(6), (7) and (8) we have easily, if the C's are cancellable elements of S,

Transcription

1 VIOL. 23, 1937 MA THEMA TICS: H. S. VANDIVER 555 where the a's belog to S'. The R is said to be a repetitive set i S, with respect to S', ad with multiplier M. If S cotais a idetity E, the if we set a, = a2 =... = ak = E we obtai a set which we call simply a repetitive set i S with multiplier M. If S cotais o idetity we give a special defiitio to cover the case where o a's appear i (6). The multipliers Ci of the co-sets i S (assumed commutative) metioed i Theorem 2 may be show to form a repetitive set with respect to G ad with ay elemet of S as multiplier. This idea is the abstract form of may relatios i umber theory. With the otatio ad defiitios of (6), (7) ad (8) we have easily, if the C's are cacellable elemets of S, k Mk IIa8. (9) s-i I particular this result may be applied to the theory of algebraic fields to obtai kow theorems cocerig quadratic ad higher residues of ideal moduli. If S is the semi-group of ideal classes moduli p, a ratioal prime, cosistig of the elemets Ci, C2,..., 1 ad G is the icluded group cosistig of C1 ad Cp_., the (9) gives Gauss' lemma i the theory of quadratic residues. 1 These PROCEEDINGS, 20, 579 (1934); Bull. Am. Math. Soc., 40, 916 (1934). ON GENERALIZA TIONS OF THE NUMBERS OF BERNOULLI AND EULER By H. S. VAIMIVER DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TEXAS Commuicated September 8, 1937 I the preset paper we shall develop a umber of geeralizatios of the Beroulli ad Euler umbers ad results cocerig the same. Full proofs of the results will appear elsewhere. Our poit of departure is the examiatio of the expressio (mb + k) = b(m, k). (1) Where m ad k are itegers, m s 0 ad we expad the left-had member by the biomial theorem ad substitute bi for bl i the result, where b, defied by the recursio formula (b+1)a=b.; a > 1; bo= 1. (2) is

2 556 MA THEMA TICS: H. S. VANDIVER PRtOC. N. A. S. The umbers so obtaied from (1) have arithmetical properties aalogous to those kow for the ordiary Beroulli umbers (2) ad which are little more complicated. For example, we have the followig geeralizatio of the vo-staudt-clause theorem: THEOREM 1. We have for eve, m ad k itegers with m o0, b(m, k) b = A- ~~~s13(3) i=1 pi, where the p's are the distict primes which are prime to m ad such that 0 (mod pi - 1); A beig some iteger. For odd, b%(m, k) is a iteger, except for = 1 with m odd. Note the peculiarity that this result is idepedet of k ad that it reduces to the ordiary vo-staudt-clause theorem for m = 1, k = 0. It may be proved by takig the kow relatio (mb + k +mp)+l- (mb + k)+' P1 ( ( + 1)m 5=0 for p prime, with 00 = 1, expadig the left-had member i ascedig powers of p, ad the extedig ad modifyig the argumet employed by vo-staudt ad Frobeius' for provig the vo-staudt-clause theorem. A advatage from some stadpoits i usig (1) istead of (2) is the fact that b,(m, k) is ot always zero for odd. We shall call b,(m, k); m $ 0, a geeralized Beroulli umber of the first order: ad a umber of the form, for r 2 1, (mrb(r) + Mr-lb("-') mlb' + m0) = b.(mr, Mr-I)... I, mo),(5) where this expressio is expaded i full by the multiomial theorem ad ba substituted for b( ); i = 1, 2,... r ad where the m's are itegers, mi $ 0; i = 1, 2,... r; a Beroulli umber of the r-th order. This is a extesio of the defiitio of Lucas of Ultra-Beroulli umbers. It is ow atural to iquire if a geeralizatio of the vo-staudt-clause theorem exists for umbers of the type (5). For the case mo = 0, r = 2 we may employ formula (3) of aother paper,2 treat it as i the derivatio of (6) of that paper except that we differetiate a times, a arbitrary, set v = 0, the substitute x = xevk but ot as before y = ye0 so that we get rid of (Xj _ yk) as a factor of the left-had member whe we differetiate oce more. The set v = 0, take the result modulo p3 ad set x = y = 1 i the resultig cogruece, ad divide through by p2, usig the relatio, (which may be derived from (4)) p-1 E (ms + k) - p(mb + k)(mod p2); p >, (6) s-0

3 VOL. 23, 1937 MA THEMA TICS: H. S. VANDIVER 557 where 00 = 1. Our result, which is a equality as well as a cogruece sice the terms are idepedet of p, expresses (kb' + jb')a (7) as a liear combiatio of expressios of the form (kb + 1)" ad (jb + j); = a, a - 1. I view of Theorem 1 we are able to determie directly the fractioal part of (7) ad this is aalogous to the result of Theorem 1. I cojecture that a similar result holds for (5), but the questio is much more complicated for higher values of r. The method just employed is quite geeral i coectio with derivig formulas cocerig Beroulli umbers. The simplicity of the formula (6), especially for k = 0, m = 1, ad the fact that (mb + k) is idepedet of p, eables us to obtai may equalities ivolvig the geeralized Beroulli umbers merely by takig simple idetities cotaiig expressios such as xvr - 1 x-1 where x is a idetermiate, settig x = eavx ad differetiatig with respect to v, fially reducig the result to a cogruece modulo p, which usually gives a equality because the terms therei are idepedet of p. As aother example, from formula (11) of my paper already referred to we may obtai a formula closely related to (12) of that paper, which gives, after substitutig x = y = 1, cogrueces from which we may evetually derive equatios ivolvig b(2, 1) ad more geerally b(m, k) as idicated by the remarks below (12). Now the geeralizatio of the Beroulli umbers as give by (1) suggests aalogous extesios for certai types of related umbers. We ote that (mb + k) = m(b + - m"(b + s)(mod d) where d is a iteger prime to m ad k, ad s is the least positive solutio of sm k(mod d2). Usig (4) for m = 1, k = 0 we obtai where (mb + k) - mb = m"sl ( md2)(mod d) IS1(k From this we have d2= l (sm (mb + k) - (mb + 1)"f= msx1(k-di - mis-.1 (1d2)

4 558 MA THEMA TICS: H. S. VANDIVER modulo d. Let d = di, the if we set we have e -1(m, k, 1) - (mb + k) - (mb + l) (8) e _i(m,.k, 1) ms_1 (kd2)- S-(- d ) (9) modulo dl, where m, k ad I are prime to di. We shall call e(m, k, 1) if m, k ad I are itegers with m # 0, k $ 1; a geeralized Euler umber but shall ot at this time refer to its order, as differet geeralizatios3 of the Euler umbers have bee proposed. The relatio of (8) to the ordiary Euler umbers is give by e(4, 3, 1) = 2e, where (-e2)= E. Our e's i (8) are ot always itegers but the prime factors of the deomiators always divide m. However, it appears that most of the kow arithmetical properties of the E's have aalogos ivolvig the e's, ad a umber of them may be obtaied from (9). For example, the theorem I gave i aother paper4 may be applied to (9) to obtai results such as that give at the top of page 422 of that paper with e(m, k) takig the place of ba/ as there give. A aalogous extesio of the Geocchi umbers is give by G(m, k) = -((mb + k)- b)m, (10) which reduces to the ordiary Geocchi umbers for k = 0, m = 2. There is aother type of umber, however, which has appeared repeatedly i ivestigatios cocerig Beroulli umbers, ad which deserves special cosideratio sice its properties differ i may respects from those of the Beroulli umber itself. It is (M 1)b (11) - of which a atural geeralizatio appears to be For m = 2, (mb + k)8 - b = t(m, PROC. N. A. S. k). (12) (11) is closely related to the so-called taget coefficiet 2(2-1) b, (13)

5 VOL. 23, 1937 MATHEMATICS: H. S. VANDIVER 559 that is, the coefficiet of x-i/(f - 1)! i the power series expasio of ta x. As is also kow, the umber (11) appears explicitly as the coefficiet of x-l/(l - 1)! i the expasio of d erz 1 - log dx el 1 as a power series i x. Although (12) is ot always a iteger we prefer its use to that of m (mb + k)-b which is a iteger divisible, i geeral, by a power of m close to m'. Cocerig (12) we have foud a reduced form for (t'(m1, ki) + t(2)(m2, k2) t(r)(mr, kr))?, where this meas that the expressio is to be expaded by the multiomial theorem ad t4 (mi, ki) substituted for (t0 )(mi, ki))8; i = 1, 2,..., r; which is much simpler tha those foud for (5). 1 Sitsugsberichte Berli Akad., (1910). 2 A. Math., 29, (1928). 3 Sylvester, Compt. Red., 52, 163; D. H. Lehmer, A. Math., 36, (1935). 4 Bull. Amer. Math. Soc., 43, (1937).