#A79 INTEGERS 16 (2016) ROUND FORMULAS FOR EXPONENTIAL POLYNOMIALS AND THE INCOMPLETE GAMMA FUNCTION

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1 #A79 INTEGERS 6 (06) ROUND FORMULAS FOR EXPONENTIAL POLYNOMIALS AND THE INCOMPLETE GAMMA FUNCTION St Wgo Deprtmet of Mthemtcs d Computer Scece, Mclester College, St. Pul, Mesot wgo@mclester.edu Receved: 5/0/6, Accepted: //6, Pulshed: /3/6 Astrct The th prtl sum of the Mclur seres for e /, where d re tegers, ecomes teger whe multpled y!. Ths teger s relted to my comtorl propertes of terest d s lso drectly ted to exct computto of,, where s the complete gmm fucto. Ths pper presets very short formuls tht gve ths teger exctly whe pple. For lrger the method exteds d whle ot s fst s the smller cses, t s mprovemet o exstg computtol methods. The cses = ± were kow for my -vlues. The pproch here exteds the geerl de to ll rtols y mkg use of cogruece to overcome the error heret the tructo of the Mclur seres. A sde-e ect of the vestgto s ew lytc lower oud o the umer of tmes prme ppers fctorl: log ( + ).. Itroducto Let e (z) = P k=0 zk k! e the prtl sum for the Mclur seres of e z ; [ ] s the roudg (or erest teger) fucto where hlf-tegers re rouded to the erest eve teger; d lwys deote tegers, wth 0 d gcd(, ) = ; deotes oegtve teger. Clerly!e s teger cll t K, ofte revted to just K sce ll deomtors the sum re clered. Here we vestgte short d e cet formuls tht gve K s exct vlue. Tle shows severl of these sequeces together wth ther locto the O-Le Ecycloped of Iteger Sequeces []. The m result here s tht very smple roud formuls for K exst whe = ± or ±, s y ozero teger, d s y postve teger. Whe the umertor s ±, the formul s remrkly smple:!e ±/ ; for = ± t s lttle more complcted ut c stll e clled oe-ler. I these cses the computtol lod s the sgle hgh-precso evluto of the rtol power of e. There re extesos to = ±3 d greter, ut ther mprovemet over the method of rute-force ddto s modest: -step lgorthm s reduced to oe log usg out steps. Here the computtol lod ecomes the evluto of lrge cogruetl resdue.

2 INTEGERS: 6 (06) My specl cses whe = ± were kow (e.g., K y S. Plou e [3] d M. v Hoej [, A000354]), ut we here provde geerl method for ± ± K s well s smlrly e cet formul for K. By the clssc detty!e (z) = e z ( +, z), where s the complete gmm fucto (defed s (, z) = R e t t dt), y formul for K z yelds smlr formul tht expresses +, symolc form s K e /. = 0 = = = 3 = 4 = 5 = 6 OEIS ctlog A A A A A A A A A A A A A A A A A A Tle.! e for pple 3, pple pple 6, d 0 pple pple 6. The teger K ofte hs terestg comtorl terprettos. For exmple, let T () e the umer of permuttos of {,..., } hvg t lest oe trsposto [, A00066]. Let T c () cout the permuttos hvg o trsposto: T c () =! T (). The, wth m =, T c () =! P m k=0 ( )k k k! =

3 INTEGERS: 6 (06) 3! m m! K m, where the frst equlty s stdrd exercse usg the prcple of cluso d excluso []. Here s smplg of some others (see the correspodg etres t [], s lsted Tle ). K () couts the totl umer of ordered tuples usg dstct elemets of {,,..., }, where the tuple s legth s from 0 to, clusve. K ( ) gves the umer of dergemets of -elemet set: permuttos wth o fxed pot (lso kow s sufctorl ()). K s the umer of wys to sort spredsheet wth colums. K s the umer of wys to choose permutto of {,,..., } d choose k elemets (0 pple k pple ) tht re ot fxed pots of the permutto. Here re some geerl propertes of K; we do ot use these, ut lst them for completeess. Asymptotcs: K (z)! e z. (Ths s ecuse e (z) s symptotc to e z.) Recurrece: K the defto.) = K +. (Proof: Follows drectly from the sum Itegrl chrcterzto: K = +, = e / R / e t t dt. (Proof: Itegrto y prts shows tht the tegrl stsfes the sme recurrece s K.) The expoetl geertg fucto of the sequece K 0 x (f) = f P! ( x) k ( x) k=0 k! k ; t x = 0 ths s! P k=0 k! s f = ex x. K () s the permet of the mtrx wth + t ech dgol etry d ll other etres. (Proof: Use the defto of the permet d clssfy the permuttos ccordg to how my fxed pots ech hs.) The strtg pot s smple theorem showg tht, uder cert codtos, the teger K c e expressed y very short formul. Theorem. Suppose Z,, N,, d ( + ) >, d. The (Roud Formul) K =!e =!e / k.) + (+) pple Proof. The tl of the Mclur seres for e z fter the th term s P j= z+j (+j)!. Multplcto y! gves P j=!z+j (+j)!, whch s strctly less th P j= z+j (+). j Ths lst, settg z = /, s coverget geometrc seres wth sum + (+). Therefore!e /!e < + (+) pple. Becuse!e s teger, the proof s complete.

4 INTEGERS: 6 (06) 4. The Cse = ± Whe = ±, the codto of Theorem holds lmost ll cses. Note tht tl cses ( pple ) re qute trvl: K 0 =, K = +, d K = ( + ) +. Corollry. Suppose = ±, s postve teger, N wth 6= 0, d (, ) 6= (, ). The!e =!e / d +, =!e / e /. Proof. Use Theorem ; ecuse = ±, the restrctve equlty holds for ll. The excluded cse mes tht ether or, d for such vlues the crtcl qutty + (+) = (+) s less th. ± To compute K Mthemtc whe, oe just uses Roud! E /. ± Ad s gve y Roud! E / - E -/. We c ow compute stteously the exct umer of permuttos wth o trsposto, s frst oted [3] d [, A00066, A000354]. Corollry 3. If m = h, the T c () =! m m pm! m! e. Proof. Use Corollry wth = d =. The T () =! m P k= =! m P k= ( ) k+ k k! =!! m P =!! k=0 =!! m m! k k! k k! m! K m m h m m! p e 3. The Cse = ± Next we tur to the cse = ± wth odd. The teger K hs cert rthmetc propertes tht wll llow us to overcome the ucertty cused y the Mclur seres tructo. As Theorem s proof, the Mclur error s ouded solute vlue y + (+), or + (+) ; the drecto of the error s determed y the sg of d the prty of : t s egtve f d oly f < 0 d s eve. So ths determes tervl whch K les; whe =, the tervl hs legth t most /. For lrger the error c e much lrger, ut for = ±, we c ppel to smple cogruece codto o K to overcome the error.

5 INTEGERS: 6 (06) 5 Whe s prme, we use p () for the expoet of!. We strt y vestgtg upper d lower lytc ouds o p (). The proof of Lemm 4 whe 3 ws provded y Roert Isrel (Uv. of Brtsh Colum) d s cluded wth hs permsso. l m Lemm 4. If s postve teger d s prme, the log ( + ) pple j k p () pple. Proof. Let D = {d j } e the set of se- dgts of ; the D = L, where L = + log c d L pple < L. Let = D. Recll the Legedre formul [5] tht p () =. Ths mmedtely gves the upper oud. The lower oud follows from pple ( ) log ( + ), whch s the sme s /( ) pple +. Defe F, cosdered s rel-vlued fucto of the d j, to e F (d,..., d L ) = ( ) log + L j=0 d j j L j=0 d j. The the equlty we seek s equvlet to F (d,..., d L ) 0. But F s cocve fucto d so ts mmum must occur t extreme pot of [0, ] L,.e., t the vector where ech etry s ether 0 or. Now, f such vector hs oe or more etres equl to 0, the the clmed equlty holds ecuse /( ) pple L pple, whch mes tht F 0. If oe re 0 the ll re, whch mes = L ; the /( ) = L = +, whch g mes F 0. Hece F 0 holds ll cses.! - - ( + ) Fgure. The ouds o the power of! Lemm 4 re frly shrp. Next we fd uversl cogruece codto for K.

6 INTEGERS: 6 (06) 6 Lemm 5. For N d prme Z wth gcd (, ) =, let p = p (), q =!, d L = mod ( q, ). The K p K 0 (mod p ). L p mod p+ ; t follows tht Proof. We hve K = P k=0 k k! k!. The proof wll proceed y showg tht the frst term the sum s cogruet to L mod p p+ whle the others re ech dvsle y p+. The frst term s! = p q. The! L p = q p L p = p ( q L), whch s dvsle y p+ y the defto of L; therefore! L p mod p+, s clmed. For the secod result, t su ce to show tht k hs t lest oe more t th does k!. Ths follows from the upper oud of Lemm 4: p (k) pple k pple k. ± Now we c use the two lemms to derve exct formul for K. The m pot s tht the Mclur error s out + whle the smple modulus p from Lemm 4 s out. For the modulus to exceed the error, we eed to ether fd lrger modulus for cogruece codto or reduce the Mclur error. The ltter pproch works well, where we smply dd oe more term to the prtl sum to etter estmte the tructo error. Theorem 6 (The cse = ±) Assume = ± d, N wth d (, ) 6= (, 3). Let r = pple!e / + ( + ), 3, odd, m = d log (+)e, d s = sg(). The K = r s mod m (sr) d +, = e / (r s mod m (sr)). Proof. By Lemm 4, m s lower oud o the power of!; therefore K 0(mod m). The theorem strts wth r d the djusts t, the proper drecto, so tht t ecomes dvsle y m. Note tht the djustmet drecto depeds oly o the sg of d ot o the prty of. Now, the tructo error oud (see Thm. ; the ecessry equlty ( + ) > lwys holds) whe usg!e ±/ for ±!e s + (+). But the ehced form Theorem 6 s sttemet usg oe ddtol seres term mproves the oud to P B = ++j + j= (+) j pple + (+)((+) ). Therefore B + ouds the d erece etwee r d K. Becuse y tervl of legth m cots exctly oe teger dvsle y m, the result follows from B + pple m, whch tur follows from the specl cse where =. By replcg d log ( + )e wth log ( + ) m, t s esy to see tht, for = d y 4, we hve B + pple m. Detls: B pple m follows from log (+) + (+) +, whch follows from log (+) +, whch s equvlet to log ( + ) + ( log ), whch smplfes to pple log +

7 INTEGERS: 6 (06) 7 or 4 pple +. Ths lst s equvlet to 5. The equlty B + pple m s esly checked whe s 3 or 4 except for the cse (, ) = (, 3), where t fls. Oe c prove verso of Theorem 6 where the tl pproxmto s the smpler r =!e ±/ y usg Lemm 5 s stroger cogruece codto ± K p mod p+, where p = p () (ote tht p p mod p+ ). But ths requres the exct computto of p, whch s voded Theorem 6. Asde: Usg the ltertg error for the cse gves smller B of + (+)(+), ut ths hs o essetl mpct. Exmple. Suppose =, =, = 0. The r = d m = d0 log e = The rouded Mclur-polyoml-plus-oeterm error oud s out , well uder m. The true vlue of ths error s!e /! e The mod-m resdue of r s 9940; ths grees to the erest teger wth the true error. Ideed, ths s the crux of the whole method: the rouded error equls the mod-m vlue of r. Now r mod m r = d so we coclude tht the exct vlue of e 0 (), the Mclur polyoml, s e 0 () = , where the deomtor s 0!. I lowest terms ths s e 0 () = , whch grees wth the sum of the terms defg e 0 (). Ad ths gves (, ) = e. All ths works qute quckly whe s lrge: K () s teger wth dgts d the formul fds t well uder oe secod. For K 0 6() the formul works 30 secods whle the ve sum tkes mutes. All tmgs here re usg Mthemtc o Apple Mc wth 4 GHz processor. ± Here s Mthemtc mplemetto of the formul for K, where 4 d s odd. Note tht mod( ) works o rel umers d sce r mod m r s dvsle y m t follows tht oe c elmte the Roud operto from the formul, s well s from the code tht follows. r - s Mod r s, d - Log[+]e /. r!! E / - (+) +,s! Sg[] 4. The Cse 3 For prme -vlues eyod the method tht works so well whe pple fls ecuse the tructo error s roughly whle the power of!, the crux of

8 INTEGERS: 6 (06) 8 the cogruetl method whe pple, s oly out /. We c stll exted the method so tht t mproves clssc lgorthms, ut the g s less strkg th the smller cses. The followg method pples to y rtol, cludg composte. For smplcty we wll use the Lgrge error oud, whch s vld ll cses: B = e / + (+). We eed modulus m B. If we try for oe of the form m = w!, the the smllest w s esy to fd d t s geerlly (ut ot lwys) less th. For exmple, ths method for K (3) hs w = 969 s the smllest possle modulus, well uder. But for K 900 (000) the smllest w tht works s 87; eg lrger th, t s useless. A good estmte of w ecuse l(!) > l, ths estmte s ever less th the smllest w d so c e used drectly lgorthm s e +W (log(b)/e), where W s the Lmert W -fucto (whch, comed wth Strlg s pproxmto l(!) l, c serve s pproxmte verse to the fctorl). Ths pproxmto gves 969. d 87.5 for the two precedg log. exmples. Some symptotc lyss shows tht w To fsh, we eed R, the mod-w! resdue of K (/). Wth w d R hd d usg s = sg() + d r =! e /, we the hve sgle formul tht pples to ll cses: K = r s mod w! (s(r R)). Becuse ech of the tl w + terms of the sum defg K re dvsle y w!, computg R requres summg oly the terml w terms of the sum, d ths s where the tme g rses. The sum computto for K 00,000 (3) requres ddtos, so the reducto to 969 terms s susttl svgs. We summrze the method the followg theorem. Theorem 7. For y tegers,, wth, 0, d d reltvely prme, K = r s mod w! (s(r R)), where s = sg() +, =, B 0 = / + ( + ) l l l( + ), r =! e /, w = e +W ( B0/e), d R = mod w!! P k= w+ k k!. Whle some reducto the modulus s possle y tkg ddtol cogrueces to ccout, the g s smll. The smple lgorthm just descred requres out oe secod to compute K (3). Drect computto of the Mclur polyoml tkes te secods. Here s Mthemtc code tht works for ll rtols (though whe pple the work of the erler sectos should e used sted). It s frly smple d s fster th kow lgorthms. K[,,] computes K y smple ddto of + terms. y ddto of the rel- KTlModFctorl[,,,w] computes mod w! K evt tl terms oly. KGeerl[,,] computes K (ssumg gcd(, ) = ) y usg fctorl cogruece to overcome the tructo error. If < 0, the ltertg seres error

9 INTEGERS: 6 (06) 9 s used; otherwse ether the geometrc seres oud s used (whe t coverges) or the Lgrge oud. Ad the Roud the defto of r hs ee dropped, sce the modulr reducto of oteger vlues of r yelds the sme thg. K[,, ]:=K h Numertor h, Deomtor h, /;GCD[,] 6= ; K[,, ]:=Module {sum, t}, sum = t =! ; Do[sum+=(t*=/( k)), {k,}]; sum ; h KTlModFctorl[,,, w ]:=Module {0, t, =0}, 0 = -w+;t=powermod[, 0, w!]! w-. 0!; Do[ +=t;t*=/((k+)), {k, 0, }]; Mod[, w!] ; h KGeerl[,, ]:=Module = As[],s=Sg[] + o,l,logb,w,r, L=(+)Log[]; logb = Whch[<0,L-Log[(+)], (+), / +L-Log[(+)], True, L-Log[(+)-] ; m le +ProductLog[logB/E] ; h If LogGmm[]<logB, K[,,],w= s=sg[] + ; r=!e / ; r-s Mod[s(r-KTlModFctorl[,,,w]),w!] /; GCD[,]== KGeerl[3,,00000]//Short//Tmg {.787, << >> } For K 0 6(3) the precedg code tkes 90 secods to fd the 5.5-mllo-dgt teger; the modulus defg the umer of terms the tl sum s w = 040. There re umer of trcks tht c e used to mprove the lgorthm. Oe c reduce the tructo error tervl y ddg more terms or cosderg lower oud o the error s well, d oe c cosder ddtol cogrueces (such s, whe s prme, the oe from Lemm 5); ut these des mprove thgs oly lttle. So the questo rems whether there re formuls or lgorthms tht would compute the symolc vlue of K (z) for y rtol z more quckly th the modulr method of Theorem 7. Ackowledgemets. The uthor thks Smo Plou e d the lte Joth Borwe for some vlule suggestos, d Roert Isrel for fdg the key de the proof of Lemm 4 the geerl cse. Refereces. The O-Le Ecycloped of Iteger Sequeces, Lrry Crter d St Wgo, The Mes Correctol Fclty, preprto.

10 INTEGERS: 6 (06) 0 3. Smo Plou e, Exct formuls for teger sequeces, Notes (993), 4. Erc Wesste, Lmert W -fucto, from MthWorld A Wolfrm we resource, 5. A. M. Legedre, Théore des Nomres, Frm Ddot Frères, Prs, 830, 0.