On a q-analogue of the p-adic Log Gamma Functions and Related Integrals

Transcription

1 Joural of Number Theory 76, (999) Article ID jth , available olie at httpwww.idealibrary.com o O a q-aalogue of the p-adic Log Gamma Fuctios ad Related Itegrals Taekyu Kim Departmet of Mathematics, Korea Mijok Leadership Academy, 334 Sosa, Aheug, Hoegsog, Kagwo Provice, South Korea Commuicated by P. Roquette Received November 29, 998 We show that Carlitz's q-beroulli umber ca be represeted as a itegral by the q-aalogue + q of the ordiary p-adic ivariat measure, whece we give a aswer to a part of a questio of Koblitz. 999 Academic Press. INTRODUCTION Throughout this paper Z, Q, Z p, Q p, ad C p will respectively deote the rig of ratioal itegers, the field of ratioal umbers, the rig of p-adic ratioal itegers, the field of p-adic ratioal umbers ad the completio of the algebraic closure of Q p. Let v p be the ormalized expoetial valuatio of C p such that p p = p &v p ( p) = p &.Ifq # C p, we ormally assume q& p <p &( p&), so that q x =exp(x log q) for x p. We use the otatio Hece, [x]=[x q]= &qx &q. lim [x q]=x q for ay x with x p i the preset p-adic case. We defie i the sequal a q-extesio B m (q) of the Beroulli umber with lim B m (q)=b m, q where B m is the usual Beroulli umber X Copyright 999 by Academic Press All rights of reproductio i ay form reserved. 320

2 p-adic LOG GAMMA FUNCTIONS 32 I [7], Koblitz costructed a q-aalogue of the p-adic L-fuctio L p, q (s, /) ad suggested two questios. Questio () was solved by Satoh [8], but questio (2) still remais ope. I this paper, we prove that the q-aalogue of Beroulli umbers occur i the coefficiets of some Stirlig type series for p-adic aalytic q-loggamma fuctios, which is a aswer to a part of Koblitz's questio (2) [7], ad we treat some applicatios of I q -itegratio. 2. q-analogue OF p-adic LOG GAMMA FUNCTIONS Let d be a fixed iteger ad let p be a fixed prime umber. We set X= N X*=. 0<a<dp (a, p)= (Zdp N Z), a+dp Z p, a+dp N Z p =[x # X x#a mod dp N ], where a # Z lies i 0a<dp N. For ay fixed positive iteger d we easily see that p& [ p q dp N ] q idpn =. Thus we ca defie a q-aalogue + q of the usual basic distributio + 0 as follows. For ay positive N we set qa + q (a+dp N Z p )= qa [dp N ] = [dp N q], ad this ca be exteded to a distributio o X as below. For the ordiary p-adic distributio + 0 defied by we see + 0 (a+dp N Z p )= dp N, lim + q =+ 0. q

3 322 TAEKYUN KIM We show that + q is distributio o X. For this it suffices to check that p& The left had side is equal to p& + q (a+i dp N +dp N+ Z p )=+ q (a+dp N Z p ). [dp N+ ] qa+idpn = Sice we see that Therefore we have p& p& [dp N+ ] qa p& q a+idpn = [dp N+ ] N+ [dp N+ ]= &qdp =[dp N ][ p q dp N ]. &q + q (a+i dp N +dp N+ Z p )= qa p& [dp N+ ] = qa [dp N ] q idpn [ p q dp N ] p& q idpn = qa [dp N ] =+ q(a+dp N Z p ). q idpn. This distributio yields a itegral for each o-egative iteger m i the case d=, [a] m d+ q (a) N p N & a=0 =I q ([a] m ), qa [a] m [ p N ] which has a sese as we see readily that the limit is coverget. We defie a q-beroulli umber B m (q)#c p by makig use of this itegral Note that I q ([a] m )=B m (q). lim B m (q)=b m, q where B m is the mth Beroulli umber.

4 p-adic LOG GAMMA FUNCTIONS 323 The geeratig fuctio F q (t) ofb k (q), is give by F q (t)= F q (t) \ k=0 B k (q) tk k!, [ p \ ] p \ & q i e [i]t, which satisfies the q-differece equatio F q (t)=qe t F q (qt)+&q&t. If q=, the we have log et F q (t)= e t & = t e t & =ebt. Let / be a primitive Dirichlet character with coductor d # Z +, the set of atural umbers. The we also defie a geeralized q-beroulli umber B m, / (q) as Thus we have B m, / (q)= X /(a)[a] m d+ q (a) N =I q ([a] m /(a)). dp N & a=0 lim B m, / (q)=b m, /, q [a] m /(a) q a [dp N ] where B m, / (q) isthemth geeralized Beroulli umber. The q-beroulli polyomials i the variable x i C p with x p are defied by B (x q)= [x+t] d+ q (t). These ca be writte as B (x q)=(q x B(q)+[x]).

5 324 TAEKYUN KIM Ideed, we see [x+t] d+ q (t)= ([x]+q x [t]) d+ q (t) For the itegral I q we first see = k=0\ k+ [x]&k k=0\ k+ [x]&k = =(q x B(q)+[x]). q kx [t] k d+ q (t) q kx B k (q) Theorem. For m0, we have Proof. We see B m (q)= (&q) m m j=0\ m j+ (&) j j+ [ j+]. &q &q p p & a=0 [a] m q a = &q &q p p & (&q) m a=0 m = (&q) &q j=0\ m p j + m j=0\ m j+ (&) j q aj q a Sice lim q p = for &q p <, our assertio follows. p &q ( j+) j (&). &q ( j+) Let Q be a algebraic closure of Q. Ifq # Q & C p,theb m (q) is the same as Carlitz's q-beroulli umbers i [, p. 993, (5.2)]. Example 2. B (q)=& q+ =& [2], B q 2(q)= (+q)(+q+q 2 ) = q [2][3], &q(q&) B 3 (q)= (+q+q 2 )(+q)(+q 2 ) =q(&q) [3][4] }}}. The proof of the distributio relatios for q-beroulli polyomials i the complex case was foud i [, 7], ad a simple proof i the p-adic case ca be give by usig I q -itegratio of q-beroulli umbers as follows.

6 p-adic LOG GAMMA FUNCTIONS 325 Lemma. For each m # Z +, the set of atural umbers, we have X [a] m d+ q (a)= [a] m d+ q (a). Proof. For d # Z +, we see lim &q &q dp dp & a=0 [a] m q a (&q) (&q) m &q dp m &q p j=0\ m j+ j=0\ m j+ dp &q ( j+) j (&) &q j+ p j &q ( j+) (&). &q j+ Sice lim q p = for &q p <, our assertio follows. Theorem 2. for all k0. Proof. For ay positive iteger m, we have [m] k& q i B k\ x+i m ; qm+ =B k(x; q) From the above lemma, we ca write [x+t] k d+ q (t) \ [mp \ ] mp \ & =0 \ [m] [ p \ q m ] q [x+] k \ [m] [ p \ q m ] [m] = =[m] k& q i [m] k _ x+i m p \ & =0 p \ & q i =0 q i B k\ x+i m ; qm+. q i+m [x+i+m] k q m k +t d+ qm& q m(t) \_ x+i qm& k m + [m] + If q # Q & C p, the B k (x q) is the same as Carlitz's q-beroulli polyomial i [, Eq. (5.9)].

7 326 TAEKYUN KIM Next we defie the fuctio G p, q (x) oc p "Z p by G p, q (x)= [(x+[z]) log(x+[z])&(x+[z])]d+ q (z) N p N & =0 for x p >. The we easily see [(x+[]) log(x+[])&(x+[])]+ q (+ p N Z p ) lim G p, q (x)=g p (x), q where G p (x) is the Diamod gamma fuctio [2]. The fuctio G p, q (x) is locally aalytic o C p "Z p. This fact ca be show by the same method as i [2, 5, 6]. Now we treat Koblitz's questio (2) i [7]. Theorem 3. For x # C p with x p > we have G p, q (x)= [2]+ \x& log x&x+ = (&) + (+) B +(q) x. Proof. We fie that (x+[z]) log(x+[z])&(x+[z]) =(x+[z]) {log \ +[z] x + +log x = &(x+[z]) =[z]+x = (&) + (+) [z] + x + +(x+[z]) log x&(x+[z]). Thus we have the p-adic Stirlig asymptotic formula for G p, q (x) i terms of B(q)-umbers as G p, q (x)= [(x+[z]) log(x+[z])&(x+[z])] d+ q (z) = [z] d+ q (z)+x = (&) + (+) x + [z] + d+ q (z) +log x (x+[z]) d+ q (z)& (x+[z]) d+ q (z)

8 p-adic LOG GAMMA FUNCTIONS 327 =B (q)+x = (&) + (+) +B (q) log x&b 0 (q)x&b (q) =(x+b (q)) log x&x+ x + B +(q)+xb 0 (q) log x = This completes the proof of our assertio. (&) + (+) x B +(q). 3. APPLICATION OF I q I this sectio we give some applicatios of the itegral I q. Propositio. For m, 0 ad x # Z p, we have Proof. I q (q x [x] m )= j=0\ j+ (&) j B m+ j (q)(&q) j. Ideed, we see I q (q x [x] m )=I q ((&(&q x )) [x] m ) = k=0 \ It ca be easily proved [5] that Hece we have k+ (&)&k I q\ (&qx ) &k = k=0\ k+ (&)&k I q\\ &qx &q+ \ &qx m &q+ + m+&k+ (&q)&k = k=0\ k+ (&)&k B m+&k (q)(&q) &k. d+ 0 (x)=& q&x log q d+ q (x). &q I 0 ([x] m )=& log q &q q &x [x] m d+ q (x). Let UD(Z p, C p ) be the set of uiformly differetiable fuctios o Z p with values i C p ad let 2 be the differece operator as usual defied by 2f(x)= f(x+)&f(x).

9 328 TAEKYUN KIM The we kow that for f, g # UD(Z p, C p ) ad g(0)=0 we have I 0 ( f2g)=i 0 ( fg & ), where g & (x)=g(&x) [4]. Now the covolutio for f, g # UD(Z p, C p ) is defied by f g()= k=0 f(k) g(&k), for [] ad ( fg)(x)=lim x ( fg)() forx # Z p. Cosequetly we see easily that I 0 (q &x ( f2g))=i q (q &x ( fg & )). I particular we take f(x)=q mx [x] m, g(x)=[&x] for m,. The we have I q (q &x (q mx [x] m ([&x&] &[&x] ))) =I q ([x] m+ q ()x ) = j=0 \ Therefore we obtai the followig j + (&) j B m++ j (q)(&q) j. Propositio 2. For m,, we have \ j + (&) j B m++ j (q)(&q) j j=0 & = k=0\ k+ [&]&k I q (q &x (q mx [x] m q &(&k)x [&x] k )). If q=, we obtai a theorem of C. F. Woodcock []. ACKNOWLEDGMENT The authors ackowledge the fiacial support of the Korea Research Foudatio made i the program year of 998 ad partial support of Jagjeo Research Istitute for Mathematical Sciece. REFERENCES. L. Carlitz, q-beroulli umbers ad polyomials, Duke Math. J. 5 (948), J. Diamod, The p-adic log gamma fuctio ad p-adic Euler costats, Tras. Amer. Math. Soc. 233 (977), K. Ikeda, T. Kim, ad K. Shiratai, O p-adic bouded fuctios, Mem. Fac. Sci. Kyushu Uiv. 36 (992), T. Kim, A aalogue of Beroulli umbers ad their cogrueces, Rep. Fac. Sci. Egrg. Saga Uiv. Math. 22 (994), 73.

10 p-adic LOG GAMMA FUNCTIONS N. Koblitz, q-extesio of the p-adic gamma fuctios, Tras. Amer. Math. Soc. 260 (980), N. Koblitz, A ew proof of certai formulas for p-adic L-fuctios, Duke Math. J. 46 (979), N. Koblitz, O Carlitz's q-beroulli umbers, J. Number Theory 4 (982), J. Satoh, q-aalogue of Riema's `-fuctio ad q-euler umbers, J. Number Theory 3 (989), K. Shiratai ad S. Yamamoto, O a p-adic iterpolatio of the Euler umbers ad its derivatives, Mem. Fac. Sci. Kyushu Uiv. 39 (985), K. Shiratai, A applicatio of p-adic zeta fuctios to some cyclotomic cogrueces, Kyugpook Math. J. 34 (994), C. F. Woodcock, A two variable Riema zeta fuctio, J. Number Theory 27 (987), 2222.