EXPLICIT CONGRUENCES FOR EULER POLYNOMIALS

Transcription

1 EXPLICIT CONGRUENCES FOR EULER POLYNOMIALS Zhi-Wei Sun Departent of Matheatics, Nanjing University Nanjing 10093, People s Republic of China zwsun@nju.edu.cn Abstract In this paper we establish soe explicit congruences for Euler polynoials odulo a general positive integer. As a consequence, if a, Z and - then k+1 x + a E k 1a E k x Z[x] for every k = 0, 1,,..., which ay be regarded as a refineent of a ultiplication forula. Keywords: Euler polynoial, Congruence, q-adic nuber. 000 Matheatics Subject Classification: Priary 11B68; Secondary 11A07, 11S Introduction Congruences for Bernoulli nubers have been a very intriguing objective of research since the tie of L. Euler, and they recently got revived in connection with p-adic interpolation of L-functions. Congruences for Euler nubers, being cognates of Bernoulli nubers, have also received uch attention fro the sae point of view of p-adic interpolation. In [S4] the author deterined Euler nubers odulo powers of two, while Euler nubers odulo any odd integer are essentially trivial. As a natural further step, we are led to consider congruences aong Bernoulli and Euler polynoials, the latter of which will be our ain concern in this paper. We prove the integrity of coefficients of f k x; a, defined by 1.7, which are related to the suands in the ultiplica- Supported by the National Science Fund for Distinguished Young Scholars No and the Key Progra of NSF No in China. 05 Nuber Theory: Tradition and Modernization, pp W. Zhang and Y. Tanigawa, eds. c 006 Springer Science + Business Media, Inc.

2 06 Z.-W. Sun tion forula 1.6 for Euler polynoials, and establish nuber-theoretic generalizations thereof Theores 1. and.1. Hereafter, the labelled forulae with star indicate those known ones which have their counterparts for Bernoulli or Euler polynoials. Hopefully, these will serve also as a basic table for these polynoials for ore inforation, the reader is referred to [AS], [E], [S1]. In referring to the, we oit the star sybol. Euler nubers E 0, E 1, E,... are defined by E 0 = 1 and n k=0 k n E n k = 0 for n N = {1,, 3, }. k 1.1 It is well known that they are integers and odd-nubered ones E 1, E 3, E 5, are all zero. For each n N 0 = {0, 1,,...}, the Euler polynoial E n x of degree n is given by n n Ek E n x = k k x 1 n k. 1. Note that k=0 E n = n E n 1/. Here are basic properties of Euler polynoials: and E n x + E n x + 1 = x n, E n x + y = k 1 n k=0 n E k xy n k, k x + a 1 a E k = E k x. 1.6 a=0 Fro now on we always assue that q is a fixed integer greater than one, and let Z q denote the ring of q-adic integers see [M]. For α, β Z q, by α β od q we ean that α β = qγ for soe γ Z q. A rational nuber in Z q is usually called a q-integer. In this paper we ai at establishing soe explicit congruences for Euler polynoials odulo a general positive integer. We adopt soe standard notations. For exaple, for a real nuber α, α stands for the greatest integer not exceeding α, and {α} = α α the fractional part of α, a, b the greatest coon divisor of a, b Z, and P x the difference P x + 1 P x of a polynoial P x. For

3 Explicit congruences for Euler polynoials 07 a prie p, n N 0 and a Z, we write p n a if p n a and p n+1 a. For a, b Z\{0}, by a b we ean that both n a and n b for a coon n N 0. For convenience we also use the logical notations and, or, if and only if, and the special notation: { 1 if A holds, [A] = 0 otherwise. By 1.1 and 1. it is easy to get E 0 x = 1, E 1 x = x 1 and E x = x x, and verify that the polynoial f k x = f k x; a, = k+1 x + a E k 1a E k x 1.7 has integral coefficients for k = 0, 1,. This phenoenon is not contingent but universal as asserted by the following general theore. Theore 1.1. For each k N 0 and a, Z with, we have f k x = f k x; a, = k+1 x + a E k 1a E k x Z[x]. This result is rearkable in that 1.6 can be expressed as the vanishing arithetic ean a f k x; a, = 0. a=0 Theore 1.1 follows fro the following ore general result whose proof will be given in Section. Theore 1.. Let k N 0, d, N and d. Let c be a real nuber, and let P x denote the polynoial [ q] + d [ q] k E k 1 c d,q d, q k+1 d k d Ek d, q x c d, q.

4 08 Z.-W. Sun Then P x Z q [x]. Furtherore, if q is odd, then P x if q is even, then q 1 -d 1j+ c+jd q 1 j x + j k od q; 1.8 q 1 c + jd P x 1 j 1 x + j k + [ d j] 1 c+jd q q q [ 4 d + 1] xk od q if k, q + [4 d + 1] xk + [4 q] x k 1 od q if k, q [d ][ ] + [ q]x k 1 od q if k 1. Now we derive various consequences of Theore Corollary 1.1. Let k N 0 and N, and let x be a q-integer. If q is odd, then k E k 1 x,q, q k+1 { } x E k, q q 1-1j+ x+j q 1 j x + j k od q. If q is even, then k+1 E k 1 x,q, q k+1 { } x E k, q q 1 x + j 1 j 1 x + j k + [ j] 1 x+j q q q [ ] + [ q]xk 1 od q if k and, + 0 od q if k = 0, q [4 + 1 k = 1 k q] od q otherwise.

5 Explicit congruences for Euler polynoials 09 Proof. Just apply Theore 1. with c = x and d =, and note that x od q if k > 0 cf. [S3, Lea.1]. q xk q Corollary 1.. Let a Z, k N 0, Z +, q and, q = 1. Then f k x; a, q 1 1 j 1 a + j q x + a + j k + q x [4 + 1] k + [ k 4 q] x k 1 od q. Proof. Clearly and x+a k x k Z[x]. Applying Theore 1. with c = a, d =, and x replaced by x + a, we obtain the desired congruence. Proof of Theore 1.1. Suppose first that > 0. Then by Corollary 1., f k x; a, Z [x]. If p is an odd prie, then E k x/ Z p [x], and also by 1.5 k+1 x + a E k = k l=0 k x + a k l E l 0 k Z p [x]. l Thus every coefficient of the polynoial f k x = f k x; a, is a p-integer for any prie p, which aounts to f k x Z[x]. For the negative odulus case > 0, using E k 1 x = 1 k E k x 1.10 and, we ay express f k x; a, as k+1 x + a + E k + 1a+ E k x. Thus the positive odulus case applies and the proof is coplete. In the spirit of Sun [S3], Theore 1. can also be used to deduce soe general congruences of Kuer s type for Euler polynoials. However, in order not to ake this paper too long, we will not go into details.

6 10 Z.-W. Sun. Proof of Theore 1. We introduce the Bernoulli polynoials B n x n N 0 by the generating power series ze xz e z 1 = n=0 B n x zn. z < π.1 n! Their values B n 0 at x = 0 are rational nubers, called Bernoulli nubers and denoted by B n ; it is well known that B k+1 = 0 for k = 1,, 3,.... Raabe s ultiplication forula counterpart of 1.6 reads 1 x + r n 1 B n = B n x for any N.. r=0 Other properties include B n x = nx n 1.3 and E n x = B n+1 x n+1 B n+1,.4 n + 1 the last one links the Euler and the Bernoulli polynoials. Lea.1. Let k be a positive integer and y be a real nuber. Then { y } ke k 1 x + {y} = 1 y B k x + {y} k B k +..5 Proof. Observe that { y < } 1 { y } {y} = y y = y..6 Hence, if y, the right hand side of.5 is x + {y} B k x + {y} k B k, which coincides with the left hand side of.5 in view of.4. Now, if y, then the right hand side of.5 is x + {y} + 1 B k x + {y} k B k. We ay express B x+{y}+1 k as 1 k B k x + {y} B x+{y} k by.. Then what reains is k B x+{y} k Bk x + {y} which equals the left hand side of.5. This copletes the proof.

7 Explicit congruences for Euler polynoials 11 Lea.. Let k N 0 and Z \ {0}. Then k + 1 k E k x/ Z[x]. Furtherore, k + 1 k E k x k x k x + k x k Proof. First note that by.4, Z[x]. l + 1E l 0 = 1 l+1 B l+1 for any l N 0. Hence, if l is even then l + 1E l 0 = [l = 0]; while for odd l, we see that in the expression l + 1E l 0 = l+1 1 l+1 1 B l p p 1 l+1 l+1 1, p p 1 l+1 1 p the third and fourth ters are integers by the von Staudt Clausen theore cf. [IR, pp ] and Ferat s little theore, respectively, so that l + 1E l 0 is an odd integer. Using 1.5 and writing k + 1 k l as l + 1 k+1 l+1, we find that k + 1 k E k = lies in Z[x] and that k + 1 k E k Now, since k l=1 -l k k l=0 k + 1 l + 1E l 0 l x k l l + 1 E 0 0x k k l=1 -l k + 1 l x k l Z[x]. l + 1 k + 1 l x k l = x + k+1 + x k+1 xk+1 l + 1 = x + k+1 x k+1 k + 1 x + k 1

8 1 Z.-W. Sun k+1 k + 1 j x + k+1 j, j j= the third ter lying in Z[x], we conclude that what we should subtract fro k + 1 k E k is k + 1 E 0 0x k + x + k+1 x k+1 k + 1 x + k 1 1 = x k + x k x + k x k, as asserted, and the proof is coplete. Lea.3 [S3, Theore 4.1]. Let k N 0, d,, n N, d n, qn, and d or q or qn. Put d = d, qn/ and =,. Then, for any real nuber y, the polynoial Lx, y = 1 d k B k+1 k + 1 d ȳ k B k+1 x.7 is in Z q [x] and is congruent to qn/ 1 Rx, y = odulo q. + 1 d q 3 [3 d] n d qn kxk 1 y + j x + j k + q [ k[d n] n qn ] x k 1 + [ n q] x k 1.8 Now we establish a result ore general than Theore 1..9 and.10 below are generalizations of 1.8 and 1.9, respectively. Theore.1. Let k N 0, d,, n N, d n, qn, and d or q or qn. Put d = d, qn/ and =,. Then for any real nuber y we have [ qn] + d[ qn] P k x, y := k E k d k ȳ 1 y/ E k Z q [x].

9 Explicit congruences for Euler polynoials 13 Moreover, if qn/ is odd then P k x, y qn/ 1 -d 1j+ y+j if qn/ is even then P k x, y where qn/ 1 1 j x + j k + q [d n d 0, 1 od 4]kx k 1 od q;.9 1 j 1 x + j k y + j + [ d j] 1 y+j + R k x od q,.10 R k x q [d n d 0, 1 od 4 ] x k if k, q = [d n d 0, 1 od 4] x k + [4 q] x k 1 if k, q [d n][d 0, 1 od 4] + [ n qn/]x k 1 if k 1. Proof. We observe that the k + 1-degree ters in.7 cancel each other in view of d/ = d/. Hence Lx, y is of degree at ost k. Writing Lx, y = k a i yx i, i=0 we see that k L, y Z q[x], and siilarly k R, y Z q[x], and a fortiori that Lx, y k+1 L, y Z q [x] and Lx, y k+1 L, y Rx, y k+1 R, y od q..11

10 14 Z.-W. Sun By.7 we can express the left hand side of.11 in such a way that we ay apply Lea.1 to deduce that Lx, y k+1 L, y = dk E k 1 d y/ k ȳ E k x..1 On the other hand, the right hand side of.11 is congruent to Σ q 3 [3 d] n d qn k x k 1 k+1 x k 1 + rx odulo q, where rx = q [ k[d n] n qn ] x k 1 + [ n q] x k 1 and Hence Σ = qn/ 1 y + j x + j k x + j k + 1 d y + j + 1 d Rx, y k+1 R, y Σ + rx od q..15 By the counterpart of.6, the second ter on the right hand side of.14 becoes qn/ 1 x + j k y + j [ d + ] y + j in which we shall divide the su into two parts via idpoint. Then Σ = qn/ 1 qn/ 1 i=0 i x + j k y + j x + i k y + i + 1 d [ d + ] y + i

11 Explicit congruences for Euler polynoials 15 qn/ 1 i=0 i+qn/ x + i + qn k y + i + qn/ [ ] y + i d + + d. Whence, writing for j = 0, 1,..., we obtain c j y = y + j.16 qn/ 1 Σ x + j k c j y[ j] + d 1 qn/ 1 + qn/ 1 qn/ 1 x + j k [ j c j y] x + j k c j y [ qn ] + j qn/ 1 [ x + j k qn ] + j c jy + d Recobination of ters yields qn/ 1 Σ where x + j k c j y x + j k [ j] 1 od q. [ j] [ qn ] j + Σ 1 + Σ od q, Σ 1 = d 1 qn/ 1 1 j x + j k, qn/ 1 Σ = x + j [ k j c j y] [ qn j c jy + d]..17

12 16 Z.-W. Sun Thus, in view of the equality [ j] [ qn j] = [ qn ] 1j 1, we deduce that Σ Writing [ qn ] qn/ 1 1 j 1 x + j k c j y + Σ 1 + Σ od q. Σ 1 = d 1 qn/ 1 k 1 j x k + j and applying 1.4 and 1.5 successively, we obtain Σ 1 = d 1 = d 1 = d 1 k k k qn/ 1 E k E k.18 1 j E k + j 1 j+1 E k + j qn/ E k + qn 1 qn/ k l=0 k qn l E k l ; l whence separating the ter with l = 0, Σ 1 = d 1 [ qn ] k E k 1 k d 1 qn/ k l + 1 k l E k l nl q l l 1 l. 0<l k.19 Now, by [S3, Lea.1], q l 1 /l 0 od q for l >, so that if 0 < l k, then d 1 nl q l d n l = l d q ql 1 l q [d n d 0, 1 od 4] [l = 1] + [l = n q] od q,.0 and in particular, only two ters with l = 1, appear on the right hand side of.19 odulo q. Hence the su on the right hand side of.19

13 Explicit congruences for Euler polynoials 17 is congruent to d 1 + d 1 qn [k > 0]kk 1 E k 1 n q 4 [k > 1]kk 1k E k odulo q. Thus, by Lea., Σ 1 is congruent to d 1 [ qn ] k E k 1 qn/ d 1 nq [k > 0] x k 1 + x k 1 x + k 1 x k 1 + n q [k > 1]k x k + x k x + k x k 4 odulo q. By.0 with l = 1,, the second ter of the above expression is congruent to 1 qn/ rx odulo q, where Thus rx = q [d n d 0, 1 od 4 k] x + k x k + q [d n d 0, 1 od 4 k] xx + k 1 x + x k 1 + q [ q kn 4 d + 1]x + k 1 x k 1..1 Σ 1 d 1 [ qn ]k E k + rx od q.. Now, fro.11.13,.18 and., it follows that d k E k 1 d y/ k ȳ E k x Σ + rx [ qn ] qn/ 1 1 j 1 x + j k c j y + d 1 [ qn ] k E k + rx + rx qn/ 1 + x + j k [ j c j y]

14 18 Z.-W. Sun [ qn ] j c jy + d od q. With the help of the binoial theore, we can easily verify that rx + rx R k x od q. If qn/ is odd, then either q or, whence R k x q [d n d 0, 1 od 4]kx k 1 od q. and the desired results follow. Proof of Theore 1.. Just apply Theore.1 with n = and y = c/d. References [AS] M. Abraowitz and I. A. Stegun, Handbook of Matheatical Functions, Dover, New York, 197. [E] [IR] [M] [S1] [S] [S3] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricoi, Higher Transcendental Functions I, McGraw-Hill, New York, Toronto and London, K. Ireland and M. Rosen, A Classical Introduction to Modern Nuber Theory Graduate texts in atheatics; 84, nd ed., Springer, New York, K. Mahler, Introduction to p-adic Nubers and their Functions, Cabridge Univ. Press, Cabridge, Z. W. Sun, Introduction to Bernoulli and Euler polynoials, a talk given at Taiwan, 00, Z. W. Sun, Cobinatorial identities in dual sequences, European J. Cobin. 4003, Z. W. Sun, General congruences for Bernoulli polynoials, Discrete Math. 6003, [S4] Z. W. Sun, On Euler nubers odulo powers of two, J. Nuber Theory, 005, in press.