g-BERNOULLI AND ETJLERIAN NUMBERS · ç-BERNOULLI AND EULERIAN NUMBERS 333 and the coefficients are the same as those occurring in (1.4). For g=l, Ama and Hm(x) reduce to well known

g-BERNOULLI AND ETJLERIAN NUMBERSBY

L. CARLITZ

1. Introduction. In a previous paper [2] the writer defined a set of

rational functions rjm of the indeterminate q by means of

(1.1) (qr,+ l)m = r,m(m> I), m - 1, m = 0,

and a set of polynomials

Vm(x) = Vm(x, q)

in g* by

(1.2) ym(x) = ([*] 4- qxv)m, Vm(0) = tj«,,

where [x] = (qx—l)/(q—l); also

(1.3) qxßm(x) = 7jm(x) 4- (q - l)vm+i(x), ßm(0) = ßm-

For 2=1, |3m reduces to the Bernoulli number Bm, ßm(x) reduces to the

Bernoulli polynomial Bm(x) ; nm however does not remain finite for m> 1.

In the present paper we first define polynomials Ama=Ame(q) by means of

r n ™ Fx + s - 11(1.4) [*]"=E^~ (m^l),

«_i L m J

where

l-x-l (g*- l)(g-*- !)•■• (g»-«H-i- 1)

lm]~ (q - l)(q2 - I) ■ ■ ■ (qm - I)

Alternatively if we define the rational function Hm = Hm(x, q) by means of

H0=l,H1 = l/(x-q),

(1.5) (qH+ 1)™ = xHm (m>l),

then we have

m

(1.6) Hm(x,q)=Am(x,q)/YL(x-q'),»=i

where

m

(1.7) Am(x, q) = Xi™!1-1 (m ^ 1),»-i

Presented to the Society, April 25, 1953; received by the editors February 5, 1953.

332

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ç-BERNOULLI AND EULERIAN NUMBERS 333

and the coefficients are the same as those occurring in (1.4). For g=l, Ama

and Hm(x) reduce to well known functions; some of the properties of these

quantities are stated in §2 below. As Frobenius [3] showed, many of the prop-

erties of the Bernoulli and related numbers can be derived from properties

of Hm. We shall show that much the same is true in the case of the q analogues.

In [2] a theorem somewhat analogous to the Staudt-Clausen theorem

was obtained for ßm (with q an indeterminate). We now show that if p is an

odd prime and we put q = a, where the rational number a is integral (mod p),

then if a = l (mod p),

(1.8) pßm = - 1 (mod p)

provided p—l\m; otherwise ßm is integral (mod p). If a jé 1 (mod p) the situa-

tion is more complicated. In particular, if a is a primitive root (mod p2), then

ßm is integral (mod p) for p — l\m, while for p — \\m we have

1pßm = - — (mod p), (k = (a"'1 - l)/p).

k

In general the denominator of ßm may be divisible by arbitrarily high

powers of p (see Theorem 4 below).

Finally we derive some congruences of Rummer's type for Hm, etc. For

example if q = a is integral (mod p) while x is an indeterminate, then

Hm(Hw - l)r = 0 (mod pm, pre) (p'-Kp - 1) | w),

where after expansion of the left member IP is replaced by Hk. We also ob-

tain simple congruences for the numbers Ams defined in (1.4). The correspond-

ing results for r\m and ßm are more complicated.

2. Eulerian numbers. To facilitate comparison we quote the following

formulas from the papers of Frobenius [3] and Worpitzky [5].

m /% + s _ 1\(2.1) xm = JÂmA ) (m ^ 1),

«=i \ m /

(2.2) Am+i,s = (m + 2 — s)Am,,-! + sAm„

' (m + l\(2.3) Ame=Z(-l)r{ r )(s-r)»,

1 ^ / m V1(2.4) Fm= -_2(-l)—»( J Amr.

m + 1 r_i V - 1/

In the next place if we put

m

(2.5) Am = Am(x) = Â^x'-1,Ê-i


334 L. CARLITZ [March

and let

Hm = Hm(x) = (x - l)-mFm(x),

then Hm satisfies

(2.6) (H +l)m = xHm (m^l), H0 = 1.

The connection between Hm and the Bernoulli numbers is furnished by

k-l Jfx kl-mç

(2.7) Z rrBm(-) =-mHm_i(ï),r=0 \ k / 1 - f

where f*—1, f^l. An immediate consequence of (2.7) is

(2.8) k~Bm(—\ - Bm= -«E -#«-i(f),\ a / o* r — i

where f runs through the Ath roots of unity distinct from 1.

We also mention

m

(2.9) Hm=YJ(x- 1)-'A*0».r=0

3. Some preliminaries. We shall use the notation of [2]; see in particular

§2 of that paper. In addition the following remarks will be useful. Let/(w) be

a polynomial in qu of degree^m. Then the difference equation

(3.1) g(u + 1) - cg(u) = f(u) (c^qr)

has a unique polynomial solution g(u), as can easily be proved by comparison

of coefficients. To put the solution in more useful form we rewrite (3.1) as

(E — c)g(u) =/(«) and recall that

A = F - 1, A2 = (F - 1)(F - q), A* = (F - l)(E - q)(E - q2),

In the identity

1 1 2 — 2i 1 (z — Zi)(z — Zt) 1+-+ -7-r,-:-+ ■ ■ ■

t — Z I — Z\ t — Zt I — z2 (/ — Zi)(/ — z2) t — z3

(z — Zl) • • • (z - z„) 1

(/ — Zi) •■■(/ — z„) i — z

take t = c, z = E, zs = q3~1, so that we get

1 1 A A2

+ Z-7-,-T + 7-IT;-r,-- +C-l ÍC- l)ic -q) ic- l)ic - q)ic - q2)

A" 1+

(c - 1) • • • (c - g"-1) c - E


1954] Î-BERNOULLI AND EULERIAN NUMBERS 335

Hence if we take n>m, we obtain the following formula for g(u):

" As/(«)

(3.2) giu) = Z-AJ—/- •s=o (c - l)(c - q) ■ ■ ■ (c - q*)

4. The number Ama. We suppose A^ defined by means of (1.4). Using

the identity

(?m+l _ l)(?z - 1) = iqm+ls _ 1)(ç*+» - 1) + qm+ls(qs _ 1)(çz+8-m-l _ ^

and multiplying both members of (1.4) by [x], we get

[*]-«- T,A„\[m + i-s]\ l\] + im+1-°U]\x+l~ HI1 Lm 4- 1J L m + 1 J)

= H\X + SI~ Hik + 2 - s]Am,B^ + q^->[s]Am,},,L«+1 J

which implies the recursion

(4.1) Am+i,a = [m+2 - sp4m,s_i 4- qm+1-°[s]Ama.

For g=l it is evident that (4.1) reduces to (2.2). As an immediate conse-

quence of (4.1) we infer that Am, is a polynomial in q with positive integral

coefficients.

It is easy to show that Am, is divisible by g(>»-«)("»-<+»/ä. Indeed if we put

(4.2) Ama = g(—.h-^-d/i^

then (4.1) becomes

(4.3) ^*+i,s = [m+2 - s]A*m.s-i + [s]aZs,

which proves the stated property. Moreover it follows easily from (4.3) that

(4.4) deg Ama = (s — l)(m - s).

Indeed assuming the truth of (4.4), we get

deg ([« -f 2 - sp4*.._i) = (m+l- s) + (s- 2)(m + 1 - s)

= is- l)(m +1- s),

■■ deg i[s]A*ms) - (j - 1) 4- (s - 1)(« - s) = is- l)(m + 1 - s),

so that

deg^4m+i,t = (s - l)(m + I — s),

which proves (4.4).


336 L. CARLITZ

The symmetry properties

[March

(4.5)

and

(4.6)

A* - A*

A*Uq) = g<-l><—"iOg-1)

will be proved below.

Comparing coefficients of qmx on both sides of (1.4) we get

(4.7) Y,Ama = [«]! = [tn][m- l] • • • [l].

More generally if we expand both sides in powers of qx and equate coefficients

we get

(4.8) ¿¿mrg" = (W jgm'-,(«-1)/2[»»]! (Oáíá**).

The following table of .4^, l|s|má5, is easily computed by means of

(4.3).

12(5+1)

3g2+5g+3

4g3+9g2+9g+4

13g2+5g-|-3

6g4 + 16g3-f22g2+16g+6

14g3+9g2+9g+4

5. A formula for A^- It is easy to show that if fix) is a polynomial in

qx of degree ^m,

(5.1)

then

(5.2)

/(*),_o L w J

(*» ̂ 1).

cm0 = (_i)«g«(-n)/«/xo),

' r«4 n(5.3) Cm,m_r = £ (-1)' \f(r + 1 - j)g'C-»/*.

.-o L s J

Since

^ Ym + 1"1^(_l)T(.-i)/2 /(* + „+!_,) =o,—o L 5 J

we have in particular


1954] g-BERNOULLI AND EULERIAN NUMBERS 337

Ï+1 Vm + n^(_1)T(S-i)/2 /(r+i-*)-o,a«o L î J

and (5.3) yields

(5.4) Cmr = ¿ (-1)—•«<•—>.<-•*->'» P " l~\sis - r),s-o L s A

which includes (5.2) also. Thus the coefficients in (5.1) are determined.

If we take/(x) = [x]m, then Cmr = Amr and we get after a little manipula-

tion

* A Ym + 11 r .(5.5) ¿C = g'(r-l>/t£ (_l)V(«-D/2 [r _ s]m.

s=0 L 5 J

for g=l, (5.5) reduces to (2.3).

Replacing q by q~l, (5.5) becomes

* Ä I"»» 4- l~l, ,ç(<-D('»-'-U„r(g-l) = ?(r-l)m+r(r-l)/2^ ( _ l).?.(.-l>/2+« [r _ 5]m>

«=0 L S J

where

j(j - 1) - (m + l)(m 4-2) (m + I - s)(m + 2 - s)e =-h ■-—

2 2

s(í 4- 1)4-(r — s + l)m = — (r — l)m.

Hence

Atr(q) = qînm-r)Alr(q-i),

which is identical with (4.6).

In the next place we observe that exactly as in the proof of (6.2) of [2] we

have

Z ( ) [x]i+Ym~i)x-r- (qim+1)x - 1)

i-o \ i / i + l m + 1

™ rx 4- s - n= 2>-.g-H ^ , •«-i L m + 1 J

Divide both sides of this identity by [x] and then put x = 0. We find that

1 ■ • r m -y1fi»m-x——TE(-i)m-s-1?-(",-s)(m-s+1>/2 ! Ams.

[m+ 1J ,_i ]_s - 1J

Using (4.2) and (4.5) this becomes



l » r m l"1

ß~ = r , ,, Z(-i)--1 . 4[w 4- l J «=i U — 1J

l ™ rw-]-1 *= T—7TE(-1)S ¿L,

[m + l\ a=i L s J

*

the first of which may be compared with (2.4).

6. The polynomial Am(x). The polynomial Am(x) = Am(x, q) is defined in

(1.7) formel; weput^40(x) = l. Put

(6.1) (t>m(x) = II (x - qs)s=0

and apply the Lagrange interpolation formula at the points x = q\

5 = 0, 1, • • • , m. Since

s—1 m

*'(?•) = ![(?•- îo n (?• - «o¿=.0 J=«+l

= (_l)"-«?»«-«(.-i)/2(c _ 1)»[j]![»í - s]!,

we get using (4.8)

<Pm(x)

(q - 1)m Zó " \ s / x — g*

As a first application of (6.2) consider

x-m-1g-m('n+1)/20m(x) » /m\ xgs

tf>„,(x) ™ /w\ 1(6.2) ¿m(*) = / 7 Z(-i)m->( )—

(g - l)m a=o \s / x -

m_i m(m+l)/2^ / ^ „ - v

ii^r-sr1)--b—^-Ê (-D—( )■g-»(g-l)» ,_o \5/

»g-™(g - l)m i=0 \ 5 / X - g«

which gives

(6.3) x"-1g",(",_1,/24m(x-1, g-1) = AJix, q) im ^ 1).

Substituting from (1.7) in (6.3) we get

m m

qm(.m-l)liYáAm,(q-1)xm-' = Z^m.(î)*'_1.

«=1 «=1

which implies

(6.4) !"'"11"^(r1) = Am.^+iiq).

Hence by (4.2) and (4.6), (6.4) becomes

çm(m-l)/2-(»,-S)(m-S+l)/2-(,-l)(m-S)J4*s(ç-i;) = q*(,-l)l2A*m,m_e+iiq),

which is the same as (4.5).

7. The functions Hm(x) and Hm(u, x). Using (6.2) and (1.6) we get



(7.1) (g - i)-#»(*) = (*-i)E(-i)"*-( )-•=o \ s / x — qs

We remark that (7.1) implies

00

(7.2) HJix) = (x - 1)Z x-'-^[r}mr=0

for |x| >|g*|, O^s^m. It is also evident that

m /m \ r / r \ 1

(l + gff)-= (x- 1)Z( )?'(3- l)"rZ(- D"( )-r-o \ r / s=o \ s / x — q3

™/w\ 1 JE, /m- s\= (*-i)E( )-Z(-i)-8( )g'(g- i)-rs_o \ 5 / x - g* r=s \r — s /

™/m\ 1 g* / -1 \"-'

" (*~ ̂ eíVí/x-g« (g-i)Ag-1/JE, fm\ qs

-(«-«—(*-l) E (-D — 1 )—!L<-0 \ 5 / X — g-

which implies

(7.3) (1 4-gF)1» = xF™ («w > 1).

We have therefore proved (1.5). Alternatively taking (7.3) as definition of

Hm one can work back to the earlier formulas obtained for Ama above.

For some purposes it is convenient to define Hmiu; x)=Hmiu; x, q), a

polynomial in qu. We put

JE, /m\ qeu(7.4) (g- l)"Fm(M;x) = (x- l)Z(-Dm-8( )—-.

.-o \ 5 / x — q3

so that Hmi0; x) —Hmix). It follows at once from (7.4) that

(7.5) ffm(l - u; x-\ g-1) = (-g)-Fm(M; x, g)

and that

(7.6) xHmiu; x) - Fm(w 4- 1; x) = (x - l)[u]m.

We have also

JE, /m\Z( \q'Hriu;x) = Hmiu + 1; x),

which becomes, using (7.6),

(7.7) (1 4- qHiu; *))- = xFm(«; x) - (x - !)[«]•



For m = 0, (7.7) reduces to (7.3).

Clearly (7.6) implies

fc-i

(7.8) Z **"'[« + *]" = xkHmiu; x) - Hmiu + k; x),t-0

which includes (7.2) as a special case.

Since Hmiu; x) is a polynomial in qu of degree m, the remarks in §3 apply

to the difference equation (7.6). In particular, application of (3.2) leads to

(7.9)A A'[u\m ( • \

Hmiu; x) = Z -ry— (*.(*) = II (* - ?r) ),8=o yp,(x) \ r_i /

provided X7iqT, r = 0, l,y • • , »t. To simplify the right member of (7.9), we

used (2.6) and (3.1) of [2]; thus

A'[u]m = Z gr(r-1)/2<wH.Mr-ig'("_r+<'>

and (7.9) becomes after a little manipulation

m r Q$(u~ri-s)

(7.10) Hm(u; x) = Z <T(r-1)/2a»,.rZ , , N ['].[«]«•r=0 a~0 Ya(x)

If we let Gr(u) denote the inner sum it is clear from (3.2) that

xGr(u) — GT(U + 1) = [u\r.

In (7.10) put w = 0, then

flm.r[r]!(7.11) Fm(x) = Z ?r(r_1)/2

r=0 *r(*)

which for q= 1 reduces to (2.9).

Using (7.1) and (7.4) it is easy to verify the formula

m /m \(7.12) Hm(u; x) = ZÍ )?rttFr[«]—' = (guF + [«])«.

Next using (7.12), (7.11), and the explicit formula [2, (6.2)] for am,, we get

5 1 r r r 1(7.13) Fm(w;x) = Z -7VT E (-l)T(s_1>/2 [« + f - * 1",

r-0 IM«) «-0 L 5 J

which is useful later.

8. Connection with i)miu). Using the formula [2, (4.7)]

» /m\ s

(g - i) «^.(ii) = Z ( -1) --• ( ) -ft <r,



we find that

k-l

(gi-i)"-1Zr^(« + T;gM=Z(-l)m-s( )„ . , .r=o \ * / s=o \ 5 / rv -1

m-1 fm _ J\ ¡-„ktu+ku-l

Ît-l)'-1"' J—-s=o \ 5 / q' - f g 1

where

r* -1, fî.

Comparing with (7.4) we have therefore

(8.1) *]—1Er,u-(« + —» ?*) = -rJ-#m-i(¿«; fg-1),r=o \ * / i - r

and in particular for u = 0

(8.2) i-'Eri-T- ?*) = —^-F.-iifg"1),r-0 V k f 1 - f

which may be compared with (2.7).

Next using the multiplication formula (see [2, (4.12)]; note that a term

is missing in that formula)

*-1 / r \ k — [k](8.3) [ä]-1EM« + —> ?*) = n-(*«, ?) + (-!)" 7—7T-

_o \ . * / (g - l)m

together with (8.1) we get

¿[¿]m_V.(w4- y> gM - 7?m(Ä«, g) - (-1)'(g- l)"1

= —Z -Hm.i(ku;irl),q t*i i - f

and in particular for m = 0,

*- [k¡k[k]~-h,m(i-, A-vm-i-iy

(q- 1)»(8.5) W - - ■ -

= -^7—F^-iG-g-1).

By means of (1.3) it is easy to write down formulas like (8.1), • ■ • , (8.5) in-

volving ßm.

9. Multiplication formulas. For the polynomial Hm(u ; x) we have, using

(7.4),



(g* - 1)- Z rrqrtSm(u + ■?-; fg-*', g*)r-0 \ k /

m /~- \ Qkau k—1

-<rr-'-i>£<->)"(j)ÎF^£r'î

™ (m\Hf-î^K-i)™-' )-

.=0 \ Í / f

kau k—1

fr*' - qks Í* "

Qkau J _ >—knk(a+t)

gk($+l) l _ !—l„s+i

Consequently if f*-1 = l, f^l, we get

í¿felmY! c-ror'Hmlu + — : ira-*". ok\ = -

(9.1)[¿]mZrrgr,Fm(W— ; fr*'. <?*) =-9-—Hm(ku; tq-', g),

r=0 \ * / f - q'

analogous to (8.3).

In the special case x= — g-1, the polynomial em(u) of [2, §8] satisfies

(9.2) e„(«) = Hm(u; -q~\ q);

in this case (9.1) becomes (f = — 1, t = l)

(9.3) [i]-¿ (-5)^^ + ^, çA = l±iim(^>ç)r-0 \ e ./ g 4-1

for & odd; note that (8.6) of [2] requires a slight correction.

10. Staudt-Clausen theorems for ßm. In [2] a theorem analogous to the

Staudt-Clausen theorem was proved for ßm with q indeterminate. Now on the

other hand we replace g by a rational number a which is assumed to be inte-

gral modulo a fixed prime p. We shall use the representation [2, (6.2)]

m

(lo.i) 0- = E(-i)'««..k]i/[i+i].8=0

where

0-«(«-n/2 < r 5 "1(10.2) <*„,, = --—-Z(-l)'g'(r-1>/2 [*-»-]-;

si r_o LrJ

the quantity am_, is a polynomial in g and has occurred in (7.10) and (7.11)

above.

Suppose first that a = l (mod p). Then from (10.1) or [2, §7] it is clear

that the 5th term in the right member of (10.1) is of the form u,

= N„(a)/F,+i(a), where F,+i(x) is the cyclotomic polynomial and N,(x) is a

polynomial with integral coefficients. If we recall that F*(l) =p when k=pe,

eèl, but Ffc(l) = l otherwise, it is clear that u„ is integral (mod p) except

possibly when s + l=pe; the same holds also for Fk(a). Now let 54-1 —p'.

Then by a simple computation it is seen that [s]\ is divisible by exactly p/,

where



f=(p<- l)/(p -l)-e,

while the denominator is divisible by exactly pe. Since (pe—l)/(p—l)^2e

for e^2, p^3, it follows that ua is integral in this case. If e=l, p~^3, we have

first p(a—l)/(ap — l) = l (mod p). As for the numerator of mp_i, it follows

readily from (10.2) and

[7] "CD <-")p-1 / P — l\

IP- l]!am,P_i^Z(-l)r( )r»r=0 \ r /

ÇÎ 7-1 (mod/-) (/>-l|m),

r»o I 0 (mod #) (¿> - l\ m).

that

We have therefore proved

Theorem 1. Letp^3, q = a = l (mod p). Then

7-1 (mod^) (P- l|«),(10.3) */3m = <

I 0 (mod #) ip - l\ m).

For p = 2, the preceding argument shows that all terms in (10.1) are

integral (mod 2) except perhaps Mi and u3. Now

ffm.i 1Ml =

[2] a + 1

while

E(-l)vH)"PJ[3-r]'[4] (a + l)(a2 + 1)

Let 2"| (a+1), 2«+1>(a4-l); then (a24-a-f-l)2=T (mod 2«+x) and

¿(_l)rar(r-l)/2r ][3_f].

r-0 L r J

= (a2 4- a + l)"> - (a2 + a + l)(a + l)m + a(a2 + a + I)

0 (mod 2e+1) im even),

U 4- 1 (mod 2e+1) (m odd).t

Consequently u3 is integral (mod 2) for m even while for m odd 2w3= 1 (mod 2).

This yields the following supplement to Theorem 1.

Theorem 2. Let p = 2, q = a = l (mod 2); also let 2e|(a-fl), 2e+l\(a+l).



Then if e = l we have 2/3m = l (mod 2) for m even ¡± 2, 2/3i=-l (mod 2), while

ßm is integral (mod 2) for m odd ^3. If e>l then

(10.4) 2«p\» = 1 (mod 2)

for all m ̂ 1.

In particular it is evident from (10.4) that the denominator of ßm may be

divisible by arbitrarily high powers of 2.

In the next place we suppose g = a^l (mod p), p>2. It is now con-

venient to use [2, (5.3)]

JL /m \ s + 1(10.5) (g- l)-ßm = Z(-l)"-s( )t—"Tí-

*=0 \ 5 / [S + 1 J

We shall assume first that a is a primitive root (mod p2). Clearly in the right

member of (10.5) we need consider only those terms in which p— l|s-|-l.

Put a""1 = 1 +kp, p\k. Then

(Í)a(j.-l)r = 1 _|_ fkp _|_ { ^ ) tfpt _|_

a(p-l)r _ ! !

= * -f — (r - l)¿2/> -f • • • = * (mod p).rp 2

Thus (10.5) implies

(10.6) (a-l)"^„B(-iy»TE(> * ^(mod/.).* r>0 V(/> - 1) - 1/r>0 V(# - 1)

But it is known [4, p. 255] that

/ m \ 7-1 (rnod^) (P ~ l\m),

o<r(p-i)ém\r(p - 1) - 1/ (.O (mod p) (p - l\ m).

Hence (10.6) implies that p is integral when m^0 (mod p — l). This proves

Theorem 3. Let p^3, q = a, a primitive root (mod p2); then ßm is integral

(mod p) for p — l\m, while

1(10.7) pßm = - — (mod p) (p - 11 to),

w/zere k = (a"-1 — I)/p.

It is now clear how to handle the general situation. We may state

Theorem 4. Let p^3, q = a, where a belongs to the exponent e (mod p),

e>l. Put



(10.8) ae = 1 + plk ip\ k).

Then

e _ / m \(10.9) (a - l)">plßm =: — Z (-l)m-re[ ) (mod p).

k r>o \re — 1/

In particular if e = p — 1, then

i 0 (mod i) (p — l\m),

(10.10) (a - l)mplß ■ 1 .I-(mod p) (p — 11 m).I k

To prove (10.9) it is only necessary to observe that (10.8) implies

re re epl-= pl-m — (mod p).

a" - 1 (14- p'ky - 1 k

It is clear from (10.10) that the denominator of ßm may be divisible by

arbitrarily high powers of p. We also remark that theorems like Theorems 3

and 4 can be framed for rjm.

When p'\a, it is evident from (10.5) that

(10.11) ßmM Z (~l)g(m)(s+ 1) =0(modp°) (m>l).«=0 \ 5 /

11. Congruences. The formula (7.11) together with (10.2) makes it pos-

sible to derive certain congruences satisfied by Hmix, q). We observe, to

begin with, that if q = a is integral (mod p) then um— [s\\am,a satisfies, for 5

fixed and ip — l)pe~l\w,

(11.1) umiuw - l)r = 0 (mod pm, pre),

where after expansion of the left member, un is replaced by un. To prove

(11.1) we need only remark that

«">(«<» - l)r = a-3<-*-l»2J^ (-l)'a'-('-1)/2rS][5 - r]mi[s - r]» - l)\r-o L r J

If we look on x in (7.11) as an indeterminate and apply (11.1), we can

assert that

(11.2) HmiH<° - iy = 0 (mod pm, p").

We interpret this congruence in the following manner. The left member of

(11.2) is a rational function of x such that the coefficient of each term in the

numerator =0 (mod pm, pr). We may call (11.2) Kummer's congruence for

Hm. Using (7.13) we can prove like results for Hmiu; x), where u is now an

integer.



In view of (1.6) the result (11.2) can be restated in terms of Am(x):

r / T \ m+rw

(11.3) Z(-l)r_s( )Am+Ux) II (* - «0 - 0 (mod pm, p").«=0 V 5 / i=m+aw+l

We may state

Theorem 5. Let q — abe integral (mod p), x an indeterminate, and r^l;

then (11.3) holds.

In the next place (11.3) implies congruences for the Am,a of (1.7). (For the

case g = l, compare [l].) Since

m+rw (r— a)w p(y _ s)w~~\

(X - a*) = [ (-l)(.r-a)w-iai(i+l)ll+i(.m+,w) ^' \ xi¡

i=m+aw+l ¿=0 L I A

examination of the coefficient of x*-1 in (11.3) implies

¿* / r\ ^ Vi* - s)wlNE(-l)r_,( )Z (-l)(r-s)"'-^»1+»u,,it-iaiCi+1)/2+i(ra+>w)

(11.4) J=,o \s / i L i J

h 0 (mod />"*, />re)-

In order to obtain simpler results we consider some special values of the

parameters. In the first place we take r = l, so that (11.4) becomes

(11.5) Am+W,k- Z(-l)tt-i^,*-;ai(i+1)/2+i"TW] = 0 (mod p", p°).

(11.6)

Now suppose first that a = l (mod p). It is necessary to examine

(of» - 1) • • • (a«"-^1 - 1)

[:]- (a - 1) ••• (a* - 1)

We assume from now on that p>2. If we put a = l+plh, p\h, then as in the

proof of Theorem 4 we find that

[w "1 fw \il snd O

are divisible by exactly the same power of p. But if i<p>, it is clear from the

identity

(w \ w /w — 1\

*/ TV*- i/that

(w \1 = 0 (mod p*-t) (j g c).



Consequently if p'~l^k<pi and j<e, we see that (11.5) implies

(11.7) Am+w.k = Amk (mod pm, p—f).

This proves

Theorem 6. Let p¡t3, q = a = l (mod p), p'~1(p — l)\w, and p'~xSk<p',

where j <e. Then (11.7) holds.

When a^l (mod p) let a belong to the exponent t (mod p). Then it is

clear from (11.6) that we need only consider those factors in the right member

with exponents divisible by t. Thus if io is the greatest integer &i/t we need

only examine

rawith a replaced by a'. The preceding discussion therefore applies and we ob-

tain the following theorem which includes Theorem 6.

Theorem 7. Letp^3, let q = a belong to the exponent t (mod p),pe~1(p—1)\ w

and k<tp', where j <e. Then (11.7) holds.

The case k = w is not covered by the theorem. We find for example that

if w = t=p— 1 (so that a is a primitive root (mod />)), then

Lm+p—l,k

Am,k (mod p) (k < p - 1),

Am.p-i + ( — jAm.o (mod p) (k = p — I),

where (a/p) is Legendre's symbol.

Returning to (11.2) we can also consider the case in which x is put equal

to an integer (mod p), provided the resulting denominators are not divisible

by p. Now the least common denominator is evidently

m+rw

^m+rw(x) = II (X - O").s-1

It will therefore suffice to assume that xâs (mod p) for any s. We may

therefore state

Theorem 8. Let a and x be rational numbers that are integral (mod p) and

suppose that xféa3 (mod p) for any s. Let

Pe~1(P — 1) | a> and r ^ 1.

Then

(11.8) Hm(x)(Hm(x) - iy m 0 (mod pm, p").



In particular the theorem may be applied with slight changes to em(u)

= em(u, a) defined in (9.2); we have explicitly [2, (8.18)]

» (-l)'a'tm(u) = ¿^

tZ (a+l)(a2+l) ■■■ («•+14-1)

• ¿(-lFa'C-D^r5 \[u + s - r]'",r-0 L r J

which is included in (7.13). If u is an integer we have

em(u)(ew(u) - l)r = 0 (mod pm, pre)

provided p = 3 (mod 4) and a is a quadratic residue (mod p). For in this case

— 1 is a nonresidue (mod p) and therefore — 1 fea3 for any s.

12. Congruences involving r¡m and ßm. Let

If.. / r \ k- [k])(12.1) um = um,k,r = — <k[k "-V T' ?M -»)»»- (-l)m 7-77-7

m { \k / (q - l)m)

so that by (8.5)

1 f^1(12.2) um = — Z-Fm_!(rg-i),

q 1*1 1 — f

where f runs through the £th roots of unity distinct from 1. As for the de-

nominators in the right member of (12.2), note that

II («' - f) = a = «,(*_1) + • • ■ 4- aa 4- 1,r„, a» - 1

which is prime to p for all 5 provided p\k. We may therefore state

Theorem 9. If a is integral (mod p) and p\k, then

(12.3) com(ùj"" - l)r = 0 (mod ¿"lr-1, ¿>re),

wAere 03m is defined by (12.1) and pe~1(p — 1) | w.

As for ßm we have

k[k]m-Yßm(^> CM -¿

(12.4)(« + l)(g - 1) f« w f^1

=-Z--Hm(Çq-1) + — Z =-: F^-iífg-1)g t*i 1 - f Ç r*i 1 - f

analogous to (8.5). In much the same way as above (12.4) implies

Theorem 10. If a is integral (mod p) and p\k then

(12.5) Q,m(W - l)r = 0 (mod pn~l, />(^1)e),



where 17m stands for the left member of (12.4) and p"~1(p — l) | w.

Unfortunately we seem unable to obtain simpler congruences for ßm and

13. Combinatorial interpretation of am3T. Put

.-i ma — / . amsroq

(»-l)(m-«)

Z a»r=0

(r„ = sis - l)/2 4- r).

The following combinatorial interpretation of the coefficients amar was kindly

suggested by J. Riordan. The number amtr is the number of permutations of m

things requiring 5 readings and such that r = r2 + 2r3+ • • ■ +is— l)re, where

rk is the number of elements read on the ¿th reading. The following numerical

illustration for m = 4 was also supplied by Riordan.

Permutation

1234

124314234123

13241342312431423412

213423142341

143241324312

21342413243142134231

321432413421

4321

Reading

1234

123 4

12 34

1 234

12 3 4

l|23|4

1 2 34

12 3 4


350 L. CARLITZ

References

1. L. Carlitz and J. Riordan, Congruences for the Eulerian numbers, Duke Math. J. vol.

20 (1953) pp. 339-344.2. L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. vol. 15 (1948) pp. 987-

1000.3. G. Frobenius, Über die Bernoullischen Zahlen und die Eulerschen Polynome, Preuss.

Akad. Wiss. Sitzungsber. (1910) pp. 809-847.4. N. Nielsen, Sur le théorème de v. Staudt et de Th. Clausen relatif aux nombres de

Bernoulli, Annali di Matemática (3) vol. 22 (1914) pp. 249-261.5. J. Worpitzky, Studient über die Bernoullischen und Eulerschen Zahlen, J. Reine Angew.

Math. vol. 94 (1883) pp. 202-232.

Duke University,

Durham, N. C.


Documents

g-BERNOULLI AND ETJLERIAN NUMBERS · ç-BERNOULLI AND EULERIAN NUMBERS 333 and the coefficients are the same as those occurring in (1.4). For g=l, Ama and Hm(x) reduce to well known