesgue Radon Nikodym theorem with resect to weighted -adic invariant integral on Z, Abstract and Alied Analysis, 2012, Article ID , 8 ages. [6] Kim, T. (2002) q-volkenborn integration. Russ. J. Math. Phys. 9(3), [7] Kim, T. (2012) Lebesgue Radon Nikodym theorem with resect to fermionic -adic invariant measure on Z, Russ. J. Math. Phys. 19, [8] Kim, T. (2007) Lebesgue Radon Nikodym theorem with resect to fermionic q-volkenborn distribution on µ q. Al. Math. Com., 187, [9] Kim, T. (2011) A note on q-bernstein olynomials. Russ. J. Math. Phys. 18(1), [10] Kim, T. (2008) Note on the Euler numbers and olynomials. Adv. Stud. Contem. Math., 17, [11] Kim, T. (2009) Some identities on the q-euler olynomials of higher order and q-stirling numbers by the fermionic -adic integral on Z. Russ. J. Math. Phys. 16, [12] Kim, T. (2010) New aroach to q-euler olynimials of higher order. Russ. J. Math. Phys. 17, [13] Kim, T., J. Choi & H. Kim A note on the weighted Lebesgue Radon Nikodym theorem with resect to -adic invariant integral on Z. to aear in JAMI. [14] Kim, T., D. V. Dolgy, S. H. Lee & C. S. Ryoo. (2011) Analogue of Lebesgue Radon Nikodym theorem with resect to -adic q-measure on Z. Abstract and Alied Analysis, 2011, Article ID637634, 6 ages. [15] Kim, T., S. D. Kim, & D. W. Park. (2001) On Uniformly differntiabitity and q-mahler exansion. Adv. Stud. Contem. Math., 4, [16] Kim, Y. H., B. Lee & T. Kim. (2011) On the q-extension of the twisted generalized Euler numbers and olynomials attached to χ. J. of Com. Anal. and Al., 13(7), [17] Jeong, J. & S. H. Rim. (2012) A note on the Lebesgue Radon Nikodym theorem with resect to weighted -adic invariant integral on Z. Abstract and Alied Analysis, 2012, Article ID , 8 ages. 17