BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES

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1 BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit part of n. If α is a positiv intgr not divisibl by p, w show that th p-adic limit of ( 1 pα U((αp! as is a wlldfind p-adic intgr, which w call z α. Not that if p 2 or α is vn, this can b thought of as U((αp!. In trms of ths, w thn giv a formula for th p-adic limit of ( ap +c bp +d as, which w call ( ap +c bp +d. Hr a b ar positiv intgrs, and c and d ar intgrs. 1. Statmnt of rsults. Lt p b a prim numbr, fixd throughout. Th st Z p of p-adic intgrs consists of xprssions of th form x c i p i with 0 c i p 1. Th nonngativ intgrs ar i0 thos x for which th sum is finit. Th mtric on Z p is dfind by d(x, y 1/p ν(x y, whr ν(x min{i : c i 0}. (S,.g., [3]. Th prim p will b implicit in most of our notation. If n is a positiv intgr, lt U(n n/p ν(n dnot th unit factor of n (with rspct to p. Our first rsult is as follows. Thorm 1.1. Lt α b a positiv intgr which is not divisibl by p. If p > 4, thn U((αp 1! ( 1 pα U((αp! mod p. This thorm implis that d ( ( 1 pα( 1 U((αp 1!, ( 1 pα U((αp! 1/p, from which th following corollary is immdiat. Corollary 1.2. If α is as in Thorm 1.1, thn lim ( 1 pα U((αp! xists in Z p. W dnot this limiting p-adic intgr by z α. 1

2 2 DONALD M. DAVIS If p 2 or α is vn, thn z α could b thought of as U((αp!. It is asy for Mapl to comput z α mod p m for m fairly larg. For xampl, if p 2, thn z mod This is obtaind by ltting C n dnot th mod 2 n+1 rduction of U(2 n! and computing C 1 1, C 2 3, C 3 C 4 C 5 C 6 11, C 7 C 8 139, C 9 651, C 10 C , and C 12 C 13 C Similarly, if p 3, thn z mod It would b intrsting to know, as a futur invstigation, if thr ar algbraic rlationships among th various z α for a fixd prim p. ( a b Thr ar two wll-known formulas for th powr of p dividing a binomial cofficint. (S,.g., [4]. On is that ν ( a b 1 (d p 1 p(b + d p (a b d p (a, whr d p (n dnots sum of th cofficints whn n is writtn in p-adic form as abov. Anothr is that ν ( a b quals th numbr of carris in th bas-p addition of b and a b. Clarly ν ( ( ap bp ν a b. Our nxt rsult involvs th unit factor of ( ap bp. Hr on of a or b might b divisibl by p. For a positiv intgr n, lt z n z U(n, whr z U(n Z p is as dfind in Corollary 1.2. Thorm 1.3. Suppos 1 b a and {ν(a, ν(b, ν(a b} {0, k} with k 0. Thn (( ap U ( 1 pck z a mod p, bp z b z a b a if ν(a k, whr c b if ν(b k, a b if ν(a b k. Not that sinc on of ν(a, ν(b, and ν(a b quals 0, at most on of thm can b positiv. Sinc ν ( ap bp is indpndnt of, w obtain th following immdiat corollary. Corollary 1.4. In th notation and hypothss of Thorm 1.3, in Z p ( ( ap ap : lim p ν(a b ( 1 pck z a. bp bp z b z a b

3 BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES 3 Our final rsult analyzs ( ap +c bp +d, whr c and d ar intgrs, possibly ngativ. Thorm 1.5. If a and b ar as in Thorm 1.3, and c and d ar intgrs, thn in Z p ( ap ( c c, d 0, ( ( ap + c ap + c ( bp d ap ( c a b c < 0 d, : lim ( bp d a bp + d bp + d ap ( c b bp c d c < 0 c d, a 0 othrwis. Hr, of cours, ( ap bp is as in Corollary 1.4, and w us th standard dfinition that if c Z and d 0, thn ( c d c(c 1 (c d + 1/d!. Ths idas aros in xtnsions of th work in [1] and [2]. 2. Proofs In this sction, w prov th thr thorms statd in Sction 1. Th main ingrdint in th proof of Thorm 1.1 is th following lmma. Lmma 2.1. Lt α b a positiv intgr which is not divisibl by p, and lt b a positiv intgr. Lt I α, {i : αp 1 < i αp }, and lt S dnot th multist consisting of th last nonngativ rsidus mod p of U(i for all i I α,. Thn vry positiv p-adic unit lss than p occurs xactly α tims in S. Proof. Lt W α, dnot th st of positiv intgrs prim to p which ar lss than αp. Thn our unit function U : I α, W α, has an invrs function φ : W α, I α, dfind by φ(u p t u, whr t max{i : p i u αp }. Not that p t u I α, sinc p t+1 u > αp which implis p t u > αp 1. On asily chcks that U and φ ar invrs and hnc bijctiv. Sinc rduction mod p from W α, to W 1, is an α-to-1 function, prcding it by th bijction U implis th rsult. Proof of Thorm 1.1. If p > 4, th product of all p-adic units lss than p is congrunt to ( 1 p mod p. (S,.g., [4, Lmma 1], whr th argumnt is attributd to Gauss. Th thorm follows immdiatly from this and Lmma 2.1, sinc, mod

4 4 DONALD M. DAVIS p, U((αp!/ U((αp 1! is th product of th lmnts of th multist S dscribd in th lmma. Proof of Thorm 1.3. Suppos ν(b 0 and a αp k with k 0 and α U(a. Thn, mod p, (( αp +k U bp U((αp +k! U((bp! U(((a bp! ( 1 pα(+k z a ( 1 pb z b ( 1 p(a b z a b ( 1 pak z a z b z a b, as claimd. Hr w hav usd Thorm 1.1 and th notation introducd in Corollary 1.2. Also w hav usd that ithr p 2 or a α mod 2. A similar argumnt works if ν(b k > 0 (and ν(a 0, or if ν(a b k > 0 (and ν(a ν(b 0. Our proof of Thorm 1.5 uss th following lmma. Lmma 2.2. Suppos f is a function with domain Z Z which satisfis Pascal s rlation (2.3 f(n, k f(n 1, k + f(n 1, k 1 for all n and k. If f(0, d Aδ 0,d for all d Z and f(c, 0 Ar for all c < 0, thn A ( c c, d 0, d A ( c f(c, d d r c < 0 d, A ( c c d (1 r c < 0 c d, 0 othrwis. Th proof of this lmma is straightforward and omittd. It is closly rlatd to work in [5] and [6], in which binomial cofficints ar xtndd to ngativ argumnts in a similar way. Howvr, in that cas (2.3 dos not hold if n k 0. Proof of Thorm 1.5. Fix a b > 0. If f (c, d : ( ap +c bp +d, whr is larg nough that ap + c > 0 and bp + d > 0, thn (2.3 holds for f. If, as, th limit xists for two trms of this vrsion of (2.3, thn it also dos for th third, and (2.3 holds for th limiting valus, for all c, d Z. Th thorm thn follows from Lmma 2.2 and (2.4 and (2.5 blow, using also that if d < 0, thn ( ( ap bp +d ap (a bp + d, to which (2.4 can b applid.

5 BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES 5 If d > 0, thn ( ( ap ap ((a bp ((a bp d + 1 (2.4 0 bp + d bp (bp + 1 (bp + d in Z p as, sinc it is p tims a factor whos p-xponnt dos not chang as incrass through larg valus. Lt c m with m > 0. Thn (2.5 ( ( ( ap m ap ((a bp ((a bp m + 1 ap a b bp bp ap (ap m + 1 bp a, in Z p as, sinc ((a bp 1 ((a bp m + 1 (ap 1 (ap m mod p [log 2 (m]. Hr w hav usd that if t < and v is not divisibl by p, thn (a bp vp t p t. Rfrncs ap vp t 1 mod [1] D. M. Davis, For which p-adic intgrs x can k ( x 1 k b dfind?, J. Comb. Numbr Thory (forthcoming. Availabl at [2], Divisibility by 2 of partial Stirling numbrs, Funct. Approx. Commnt. Math. (forthcoming. Availabl at [3] F. Q. Gouva, p-adic Numbrs: an Introduction, Springr-Vrlag, Brlin, Hidlbrg, [4] A. Granvill, Binomial cofficints modulo prim powrs, CMS Conf. Proc 20 ( [5] P. J. Hilton, J. Pdrson, Extnding th binomial cofficints to prsrv symmtry and pattrn, Comput. Math. Appl. 17 ( [6] R. Sprugnoli, Ngation of binomial cofficints, Discrt Math. 308 ( Dpartmnt of Mathmatics, Lhigh Univrsity, Bthlhm, PA dmd1@lhigh.du