Journal of Number Theory. On Euler numbers, polynomials and related p-adic integrals

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1 Journal of Number Theory Contents lsts avalable at ScenceDrect Journal of Number Theory On Euler numbers, polynomals and related p-adc ntegrals Mn-Soo Km Natonal Insttute for Mathematcal Scences, Doryong-dong, Yuseong-gu, Daejeon , South Korea artcle nfo abstract Artcle hstory: Receved 10 October 008 Avalableonlne3January009 Communcated by Davd Goss MSC: 11S80, 11B68, 11M99 Keywords: p-adc ntegrals Euler numbers Euler polynomals In ths note we gve a new proof of Wtt s formula for Euler numbers, whch are related to some nown or new denttes nvolvng the Euler numbers. We also obtan a bref proof of a classcal result on Euler numbers modulo of two due to M.A. Stern usng the approach of p-adc ntegraton, whch was recently proved by G. Lu, and Z.-W. Sun. Fnally some explct formulas for Genocch numbers are proved and applcatons are gven. 009 Elsever Inc. All rghts reserved. 1. Introducton Euler numbers E m,m 0 are ntegers gven by cf. [13,4] E 0 1, E m m 1 m m E for m 1,, The Euler polynomal E m x s defned by see [15, p. 5]: E m x m m E x 1 m, 1. E-mal address: msm@nms.re.r X/$ see front matter 009 Elsever Inc. All rghts reserved. do: /j.jnt

2 M.-S. Km / Journal of Number Theory whch holds for all nonnegatve ntegers m and all real x, and whch was obtaned by Raabe [19] n Further, n vew of the p-adc ntegral, ths dentty can be obtan n Secton, Eq..17 below. Settng x 1/ and normalzng by m gves the Euler numbers E m m E m 1, 1.3 where E 0 1, E 1, E 4 5, E 6 61,... Therefore, E m E m 0, n fact [4, p. 374,.1] E m 0 1 m+1 B m+1, 1.4 m + 1 where B m means the Bernoull numbers. The Euler numbers and polynomals so-named by Scher n 185 appear n Euler s famous boo, Insttutones Calcul Dfferentals 1755, pp and p. 5. Let p be an odd prme number. For all m Z \{0}, we denote ord p m the greatest nteger 0 such that p dvdes m n Z. If m 0, we agree to wrte ord p 0. For any ratonal number x m/n, defne ord p x to be ord p m ord p n. Further defne a map p on Q as follows: { p x p ord p x, f x 0, 0, f x 0. It s well nown that p s a norm over Q, called the p-adc norm over Q, whle ord p s called the p-adc ordnal over Q. Let Q p be the topologcal completon of Q wth respect to the metrc topology nduced by p. Let C p be the feld of p-adc completon of algebrac closure of Q p. Let be the topologcal closure of Z. We have {x Q p x p 1}. We say that f : C p s unformly dfferental functon at aponta, and we wrte f UD, f the dfference quotents Φ f : C p such that Φ f x, y f x f y x y 1.5 have a lmt f a as x, y a, a x and y remanng dstnct. Set a + p N {x Q p x a p < p N }. Defne μa + p N 1 a. Ths extends to a dstrbuton on, snce μa + p N b0 μa + bpn + p N+1. For f UD, the p-adc ntegral on was defned by I f p N 1 f a dμa lm N f a 1 a 1.6 cf. [6 8,1,16,18,1,]. In vew of 1.6, for f UD we get I f 1 + I f f 0, 1.7 where f 1 x f x + 1. The relaton 1.7 were studed n great detal by T. Km [8], who was partcularly nterested n ther relatonshp to Euler numbers. More recently, many authors nvestgated some nterestng ntegral equatons related to q-analogue of 1.6 cf. [5 8,10 1,16 18,0 ]. In order to consder p-adc and complex cases smultaneously we wll use an somorphsm σ, between the algebrac closure of the ratonal numbers n C p and the algebrac closure of the ratonal numbers wthn the complex numbers C. So we shall consder σ as fxed throughout ths note and use σ to dentfy p-adc algebrac numbers wth complex algebrac numbers. We shall wrte x y when x C p, y C and y σ x. From 1.7, we derve

3 168 M.-S. Km / Journal of Number Theory e at dμa e t + 1 E m 0 tm m!. 1.8 Here E m 0 are the Euler polynomals wth x 0 n 1.. One, n [8,13,4,6], can fnd the fully detaled study of the numbers E m 0 for m 0. Namely, Wtt s formula Wtt s p-adc characterzaton for the numbers E m 0 s proved by n [8] usng the ntegral equaton 1.7. Also, varous applcatons and some denttes for Euler numbers can be founded n some resent wors for example [,8,0, 4 7]. In ths note we gve a new proof of Wtt s formula for Euler polynomals and Euler numbers. Also we gve a smple treatment to some nown or new denttes nvolvng the Euler numbers. We obtan a bref proof of a classcal result on Euler numbers modulo of two due to M.A. Stern [3] asserts that E m E m mod f and only f n m mod, 1.9 whch was recently proved by G. Lu [13] and Z.-W. Sun [4]. Fnally, some explct formulas for Genocch numbers are proved and applcatons are gven. Even though some results are not really new, the wrter beleves that all the proofs n the paper are new.. Some results From 1.8, the Euler polynomals E m x, 0 m <, are defned by means of the followng generatng functon: e xt e at dμa E m x tm m!..1 It wll be shown n the sequel that, ndeed, m E m x s a polynomal n x of degree m. Now consder the seres f t et + 1 et + 1 et 3 +, 3. whch converges for 1 e t < and hence small values of t. Moreover, we can wrte f t 1 e t et e t or f t 1 1 j e jt..4 j j0 We may now expand the expresson n.4 usng.1,. and.3. We obtan for small values of t > 0therelaton E m x tm m! f t m m 1 text 1 j j + x m..5 m! j Eq..5 t follows that E m x can always be wrtten n the closed form. j0

4 M.-S. Km / Journal of Number Theory Lemma.1. Let m 0 be ntegers. Then m 1 E m x 1 j j + x m. j In partcular, f m 0 be ntegers, then m E m x Z[x] and m E m 0 Z. j0 Tang x 1 n Lemma.1 and nothng that E m m E m 1 we deduce: Corollary.. Let m 0 be ntegers. Then E m m 1 1 j j + 1 m. j j0 We use the followng notatons. If f x a 0 x 0 + a 1 x 1 + +a m x m s a polynomal, then by f Ex we mean the polynomal a 0 E 0 x + a 1 E 1 x + +a m E m x. Analogously, f f x, t s a power seres of the form f mxt m, where f m x s a polynomal, then by f Ex, t we means the seres f mext m. Usng ths notaton,.1 can be wrtten n the form and hence we have e Ex+1t + e Ext e xt, Ex m + Em x x m, m 0..7 Therefore by.7, we fnd that Euler polynomals are related to the recurrence relaton E 0 x 1, E m x x m 1 m 1 m E x.8 for m 1. Now, from.6, we have e Ex+ρ+1t + e Ex+ρt e x+ρt,.9 where ρ s a nonnegatve nteger. The dentty.9 s equvalent to m t m Ex + ρ + 1 m! + m t m Ex + ρ m! x + ρ m tm m!..10 Clearly ths mples Ex + a m + Ex + a 1 m x + a 1 m, a 1,,...,ρ..11 Alternatng addng and subtractng ths dentty wth a 1,,...,ρ for each case, gves the formula m Ex + ρ + Em x 1 a x + a m..1

5 170 M.-S. Km / Journal of Number Theory Settng ρ p N n.1 we obtan and hence p N 1 Ex + p N m + Em x 1 a x + a m,.13 m 1 E m x + p N 1 p m E N 1 xp m 1N 1 a x + a m Let N, then p N 0nQ p, and thus by 1.6 and.14, we have Lemma.3. Let m 0 and p 3. Then p N 1 E m x lm 1 a x + a m x + a m dμa. N Corollary.4. Let m 0 mod p 1 wth p 3. Then E m 0 a m dμa 0 mod p. Proof. Note that p N 1 a 0 mod p p N 1 1 a 1 a p N pb b0 It s clear from.15 that for m 0 mod p 1, p N 1 1 a a m p N 1 a 0 mod p 1 a 0 mod p..16 By substtutng x 0 nto Lemma.3 and.16, we get at once E m 0 lm N 0 0 mod p. Ths completes the proof. Theorem.5 Wtt s formula of Euler numbers. E m a + 1 m dμa.

6 M.-S. Km / Journal of Number Theory Proof. From 1.3,.1 and Lemma.3 we obtan the formula a + 1 m t m dμa m! m a + 1 m dμa tm m! Z p 1 t m E m m! t m E m m!. We therefore obtan the theorem. Theorem.5 can be regarded as Wtt s p-adc characterzaton of the Euler numbers see [3]. Remar.6. By Lemma.3 and Theorem.5, we have E m x m x + a m dμa m x 1 + a + 1 m dμa m m x m 1 m a + 1 dμa m m m x 1 m E m m x 1 x 1 m E m m E x 1 m,.17 whch s the same as Eq. 1.. For a dfferent approach of.17 see [15, p. 5]. Snce the Euler numbers are all ntegers, there s no analogue for them of the von Staudt Clausen theorem. But Kummer s congruence has an analogue, due also to Kummer cf. [1,6]: Theorem.7. If m 1 and p 3,then E m E m+ mod p. Proof. Let ρ 1 be ntegers. Applyng.1 wth x 1, we fnd that

7 17 M.-S. Km / Journal of Number Theory and thus Clearly m m a + a E + ρ + E m m m 1 1 E ρ m + E m m m 1 1 m E m ρ m + E m m m m ρ m E + m E m m m ρ m E + E m 1 a 1 + a m. E m 1 a 1 + a m mod ρ.18 snce E m Z. Wrte ρ p. Taen modulo p, we have and E m 1 a 1 + a m mod p E m+ 1 a 1 + a m+ mod p. Now t suffces to show that 1 + a m+ 1 + a m mod p. In fact, 1 + a m+ 1 + a m 1 + a 1 + a m mod p for 0 a 1 and + 1 a p 1 by Fermat s Lttle Theorem. Clearly 1 + a m+ 1 + a m 0 mod p when a. Hence we have E m E m+ mod p. Ths completes the proof. By.18, we nown that mod p 1, wehave E m 1a 1 + a m mod ρ. Tang ρ p and m 0

8 M.-S. Km / Journal of Number Theory E m 1 a 1 + a m mod p 1 1 a + a +1 1 by Fermat Lttle Theorem. We have also the followng theorem. a mod p p+1 mod p.19 Theorem.8. See [7]. If p 3 and m 0 mod p 1, then E m mod p. Lemma.9. See [0]. If m s a nonnegatve nteger, then E m 0 a m dμa G m+1 m + 1, where G m are the Genocch numbers, whch are defned by G 0 0, G 1 1, G + 1 m + G m 0 f m, symbolcally. Remar.10. It follows that G m+1 0 m 1. The Genocch numbers G m are drectly connected to the Bernoull numbers by G m 1 m B m, whle to E m 1 0 by G m me m 1 0 Z. Now we gve a new proof of a classcal result due to M.A. Stern n [3], see also [13] and [6]. Theorem.11. Let n m 0 be ntegers, and n m mod. Then E n E m n + m mod +1. Proof. If n m mod, then n l + m for some l. From Theorem.5 we have It follows from Lemma.9 that E n a + 1 l+m dμa l dμa a + 1 m a + 1 l l a + 1 m a dμa 0 l a + 1 m l dμa + a + 1 m a dμa. Z 1 p

9 174 M.-S. Km / Journal of Number Theory a + 1 m a dμa m m j j0 j + a j dμa m m j j0 j G + j+1 + j + 1. Therefore we can rewrte E n as l m l m E n E m + + j G + j+1 j + j j0 E m + +1 l G + l m m j j1 j+1 G l j+ j + + m l j0 m j + j G + j+1 + j + 1. By ord m j j+1 j + m j+1 ord + ord 1 j j + for j 1,...,m, and ord l m + j j + j + 1 l m + j ord + ord + ord + 1 j + j + 1 for,..., l and j 0,...,m, and note that G 1, G m Z, we have E n E m l mod +1. Ths completes the proof. Remar.1. For n m +, Theorem.11 mmedately yelds a result due to Wagstaff see [6]. Corollary.13. If m s a postve nteger, then E 4m 1 mod 4. Proof. Note that and ord 4m E 4m + 1 a + 1 4m dμa 4m 4m 0 4m 4m ord 4m a dμa G m ord + 1 for,...,4m. Ths completes the proof.

10 M.-S. Km / Journal of Number Theory Recall that the Genocch polynomals G m x defned by It satsfes G m x m m G x m..0 n 1 x + n m 1 1 G m G m x, f n s odd..1 n Note that G m 0 G m, where G m are Genocch numbers. Ther nvestgaton goes bac to L. Euler whle the term Genocch numbers, named after A. Genocch, was presumably ntroduced by E. Lucas n [14]. In [0], Rm et al. constructed the Hurwt s-type Genocch zeta functon whch nterpolates Genocch polynomals at nonpostve ntegers; see also [9] and [11]. Intensve wors on these numbers were done n [5,11,0] and []. Relatons between E n and G n are the followng: Corollary.14. Let m 0. Then E m m m G Proof. By Lemma.3 and.9, we have E m 1 m 1 + a dμa m 1 + a m dμa m m m G Ths together wth 1.3 yelds the result. Corollary.15. Let m 0. Then G m+1 m + 1 m m m 1 E m. Proof. By Theorem.5 and Corollary.9, we have m G m+1 m + 1 a m dμa a m dμa m m 1 a + 1 m dμa m m 1 E m. Ths completes the proof.

11 176 M.-S. Km / Journal of Number Theory The followng theorem s a trval consequence of the power seres defnton of the sequence E m x. We gve another proof here, however, whch depends only upon.17, Corollary.4 and Lemma.9. Theorem.16. If m s an even postve nteger, then E m 0 E m 1 0. Also, f m s a postve nteger, then G m+1 0. Proof. Let x 0 and x 1 n.17. Thus we have 1 m E m 0 m m m 1 E m m m E E m 1,. snce E In fact, G 0 0 and G 1 1. By Corollary.4 and Lemma.9, we have G m+1 m + 1 a m dμa E m 0 0 mod p.3 for m 1 and nfntely many prmes p. Combnng.3 and. we arrve at the desred results. We can now prove the followng, whch corresponds to Eq..18. Theorem.17. Let m and ρ be postve ntegers. Then G m 4m 1 a a m 1 mod ρ. Proof. From.0, we can be wrtten symbolcally as e Gx+ρ+1t + e Gx+ρt te x+ρt. Ths gves Gx + a m + Gx + a 1 m mx + a 1 m 1, where m 1, a 1,...,ρ and Gx + a m G m x + a. Alternatng addng and subtractng ths dentty wth a 1,,...,ρ for each case, gves the formula Clearly m Gx + ρ + Gm x m 1 a x + a m 1. mg m 1 ρ + G m 4m 1 a a m 1 mod ρ. As G m 1 0, by the above we are done.

12 M.-S. Km / Journal of Number Theory In proof of Theorem.17, we have G m m 1 a a m 1 mod ρ..4 a1 Tang ρ p and m p + 1, we fnd that the congruence G p+1 1 a a p mod p a1 1 a a mod p a1 1 a 4 a + 1 mod p by Fermat Lttle Theorem. We have also the followng theorem. a1 1 mod p.5 Theorem.18. If p 3,then G p+1 1 mod p. Theorem.19. If p 3,then G m 4m p a m 1 mod p. a1 Proof. Let p 3. Then p 1 does not dvde m 1. From Theorem.17, we obtan G m m 1 a a m 1 mod p a1 a m 1 p a m 1 mod p a1 p a m 1 mod p, a1 because a1 am 1 0 mod p snce p 1 does not dvde m 1 see [4, p. 35, Lemma ]. Remar.0. Snce G m 1 m B m, as examples, tang p 5nTheorem.19wefnd 1 m B m 5 a m 1 3 m m 1 mod 5..6 m a1

13 178 M.-S. Km / Journal of Number Theory Now from.6 we have, wth m replace by + 1: 1 4+ B by Fermat Lttle Theorem. Hence mod B mod Tang p 3nTheorem.19wefnd 1 m B m 1 3 a m 1 1 mod 3.8 m a1 and 4+ 1B mod References [1] L. Carltz, J. Levne, Some problems concernng Kummer s congruences for the Euler numbers and polynomals, Trans. Amer. Math. Soc [] D. Cvjovć, J. Klnows, New formulae for the Bernoull and Euler polynomals at ratonal arguments, Proc. Amer. Math. Soc [3] H. Hasse, Vandver s congruence for the relatve class number of the pth cyclotomc feld, J. Math. Anal. Appl [4] K. Ireland, M. Rosen, A Classcal Introducton to Modern Number Theory, second ed., Grad. Texts n Math., vol. 84, Sprnger- Verlag, New Yor, [5] L.-C. Jang, T. Km, q-genocch numbers and polynomals assocated wth fermonc p-adc nvarant ntegrals on,abstr. Appl. Anal , Art. ID 3187, 8 pp. [6] M.-S. Km, J.-W. Son, Analytc propertes of the q-volenborn ntegral on the rng of p-adc ntegers, Bull. Korean Math. Soc [7] T. Km, On a q-analogue of the p-adc log gamma functons and related ntegrals, J. Number Theory [8] T. Km, On the analogs of Euler numbers and polynomals assocated wth p-adc q-ntegral on at q 1, J. Math. Anal. Appl [9] T. Km, On the q-extanson of Euler and Genocch numbers, J. Math. Anal. Appl [10] T. Km, An nvarant p-adc q-ntegral on, Appl. Math. Lett [11] T. Km, L.-C. Jang, H.K. Pa, A note on q-euler and Genocch numbers, Proc. Japan Acad. Ser. A Math. Sc [1] T. Km, M.-S. Km, L. Jang, S.-H. Rm, New q-euler numbers and polynomals assocated wth p-adc q-ntegrals, Adv. Stud. Contemp. Math [13] G. Lu, On congruences of Euler numbers modulo powers of two, J. Number Theory [14] E. Lucas, Theore des nombres. Tome premer: Le calcul des nombres enters, le calcul des nombres ratonnels, la dvsblte arthmetque, Gauther Vllars et Fls, Impremeurs-Lbrares, Pars, 1891; reprnt: Edtones Jacques Gabay, Pars, ISBN X, [15] N.E. Nörlund, Vorlesungen über Dfferenzenrechnung, Sprnger, Berln, 194. [16] H. Ozden, Y. Smse, Multvarate nterpolaton functons of hgher-order q-euler numbers and ther applcatons, Abstr. Appl. Anal , Art. ID , 16 pp. [17] H. Ozden, Y. Smse, A new extenson of q-euler numbers and polynomals related to ther nterpolaton functons, Appl. Math. Lett [18] H. Ozden, Y. Smse, I.N. Cangul, Euler polynomals assocated wth p-adc q-euler measure, Gen. Math [19] J.L. Raabe, Zurücführung enger Summen und bestmmtem Integrale auf de Jacob Bernoullsche Functon, J. Rene Angew. Math [0] S.-H. Rm, K.H. Par, E.J. Moon, On Genocch numbers and polynomals, Abstr. Appl. Anal. 008, Art. ID , 7 pp. [1] Y. Smse, q-analogue of twsted l-seres and q-twsted Euler numbers, J. Number Theory [] Y. Smse, I.N. Cangul, V. Kurt, D. Km, q-genocch numbers and polynomals assocated wth q-genocch-type l-functons, Adv. Dfference Equ , Art. ID , 1 pp. [3] M.A. Stern, Zur Theore der Eulerschen Zahlen, J. Rene Angew. Math

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