Q-analogue of a Linear Transformation Preserving Log-concavity

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1 Iteratoal Joural of Algebra, Vol. 1, 2007, o. 2, Q-aalogue of a Lear Trasformato Preservg Log-cocavty Daozhog Luo Departmet of Mathematcs, Huaqao Uversty Quazhou, Fua , P. R. Cha ldzblue@163.com Abstract I ths paper we prove that the lear trasformato y, 0, 1, 2,... preserves Log covety. As a applcato, we vestgate the sequece S. Keywords: Log-cocavty; Log-covety; Lear trasformatos 1 Itroducto Log-cocave ad Log-cove sequeces arse ofte combatorcs, algebra, probablty ad statstcs. There has bee a cosderable amout of research devoted to ths topc recet years. Let { } 0 be a sequece of o-egatve real umbers. We say that { } s Log-cocave Log-cove resp.) f ad oly f resp.) for all 1 relevat results ca see 2 ad 4). For our purpose, whe a sequece s sad to be Log-cocave or Log-cove we always assume that t has o teral zeros,.e., there are o three dces <<ksuch that, k 0 ad 0. Ths s a atural assumpto for sequeces sce most of the Log-cocave ad Log-cove sequeces of terest to us actually meet the codto. Thus a sequeces s Log-cocave or Logcove) f ad oly f 1 +1 or 1 +1 ) for ay 1 see 1 ). Let f),g) R, we say f) g) f ad oly f f) g) has oegatve coeffcets. We say that a lear trasformato y a,, 1, 2,... 0

2 88 Daozhog Luo preserves log-cocavty f the log-cocavty of { } mples the log-cocavty of {y } ad preserves Log-covety f the Log-covety of { } mples the Log-covety of {y }. So far there have bee foud some mportat lear trasformatos preservg log-cocavty. For eample, t s well kow that the lear trasformato y ) 1) preserves log-cocavty see 1). Recetly, Ehreborg ad Stegrmsso showed that the lear trasformato y ) 1+ 2) preserves log-cocavty see 3). Whereafter Y Wag geeralzed the result as followg: y preserves Log-cocavty see 6). From 5 we ca kow that the q-bomal coeffcet ) m + 3) +!!-! defed: where k! 12 k ad 1+q + q q 1. It s well kow that t has the followg recurso: q 1 4) ad q 1 5) The ma obect of the preset paper s twofold. frst, we gve a q-aalogue of 1). Secodly, we vestgate the sequece S.

3 Q-aalogue of a lear trasformato 89 2 The ma result Theorem 2.1 Let q 1 be a real umber. The the lear trasformato y, 0, 1, 2,... preserves Log-covety. Before the proof of Theorem 2.1 we gve two lemmas. Lemma 2.2 Let k 0 ad 1 be tegers ad let q 1 be a real umber. The we have the followg equalty: ) q +1 k ) +1 6) Proof. We argue by ducto o. If 1, the left of the equalty 6) s +1+q + ) + 1+qk + +1 ad the rght of the equalty 6) s 1+ 1+qk It s obvous that 7) 8). Suppose that the equalty 6) holds for 1. The { ) 1 +q +k 1 { 1 +q 1 { , 7) 1+q + qk q. 8) k q +1 k ) q 1 k ) + q +k+1 q )+1 ) 1 1 )+1 qk q 1 k ) + q 2+k q ) 1 + q ) 1 1 ) )+1 qk+1 q )+1 q )+1 )+1 qk+1 } )+1 qk+1 } q )+1 }

4 90 Daozhog Luo +q { 1 1 +q k 1 { + 2 q 2 k { 1 q ) q ) ) ) 1 +1 ) + 1 ) 1 1 ) qk+1 ) qk+1 q ) } } } q ) By the ducto assumpto every term of the above sum s a polyomal o wth oegatve coeffcets, we thus obta that as desred ) q +1 k ) +1, Lemma 2.3 Suppose the sequeces { } ad {a } satsfy the followg two codtos: ) ) a 0, for all 0. The a 0. Proof. It s easy to see that a a 0 + a a ) + 1 )a 0 + a a 1 ) )a 0, from whch the statemet follows. Proof of Theorem 2.1. Wthout loss of geeralty, we assume m. Let { } be a Log-cove sequece ad {y } the sequece defed by 1). Obvously, the sequece {y } s oegatve ad has o teral zeros. It remas to show that y 1 y +1 y 2 0 for each 1. For u v, let C uv be the coeffcet of u v. For coveece, set 0f<0ad C uv 0fu<0oru>v. Deote S k C,k k, r k/2 where { 0 f 0 k r k 1 otherwse

5 Q-aalogue of a lear trasformato 91 The we have 0 k 2 S k. So t suffces to show that S k s oegatve for each k. The Log-covety of { } mples k +1 k 1. The we oly eed to prove that C,k s satsfy ) of Lemma 2.3 If k 2, the S k I what follows we cosder the case k<2. Let C k C,k. r k/2 Frst, we show that C k s oegatve for all 0 k<2. Observe that the geeratg fucto of C k s { 2 +1 }{ +1 1 } { } 2 1 C k k. k0 Deote The f +1 ) f ) f )+q +1 q +1 1 q )+1. ad so C k k 1 1 k0 f 1 ) f )+q +1 q )+1 + f ) f 1 1)+q 1 q )+1 q +1 f 1 ) 1 q q 1 1 q )+1 q f ) q ) q )+1 q )+1 +1.

6 92 Daozhog Luo By the Lemma 2.2 we obta that C k 0 holds for each k. Net, we demostrate that C,k s satsfy Lemma 2.3. Whe 0 k<, 0 <k/2,k, 1 <k/2, ad <k<2, r +2 <k/2. we have C,k k 1 k +1 2 k 1! + 1!!k! +1 k +! +1! A k ad for k/2 where C,k !! 2! 2 +1! 2 A 2 A +1 k + +1 k ) k +. Thus C,k has the same sg as that of A. Note that the dervatve of A wth respect to s { )2 k) whe q 1 A l q q 1) q k+ q ) { q +1 1)2qq k+ + q ) q 1+2qq 1) } whe q > 1 3 It s obvous that A 0. Fally, we prove that C,k s satsfy Lemma 2.3. If 0 k<, for 0 k/2,c,k s decreasg ad C k r k/2 C,k 0, the C,k s satsfes Lemma 2.3. If k, C 0,k q + + q ) 2 0 ad for 1, C,k s decreasg, C k 0 k/2 C,k 0, the C,k s satsfes Lemma 2.3. If k, 1 C r,k r > 0 r

7 Q-aalogue of a lear trasformato 93 ad C r,k r + C r+1,k r r 1 r r +1 1! {r r 1 2} > 0. r + 1! r 1! For r +2 k/2,c,k s decreasg, ad C k 0 k/2 C,k 0. The C,k s satsfes Lemma 2.3. Thus we complete the proof. Remark 2.4 Let q 1we ca deduce that the lear trasformato ) y preserves Log-covety. Thus the trasformato preserves both Log-cocavty ad Log-covety. Though we gve a q-aalogue of 1), we fd that the lear trasformato 2) ad 3) do t eoy ths property. Now, we gve a applcato of Theorem 2.1. Corollary 2.5 Let q be a real umber ad suppose q 1. The the sequece S,0, 1,... s log cove. Proof. I Theorem 2.1, suppose 1. From the defto ad recurso 5) we have: S q 1 + q S + q. The S +1 S q. 9) By corollary 2.5, we kow that S 2 S 1S +1. The followg theorem s a deeper relato betwee them.

8 94 Daozhog Luo Theorem 2.6 Let q be a real umber, ad suppose q 1. The for 1, S satsfy: Proof. By corollary 2.5, we have S 1 S +1 qs 2 qs 1 S +1 qs 2 qs 1S +1. I Lemma 2.2, suppose k 1, q. We have q q q. 10) By 9) ad 10) we obta: qs +1 S +1 S ) S S +2 S +1 ). qs 2 +1 S S +2 q 1)S S Thus we complemet the proof. Refereces 1 F. Bret. Umodal, Log-cocave ad Polya Freqecy Seqeces Combatorcs, Memors of the Amerca Math. Socety, No. 413, Amer. Math. Soc, Provdece, RI F. Bret, Log-cocave ad Umodal Sequeces Algebra, Combatorcs, ad Geometry: A Update, Cotemp. Math., ): R. Ehreborg, E. Stegrmsso, The Ecedece Set of a Permutato, Adv. Appl. Math ): R. P. Staley, Log-cocave ad Umodal Sequeces Algebra, Combatorcs, ad Geometry A. New York Acad. Sc., ): R. P. Staley, Eumeratve Combatorcs. Volum 1. Cambrdge Uversty Press Y Wag, Lear Trastormatos Preservg Log-cocavty. Lear Algebra Appl ): Receved: August 17, 2006