Research Article The Peak of Noncentral Stirling Numbers of the First Kind

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1 Iteatioal Joual of Mathematics ad Mathematical Scieces Volume 205, Aticle ID 98282, 7 pages Reseach Aticle The Peak of Nocetal Stilig Numbes of the Fist Kid Robeto B. Cocio, Cistia B. Cocio, ad Pete Joh B. Aaas 2 Mathematics ad ICT Depatmet, Cebu Nomal Uivesity, 6000 Cebu City, Philippies 2 Depatmet of Mathematics, Midaao State Uivesity, Mai Campus, 9700 Maawi City, Philippies Coespodece should be addessed to Robeto B. Cocio; cocio@yahoo.com Received 8 Septembe 204; Accepted 20 Novembe 204 Academic Edito: Seka Aaci Copyight 205 Robeto B. Cocio et al. This is a ope access aticle distibuted ude the Ceative Commos Attibutio Licese, which pemits uesticted use, distibutio, ad epoductio i ay medium, povided the oigial wok is popely cited. We locate the peak of the distibutio of ocetal Stilig umbes of the fist kid by detemiig the value of the idex coespodig to the maximum value of the distibutio.. Itoductio I 982, Koutas [] itoduced the ocetal Stilig umbes of the fist ad secod kid as a atual extesio of the defiitio of the classical Stilig umbes, amely, the expessio of the factoial (x) i tems of powes of x ad vice vesa. These umbes ae, espectively, deoted by s a (, k) ad S a (, k) which ae defied by meas of the followig ivese elatios: (t) = k! [ dk dt (t) (t a) k, k () V]t=a (t a) = k! [Δk (t a) ] t=0 (t) k, (2) whee a, t ae ay eal umbes, is a oegative itege, ad s a (, k) = k! [ dk dt (t), k V]t=a S a (, k) = k! [Δk (t a) ] t=0. The umbes satisfy the followig ecuece elatios: (3) s a (+,k) =s a (, k ) + (a ) s a (, k), (4) S a (+,k) =S a (, k ) + (k a) S a (, k) (5) adhaveiitialcoditios s a (0, 0) =, s a (, 0) = (a), s a (0, k) =0,,, S a (0, 0) =, S a (, 0) = ( a), S a (0, k) =0,,. (6) It is woth metioig that fo a give egative biomial distibutio Y ad the sum X = X +X 2 + +X k of k idepedet adom vaiables followig the logaithmic distibutio, the umbes s a (, k) appeaed i the distibutio of the sum W = X + Y,whiletheumbesS a (, k) appeaed i the distibutio of the sum Z= X + Y whee X is the sum of k idepedet adom vaiables followig the tucated Poisso distibutio away fom zeo ad Y is a Poisso adom vaiable. Moe pecisely, the pobability distibutios of W ad Z ae give, espectively, by k! θ P [W =] = ( θ) s ( log ( θ)) k! ( ) k s s (, k), k! l P [Z =] = e m (e l ) k! ( ) k S m/l (, k). Fo a moe detailed discussio of ocetal Stilig umbes, oe may see []. Detemiig the locatio of the maximum of Stilig umbes is a iteestig poblem to coside. I [2], Mezö (7)

2 2 Iteatioal Joual of Mathematics ad Mathematical Scieces obtaied esults fo the so-called -Stilig umbes which ae atual geealizatios of Stilig umbes. He showed that the sequeces of -Stilig umbes of the fist ad secod kids ae stictly log-cocave. Usig the theoem of Edös ad Stoe [3] he was able to establish that the lagest idex fo which the sequece of -Stilig umbes of the fist kid assumes its maximum is give by the appoximatio K (), =+[log ( ) +o()]. (8) Followig the methods of Mezö, we establish stict logcocavity ad hece uimodality of the sequece of ocetal Stilig umbes of the fist kid ad, evetually, obtai a estimatig idex at which the maximum elemet of the sequece of ocetal Stilig umbes of the fist kid occus. 2. Explicit Fomula I this sectio, we establish a explicit fomula i symmetic fuctio fom which is ecessay i locatig the maximum of ocetal Stilig umbes of the fist kid. Let f i (x), i=,2,...,be diffeetiable fuctios ad let F (x) = i= f i(x). It ca easily be veified that, fo all 3, F (x) = f < 2 < < i N \{, 2,..., } i (x) k= f k (x). Now, coside the followig deivative of (ξ + a) whe =, 2: d dξ (ξ + a) =, d dξ (ξ + a) 2 =(ξ+a)+(ξ+a ). The, fo 3ad usig (9),weget d dξ (ξ + a) = 0 < 2 < < k= The, we have the followig lemma. (9) (0) (ξ+ a k ). () Lemma. Fo ay oegative iteges ad k,oehas d k dξ (ξ + a) k = 0 < 2 < < k k k! q= (ξ + a q ). (2) Poof. We pove by iductio o k. Fok = 0, (2) clealy holds. Fo k=, (2) ca easily be veified usig ().Suppose fo m, d m dξ m (ξ + a) = 0 < 2 < < m m m! q= (ξ + a q ). (3) The, d m+ dξ m+ (ξ + a) =m! 0 < 2 < < m m d dξ q= (ξ + a q ), (4) whee the sum has ( m ) = ( )( 2) ( m + )/m! temsaditssummad m d dξ q= (ξ + a q )= i <i 2 < <i m m q= (ξ + a i q ), (5) i q {, 2,..., m } has ( m m ) = ( m)( m )!/( m )! = m tems. Theefoe, the expasio of (d m+ /dξ m+ )(ξ + a) has a total of ( ) ( m + )( m)/m! tems of the fom m q= (ξ + (a q )/). Howeve,ifthesumisevaluatedoveallpossible combiatios 2 m such that 0 < 2 < < m, the the sum has ( m ) = ( ) ( m)( m!)/(m + )!( m )! = (/(m + ))(( ) ( m)/m!) distict tems. It follows that evey tem m q= (ξ + (a q )/) appeas m+times i the expasio of (/ )(d m+ /dξ m+ )(ξ + a).thuswehave d m+ dξ m+ (ξ + a) =m! 0 < 2 < < m m m+ q= = (m+)! 0 < 2 < < m m q= (ξ + a q ) (ξ + a q ). (6) Lemma 2. Let s(, k; a) = (/k!)lim ξ 0 (( k =0 ( )k ( k )(d k /dξ k )(ξ + a) )/k!). The s (, k; a) = Poof. Usig Lemma, s (, k; a) = k! lim ξ 0 (( k k 0 < 2 < < k q= (a q ). (7) ( ) k ( k =0 ) k! 0 < < k k q= (ξ + a q )) (k!) ). (8)

3 Iteatioal Joual of Mathematics ad Mathematical Scieces 3 Note that k q= (ξ + (a q)/) = (/ k ) k q= (ξ + a q). Hece, the expessio at the ight-had side of (8) becomes k k! lim [ ( ) k ( k ξ 0 =0 )k [ = k! [ [ k 0 < 2 < < k q= k =0 which boils dow to sice ( ) k ( k )k 0 < 2 < < k q= 0 < 2 < < k q= k (ξ + a q )] ] k k (a q )], ] (9) (a q ), (20) ( ) k ( k k! )k =S(k, k) =, (2) =0 whee S(, k) deote the Stilig umbes of the secod kid. Theoem 3. The ocetal Stilig umbes of the fist kid equal s a (, k) = s (, k; a) = Poof. We kow that k 0 < 2 < < k q= k+ 0 < 2 < < k+ q= (a q ). (22) (a q ) (23) isequaltothesumofthepoducts(a )(a 2 ) (a + k ) whee the sum is evaluated oveall possible combiatios 2 + k, i {0,,2,...,}.Thesepossiblecombiatios ca be divided ito two: the combiatios with i = fo some i {,2,..., k+}ad the combiatios with =fo all i {0,,2,..., k+}.thus i is equal to k+ 0 < 2 < < k+ q= k+ 0 < 2 < < k+ q= 0 < 2 < < k q= (a q ) (24) (a q )+(a ) k (a q ). (25) This implies that s (+,k;a) = s (,k ;a) + (a ) s (, k; a). (26) Thisisexactlythetiagulaecueceelatioi(4) fo s a (, k). This poves the theoem. The explicit fomula i Theoem 3 is ecessay i locatig the peak of the distibutio of ocetal Stilig umbes of the fist kid. Besides, this explicit fomula ca also be used to give cetai combiatoial itepetatio of s a (, k). A 0- tableau, as defied i [4]bydeMédicis ad Leoux, is a pai φ = (λ, f),whee λ=(λ λ 2 λ k ) (27) is a patitio of a itege m,adf=(f i ) λi is a fillig of the cells of coespodig Fees diagam of shape λ with 0 s ad s, such that thee is exactly oe i each colum. Usig the patitioλ=(5,3,3,2,)we ca costuct 60 distict 0- tableaux. Oe of these 0- tableauxisgiveithefollowig figue with f 4 =f 5 =f 23 =f 3 =f 42 =, f i =0elsewhee ( λ i ): (28) Also, as defied i [4], a A-tableau is a list φ of colum c of a Fees diagam of a patitio λ (by deceasig ode of legth) such that the legths c ae pat of the sequece A= (a i ) i 0, a i Z + {0}.IfTd A (h, ) is the set of A-tableaux with exactly distictcolumswhoselegthsaeitheseta= {a 0,a,...,a h },the Td A (h, ) = ( h+ ). Now, tasfomig each colum c of a A-tableau i Td A (, k) ito a colum of legth ω( c ), we obtai a ew tableau which is called A ω -tableau. Ifω( c ) = c, the the A ω -tableau is simply the A-tableau. Now, we defie a A ω (0, )-tableau to be a 0- tableauwhichiscostuctedbyfilligupthecellsof a A ω -tableau with 0 s ad s such that thee is oly oe i each colum. We use Td Aω(0,) (, k)todeote the set of such A ω (0, )-tableaux. Itcaeasilybeseethatevey( k) combiatio 2 k of the set {0,,2,..., } ca be epeseted geometically by a elemet φ i Td A (, k)with i as the legth of ( k i+)th colum of φ whee A = {0,,2,..., }.Hece,withω( c ) = a c, (22) may be witte as s a (, k) = φ Td A (, k) c φ ω ( c ). (29) Thus, usig (29), we ca easily pove the followig theoem. Theoem 4. The umbe of A ω (0, )-tableaux i Td A ω(0,) (, k) whee A={0,,2,..., }such that ω( c ) = a c is equal to s a (, k).

4 4 Iteatioal Joual of Mathematics ad Mathematical Scieces Let φ be a A-tableau i Td A (, k)with A = {0,,2,..., },ad ω A (φ) = ω ( c ) c φ = k i= (a i ), i {0,,2,..., }. If a=a +a 2 fo some a ad a 2,the,withω () = a 2, ω A (φ) = = k i= (a +ω ( i )) k a k =0 i q <q 2 < <q k i= ω (q i ). (30) (3) Suppose B φ is the set of all A-tableaux coespodig to φ such that fo each ψ B φ eithe ψ hasocolumwhoseweightisa,o ψ has oe colum whose weight is a,o. ψ has kcolums whose weights ae a.the,we may wite ω A (φ) = ψ B φ ω A (ψ). Now, if colums i ψ have weights othe tha a,the ω A (ψ) = a k i= (32) ω (q i ), (33) whee q,q 2,...,q {, 2,..., k }.Hece,(29) may be witte as s a (, k) = φ Td A (, k) ω A (ψ). ψ B φ (34) Note that fo each, theecoespod( k ) tableaux with distict colums havig weights w (q i ), q i {, 2,..., k }. Sice Td A (, k)has ( k ) elemets, fo each φ Td A (, k),thetotalumbeofa-tableaux ψ coespodig to φ is ( k )( k ) (35) times i the collectio. Cosequetly, we obtai s a (, k) = k =0 ( k )a k ψ B c ψ ω ( c ), (37) whee B deotes the set of all tableaux ψ havig distict colums whose legths ae i the set {0,,2,..., }. Reidexig the double sum, we get s a (, k) = =k ( k )a k ψ B ω ( c ). (38) c ψ Clealy, B =Td A (, ).Thus,usig(22),weobtai the followig theoem. Theoem 5. The umbes s a (, k) satisfy the followig idetity: s a (, k) = =k ( k )a k s a2 (, ), (39) whee a=a +a 2 fo some umbes a ad a 2. The ext theoem cotais cetai covolutio-type fomula fo s a (, k) which will be poved usig the combiatoics of A-tableau. Theoem 6. The umbes s a (, k) have covolutio fomula s a (m +, ) = s a (m, k) s a m (, k). (40) Poof. Suppose that φ is a tableau with exactly m kdistict columswhoselegthsaeitheseta ={0,,2,...,m } ad φ 2 is a tableau with exactly +k distict colums whose legths ae i the set A 2 = {m, m +, m + 2,..., m + }. The φ Td A (m,m k)ad φ 2 Td A 2 (, +k). Notice that by oiig the colums of φ ad φ 2,weobtaia A-tableau φ with m + distict colums whose legths ae i the set A={0,,2,...,m+ };thatis,φ Td A (m +,m+ ).Hece φ Td A (m+,m+ ) ω A (φ) elemets. Howeve, oly ( ) tableaux i B φ with distict colums of weights othe tha a ae distict. Hece, evey distict tableau ψ appeas ( k )( k ) ( =( ) k ) (36) { = { { { { { φ Td A (m,m k) φ 2 Td A 2 (, +k) ω A (φ ) } } } ω A2 (φ 2 ) } }. } (4)

5 Iteatioal Joual of Mathematics ad Mathematical Scieces 5 Note that φ 2 Td A 2 (, +k) ω A2 (φ 2 ) = m g <g 2 < <g +k m+ = 0 g <g 2 < <g k +k q= +k q= (a g q ) (a (m + g q )) (42) Now, coside the followig polyomial: s a (, k)(t+a) k. (47) This polyomial is ust the expasio of the factoial t = t(t + )(t + 2) (t + ) which has eal oots 0,, 2,..., +.Ifweeplacetby t a,weseeatocethat the oots of the polyomial s a(, k)t k ae a,a,...,a +. Applyig Newto s Iequality completes the poof of the followig theoem. Thus, =s a m (, k). Also, usig (29),wehave φ Td A (m,m k) φ Td A (m+,m+ ) s a (m+,) = ω A (φ )=s a (m, k) ω A (φ) = s a (m +, ). (43) s a (m, k) s a m (, k). (44) The followig theoem gives aothe fom of covolutio fomula. Theoem 7. The umbes s a (, k) satisfy the secod fom of covolutio fomula s a (+,m++) = s a (k, ) s a (k+) ( k,). (45) Poof. Let φ be a tableau with k mcolums whose legths ae i A ={0,,...,k }, φ 2 be a tableau with k colums whose legths ae i A 2 ={k+,...,}. The φ Td A (k, k m); φ 2 Td A 2 ( k, k ). Usig the same agumet above, we ca easily obtai the covolutio fomula. 3. The Maximum of Nocetal Stilig Numbes of the Fist Kid We ae ow eady to locate the maximum of s a (, k).fist,let us coside the followig theoem o Newto s iequality [5] which is a good tool i povig log-cocavity o uimodality of cetai combiatoial sequeces. Theoem 8. If the polyomial a x+a 2 x 2 + +a x has oly eal zeos the a 2 k a k k+a k k k+ k (k=2,..., ). (46) Theoem 9. The sequece {s a (, k)} is stictly log-cocave ad, hece, uimodal. as By eplacig t with t, theelatioi() may be witte t = ( ) k s a (, k)(t+a) k, (48) whee t = t(t + )(t + 2) (t + ). Notethat,fom Theoem 3 with a<0, s a (, k) = ( ) k k 0 < 2 < < k q= (b + q ), (49) whee b= a>0. Now, we defie the sigless ocetal Stilig umbe of the fist kid, deoted by s a (, k),as s a (, k) = ( ) k s a (, k) = k 0 < 2 < < k q= (b + q ). (50) To itoduce the mai esult of this pape, we eed to state fist the followig theoem of Edös ad Stoe [3]. Theoem 0 (see [3]). Let u < u 2 < be a ifiite sequece of positive eal umbes such that =, u i i= <. (5) i= ui 2 Deote by,k the sum of the poduct of the fist of them take k at a time ad deote by K the lagest value of k fo which,k assumes its maximum value. The K = [ u i i= i= ui 2 ( + ) +o()]. (52) u i We also eed to ecall the asymptotic expasio of hamoic umbes which is give by = log +γ+o(), (53) whee γ is the Eule-Mascheoi costat. The followig theoem cotais a fomula that detemies the value of the idex coespodig to the maximum of the sequece { s a (, k) }.

6 6 Iteatioal Joual of Mathematics ad Mathematical Scieces Theoem. The lagest idex fo which the sequece { s a (, k) } assumes its maximum is give by the appoximatio k a, =[log ( b+ )+o()], (54) b whee [x] is the itege pat of x ad b= a, a<0. Poof. Usig Theoem 0 ad by (50),weseethat s a (, k) =, k.deotigbyk a, fo which, k is maximum ad with u =b+0,u 2 =b+,...,u =b+ we have k a, =[ b+i (b+i) 2 ( + b+i ) +o()] Table : Values of s (, k). /k =[ =[ b+i (b+i)(b+i+) +o()] b+i+ +o()]. But usig (53),we see that (55) Agai, usig (53), weget k a, =[log ( a+ )+o()]. (60) a + = log (b+) log b. (56) b+i+ Example 3. The maximum elemet of the sequece {s (9, k)} 9 occus at (Table ) Fom this we get k a, =[log ( b+ )+o()]. (57) b k,9 =[log ( +8 )+o()] =[log 9+o()] (6) Fo the case i which a > 0 we will oly coside the sequece of ocetal Stilig umbes of the fist kid fo which a. Theoem 2. The maximizig idex fo which the maximum ocetal Stilig umbe occus fo a is give by the appoximatio k a, =[log ( a+ )+o()]. (58) a + Poof. Fom the defiitio, fo a,s a (, k) > 0 ad by Theoem 3, s a (, k) is the sum of the poducts (a )(a 2 ) (a k ) whee i saetakefomtheset{0,,2,..., }. ByTheoem 0, s a (, k) =, k.thuswithu =a,u 2 = a,...,u =a +we have k a, =[ a i (a i) 2 ( + a i ) +o()] =[ a i (a i)(a i+) +o()] (59) 2. Example 4. The maximum elemet of the sequece {s 0 (0, k)} 0 occus at (Table 2) k 0,0 =[log ( )+o()] =[log + o ()] 2. (62) We kow that the classical Stilig umbes of the fist kid ae special cases of s a (, k) by takig a=0.howeve, fomulas i Theoems ad 2 do ot hold whe a = 0. Hece, these fomulas ae ot applicable to detemie the maximum of the classical Stilig umbes. Hee, we deive a fomula that detemies the value of the idex coespodig to the maximum of the sigless Stilig umbes of the fist kid. The sigless Stilig umbes of the fist kid [6]aethe sum of all poducts of kdiffeet iteges take fom {,2,3,..., }.Thatis, =[ a i+ +o()]. s (, k) = i i 2 i k. i <i 2 < <i k (63)

7 Iteatioal Joual of Mathematics ad Mathematical Scieces 7 Table 2: Values of s 0 (, k). /k Table 3: Values of s(, k) fo 0 0. s(, k) Usig Theoem 0, s(, k) =, k.weusek to deote the lagest value of k fo which, k is maximum. With u =,u 2 =2,...,u = we have k =+[ =+[ i= i= i i 2 ( + i ) +o()] i= i i= =+[ i+ +o()]. i= Usig (53),we seethat ( i i+ )+o()] (64) = log log +γ+o(). (65) +i i= Theefoe, we have k = [log +γ+o()]. (66) ca be veified that the maximum elemet of the sequece { s(7, k) } 7 occus at k 7 =[log 7+γ+o()] = [ o ()] = [ o ()] 2. Moeove, whe =0,themaximumvalueoccusat k 0 =[log 0+γ+o()] = [ o ()] = [ o ()] 3. (67) (68) Recetly, a pape by Cakić et al.[7] established explicit fomulas fo multipaamete ocetal Stilig umbes which ae expessible i symmetic fuctio foms. Oe may the ty to ivestigate the locatio of the maximum value of these umbes usig the Edös-Stoetheoem. Coflict of Iteests The authos declae that thee is o coflict of iteests egadig the publicatio of this pape. Ackowledgmet The authos wish to thak the efeees fo eadig the pape thooughly. Refeeces [] M. Koutas, Nocetal Stilig umbes ad some applicatios, Discete Mathematics,vol.42,o.,pp.73 89,982. [2] I. Mezö, O the maximum of -Stilig umbes, Advaces i Applied Mathematics,vol.4,o.3,pp ,2008. [3] P. Edös, OacoectueofHammesley, Joualofthe Lodo Mathematical Society,vol.28,pp ,953. [4] A. de Médicis ad P. Leoux, Geealized Stilig umbes, covolutio fomulae ad p, q-aalogues, Caadia Joual of Mathematics,vol.47,o.3,pp ,995. [5] E. H. Lieb, Cocavity popeties ad a geeatig fuctio fo Stilig umbes, Joual of Combiatoial Theoy, vol. 5, o. 2, pp , 968. [6] L. Comtet, Advaced Combiatoics, Reidel, Dodecht, The Nethelads, 974. [7] N. P. Cakić, B. S. El-Desouky, ad G. V. Milovaović, Explicit fomulas ad combiatoial idetities fo geealized Stilig umbes, Mediteaea Joual of Mathematics,vol.0,o., pp , 203. Example 5. It is show i Table 3 that the maximum value of s(, k) whe = 7 occus at k = 2.Usig(66), it

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