The Lucas congruence for Stirling numbers of the second kind

Transcription

1 ACTA ARITHMETICA XCIV 2 The Luc cogruece for Srlg umber of he ecod kd by Robero Sáchez-Peregro Pdov Iroduco The umber roduced by Srlg 7 h Mehodu dfferel [], ubequely clled Srlg umber of he fr d ecod kd, re of he gree uly he clculu of fe dfferece, umber heory, he ummo of ere, he heory of lgorhm, he clculo of he Bere polyoml [9] I h udy, we demore ome propere of Srlg umber of he ecod kd mlr o hoe fed by boml coeffce; prculr we how h hey fy cogruece logou o h of Luc, h o: b b mod p wh p, b b p ; p, b p Ug Propoo 4 we gve oher proof for Keko recurrece formul for poly-beroull umber [] Some of he reul re mlr o hoe of Howrd [5] I cocluo, I wh o gve my be hk o he Geomery Group of he Dprmeo d Memc Pur ed Applc d Dprmeo d Meod Memc per le Sceze Applce of he Uvery of Pdov, for uppor d help gve durg he prepro of h work I prculr, I wh o hk Frk Sullv for h precou dvce d uggeo Noo d defo I h eco, we wll revew vrou defo d oo for Srlg umber of he ecod kd Le, N We e f,, f >,, f, >, f >, > 2 Mhemc Subjec Clfco: A7, B7 [4]

2 42 R Sáchez- Peregro Defo The umber repreeed by he ymbol re clled Srlg umber of he ecod kd A h oe del oly wh he Srlg umber of he ecod kd, we wll cll hem mply Srlg umber Srlg umber of he ecod kd re lo dced he lerure wh oher ymbol: S, []; S [6] The oo ued here h bee propoed by D Kuh [4]; followg h dvce we my red brcke Wh Defo he eger, re umed o-egve Neverhele, ueful o mplfy oo ll he ecery pge, gvg vlue o he umber eve whe < I h ce, we coveolly pu For, wh mple codero oe verfe h Trdolly, h he wy whch Srlg umber re roduced [6] I hould be oced, moreover, h he Srlg umber k equl o he umber of pro of he e,, o k-block [2, ] Moreover, Srlg umber hve he followg propere: f > ; 2! j j j j See [6], pge 68 d 69, for demoro 2 Addo formul A coequece of j j j j d of propery 2, we hve he followg Propoo 2 For ech prme umber p, p > 2, he Srlg umber fy p mod p for ech, p Propoo 22 ddo Le,, N The j j j j P r o o f The proof by duco o Whe, he rgh hd de of reduce o

3 Luc cogruece for Srlg umber However, he ler equl o Thu, formul hold whe Now rem o demore h f rue for url umber >, he lo rue for : j j j j j j j [ j j j j j j j j j f j j j j j j j j j j f f f f f f f f j ]

4 44 R Sáchez- Peregro 2 j f f h j j j f f f f h h f h h f f f f f If y >, y Therefore, he hrd dded of 2 we c eed he ecod um o, d 2 become f h [ h f h ] h hf f f f h h f f h h f f h hf h f f f f h f h f h hf h h f f Corollry 2 If p he p mod p p Th formul h bee demored by Becker & Rord [], ug oher mehod

5 Luc cogruece for Srlg umber 45 Lemm 2 Le,, y N The y p y p p mod p The proof by duco o If, he rue ccordg o Corollry 2 I rem o be how h f rue for url umber >, he lo hold for To do h, we c ue procedure logou o h ued o prove Newo Boml Formul Remrk 2 Th lemm mlr o Theorem 42 of Howrd [5] he ce, y < p Th lemm, ogeher wh ey duco rgume o r, gve more elemery proof of Theorem 44 of Howrd [5] Corollry 22 Le,, y N The y p y y p j mod p j Fr pproch o he Luc Theorem Propoo Le, y,, N wh y p, p The y p y y 4 mod p p p 5 P r o o f Le, from Lemm 2 follow h y p y p p y p mod p p p p y j mod p j j p j m m y m mp p mod p Added of 5 whch correpod o m 2 re ull vew of propery of Seco, becue, h ce, y m < m p Added for whch m re ull, ce m p < Hece, we coclude h he ecod um 5 reduce o oly he dded for whch m d m : 6 [ y ] y p

6 46 R Sáchez- Peregro y y p If we ue Vdermode equly [6] for boml coeffce, equo 6 reduce o y y p Obervo Wh he hypohe h y < p, 4 reduce o y p y mod p p I ueful o oce h h formul very mlr o Luc formul for boml coeffce Remrk I he ce r < p he cogruece 4 gve he formul 47 d 48 of Howrd [5] 4 The Luc Theorem Propoo 4 Le, y,, N The y p y l 7 l, l,, l k l kp k mod p l l l P r o o f The proof by duco o Fr of ll, ccordg o Corollry 22 formul 7 rue whe I rem o be how h f 7 rue for url umber, he lo rue for : y p p l l l l l l l, l,, l l, l,, l y l k l kp k p y l k l kp k p m m m m y p l k l kp k mod p [ y l k l kp k ] mod p m, m,, m y l k l kp k p 2 y m k m kp k

7 m m m m m m m m 2 m m m m m m m m m m m m m m m m m m m Luc cogruece for Srlg umber 47 y m m, m,, m k m kp k m, m,, m y m k m kp k y m m, m,, m k m kp k y m m m, m,, m k m kp k y m m m, m,, m k m kp k m m, m,, m 2 y m k m kp k y m m m, m,, m k m kp k m, m,, m y m k m kp k Applyg Propoo 4 ow gve he followg heorem Theorem 4 Luc Le, y,, y N,,, m,, y p, y y y y, y N,,, m The y y p y 2 p 2 y m p m p 2 p 2 m p m y y y y 2 y 2 y 22 y 2 y m y m y mm y m y y 2 y m y, y y 2, y 2, y 22 y m, y m,, y mm y y y 2 y m y m y 2 y p mod p 2 y m2 y 22 p 2 m y mm p m

8 48 R Sáchez- Peregro 5 A pplco o Clue Vo Sud cogruece for he poly-beroull umber For every eger k, we defe equece of rol umber B k,,, whch we refer o poly-beroull umber, by 8 z L kz z e B k! Here, for y eger k, L k z deoe he forml power ere m zm /m k, he kh polylogrhm f k d rol fuco f k Whe k, B he uul Beroull umber wh B /2 [8] Throughou h eco, ν p he drd p-dc vluo o Q The rol p-dc eger Z p Q re he rol umber r uch h ν p r We hve he followg epo of he umber B k erm of he Srlg umber of ecod kd Theorem 5 B k See [7] for demoro m m m! m k m We e m q r p r r p r l p l wh q r [, p ] d [, p ] for r,, l The m! 9 ν p m k rk pr p q r p r l r p p Remrk 5 We deoe he rgh hd de of 9 by Le p k, d le b b p b l p l be he p-dc epo of Pu l b The Remrk 52 We hve p p We eblh he followg lemm whch wll be coly ued below Lemm 5 mod p f p, p mod p f p P r o o f Wh he oo roduced fer Remrk 5, follow from Propoo 4 d Remrk 52 h p k mod p, where p k p p p

9 Luc cogruece for Srlg umber 49 We c ere h procedure d he ed we ob rp mod p wh r p p p Sce r p d by Propoo cogruece equvle o p mod p, by Propoo 2 we ob he fr equly For he ecod ce he proof he me Theorem 52 If k 2, p, d k 2 p, he p k B k pz p, e p k B k mod p P r o o f Le p be pove prme By Theorem 5 we ob p k B k p k m m! m k m m We e m q r p r r p r l p l wh q r [, p ] d [, p ] for r,, l The by 9 equo equvle o p k B k p k m m! m k p k m m! 2 m m k m r r, > p k m m! m k p p! m p r 2 Becue m eger he rgh hd de of 2 eleme of pz p Sce p prme d m eger, we c prove h he um eleme of pz p By Lemm 5 d by Wlo Theorem he ecod um equvle o mod p Thu we flly ob he ero of Theorem 52 Theorem 5 If k 2, p d k 2 p he p k B k Z p The proof of h heorem mlr o h of Theorem 52 Remrk 5 The ce k 2 of Theorem 52 gve by Keko [7] For p < k 2 he behvour of he ν p B k choc We how h he ce p 2,, k Propoo 5 If eve, he ν 2 B 4 P r o o f Ug 9 we ee h he oly ummd B m m! m m m whch hve vluo le h re! 4 2 mod 4 d 7 mod 4 d 7! 8 7, bu

10 5 R Sáchez- Peregro Propoo 52 If 2 mod, he ν B 4 P r o o f Ug 9 we ee h he oly ummd B whch gve he vluo of B 8! 9 8, bu 8! 8 9 mod Propoo 5 If 6 8α 2 or 22 8α, he ν B P r o o f By 9 he oly ummd B whch gve corbuo o he vluo re 2! 2, 5! 6 5, d 8! 6 8 We fd by Lemm 5 d by duco h 2 d, 5 mod, 8! 68α2 8 2 α2 e 5 d 8! 228α 8 α 4 h 5, wh d, e, h N I he fr ce we hve 6 8α2 4 ν ν d α 2 2 e ; he ecod ce we hve 22 8α 2 ν ν d α 2 h Propoo 54 Le 8k If 54α, he f α mod, ν B f α mod, f α 2 mod I he remg ce 2854α 2 d 4654α we hve ν B 2 P r o o f By drec clculo d by duco we ob 2 25 d, 5! 5 e, d 8! 54α 8 9α 6 f wh d, e, f N I he ce α mod d α mod we ob 7 ν B α 54α ν d f ; he ce α 2 we ob e ν B 54α m ν 2 d f, ν 8 If 28 54α 2 we ob 8! 2854α 2 8 9α2 7 g 6 wh g N d o 9 ν B α2 2854α 2 ν 2 d g If 46 54α we ob 8! 4654α 8 9α 6 h wh h N d o 7 ν B α 4654α ν 2 d h Remrk 54 I Propoo 5 d 54, mod Propoo 55 If 2 8α or 24 8α 2, he ν B

11 Luc cogruece for Srlg umber 5 P r o o f By 9 he oly ummd B whch my gve corbuo o vluo re 2! 2, 5! 6 5 d 8! 9 8 Ug Lemm 5 we ob 2 d by duco we ge 5 mod I he fr ce we ob 8! 28α 8 26b 5 c o 4 ν B 28α ν 2 b 2 c I he ecod ce we ob 8! 248α d d hu ν B 248α 2 ν 2 Propoo 56 Le 8α We hve f α mod 9, ν B 8α f α 6 mod 9, 2 oherwe P r o o f By duco d by drec clculo we ob 2, 5! 5 8 b, d 8! 8 6k 6 d The 2! ν B ν 5! 2 6 8! 8k 5 9 ν Remrk 55 I Propoo 55 d 56, mod Ackowledgme We wh o hk M Keko for h dvce o h ubjec, d we wh o epre our cere hk o he referee for h helpful comme d uggeo h led o coderble mproveme of h pper Referece [] H W Becker d J Rord, The rhmec of Bell d Srlg umber, Amer J Mh 7 948, [2] L Crlz, Weghed Srlg umber of he fr d ecod kd, I, Fbocc Qur 8 97, [] L Come, Alye Combore, Tome,, Pree Uv de Frce, Pr, 97 [4] A E Fekee, Apropo wo oe o oo, Amer Mh Mohly , [5] F T Howrd, Cogruece for he Srlg umber d oced Srlg umber, Ac Arh 55 99, 29 4 [6] C Jord, Clculu of Fe Dfferece, Chele, New York, 96 [7] M Keko, Poly Beroull umber, J Théor Nombre Bordeu 9 997, [8] N Koblz, p-dc Number, p-dc Aly, d Ze-Fuco, Grd Te Mh 58, Sprger, Berl, 977

12 52 R Sáchez- Peregro [9] R Sáchez-Peregro, Ideé de Bere pour ue foco homogèe à gulré olée, Red Sem M Uv Pdov 8 989, [], Aoher proof of Keko recurrece formul for he Beroull umber, prepr [] J Srlg, Mehodu dfferel, ve rcu de ummoe e erpolzoe ererum frum, Lod, 7 Dprmeo de Memc Pur ed Applc Uverà d Pdov V Belzo, 7 5 Pdov, Ily E-ml: chez@mhupd Receved o d reved form o