Aales Mathematicae et Iformaticae 43 2014 pp. 3 12 http://ami.etf.hu The log-cocavity ad log-covexity properties associated to hyperpell ad hyperpell-lucas sequeces Moussa Ahmia ab, Hacèe Belbachir b, Amie Belhir b a UFAS, Dep. of Math., DG-RSDT, Setif 19000, Algeria ahmiamoussa@gmail.com b USTHB, Fac. of Math., RECITS Laboratory, DG-RSDT, BP 32, El Alia 16111, Bab Ezzouar, Algiers, Algeria haceebelbachir@gmail.com or hbelbachir@usthb.dz ambelhir@gmail.com or ambelhir@usthb.dz Submitted July 22, 2014 Accepted December 12, 2014 Abstract We establish the log-cocavity ad the log-covexity properties for the hyperpell, hyperpell-lucas ad associated sequeces. Further, we ivestigate the q-log-cocavity property. Keywords: hyperpell umbers; hyperpell-lucas umbers; log-cocavity; q-logcocavity, log-covexity. MSC: 11B39; 05A19; 11B37. 1. Itroductio Zheg ad Liu [13] discuss the properties of the hyperfiboacci umbers F [r] ad the hyperlucas umbers L [r]. They ivestigate the log-cocavity ad the log covexity property of hyperfiboacci ad hyperlucas umbers. I additio, they exted their wor to the geeralized hyperfiboacci ad hyperlucas umbers. 3
4 M. Ahmia, H. Belbachir, A. Belhir The hyperfiboacci umbers F [r] ad hyperlucas umbers L [r], itroduced by Dil ad Mező [9] are defied as follows. Put F [r] = L [r] = F [r 1], with F [0] = F, L [r 1], with L [0] = L, where r is a positive iteger, ad F ad L are the Fiboacci ad Lucas umbers, respectively. Belbachir ad Belhir [1] gave a combiatorial iterpretatio ad a explicit formula for hyperfiboacci umbers, /2 F [r] +1 = + r + r. 1.1 Let U 0 ad V 0 deote the geeralized Fiboacci ad Lucas sequeces give by the recurrece relatio W +1 = pw + W 1 1, with U 0 = 0, U 1 = 1, V 0 = 2, V 1 = p. 1.2 The Biet forms of U ad V are U = τ 1 τ ad V = τ + 1 τ ; 1.3 with = p 2 + 4, τ = p + /2, ad p 1. The geeralized hyperfiboacci ad geeralized hyperlucas umbers are defied, respectively, by U [r] := V [r] := U [r 1], with U [0] = U, V [r 1], with V [0] = V. The paper of Zheg ad Liu [13] allows us to exploit other relevat results. More precisely, we propose some results o log-cocavity ad log-covexity i the case of p = 2 for the hyperpell sequece ad the hyperpell-lucas sequece. Defiitio 1.1. Hyperpell umbers P [r] defied by := ad hyperpell-lucas umbers P [r 1], with P [0] = P, are
The log-cocavity ad log-covexity properties... 5 := Q [r 1], with Q [0] = Q, where r is a positive iteger, ad P ad Q are the Pell ad the Pell-Lucas sequeces respectively. Now we recall some formulas for Pell ad Pell-Lucas umbers. It is well ow that the Biet forms of P ad Q are P = α 1 α 2 2 where α = 1 + 2. The itegers P, = 2 2 ad Q = α + 1 α, 1.4 ad Q, = 2 2, 1.5 are lied to the sequeces P ad Q. It is established [2] that for each fixed these two sequeces are log-cocave ad the uimodal. For the geeralized sequece give by 1.2, also the correspodig associated sequeces are log-cocave ad the uimodal, see [3, 4]. The sequeces P ad Q satisfy the recurrece relatio 1.2, for p = 2, ad for 0 ad 1 respectively, we have P +1 = /2 2 2 ad Q = /2 It follows from 1.4 that the followig formulas hold 2 2. 1.6 P 2 P 1 P +1 = 1 +1, 1.7 Q 2 Q 1 Q +1 = 8 1. 1.8 It is easy to see, for example by iductio, that for 1 P ad Q. 1.9 Let x 0 be a sequece of oegative umbers. The sequece x 0 is log-cocave respectively log-covex if x 2 j x j 1x j+1 respectively x 2 j x j 1x j+1 for all j > 0, which is equivalet see [5] to x i x j x i 1 x j+1 respectively x i x j x i 1 x j+1 for j i 1. We say that x 0 is log-balaced if x 0 is log-covex ad x /! 0 is log-cocave. Let q be a idetermiate ad f q 0 be a sequece of polyomials of q. If for each 1, f 2 q f 1 qf +1 q has oegative coefficiets, we say that f q 0 is q-log-cocave. I sectio 2, we give the geeratig fuctios of hyperpell ad hyperpell-lucas sequeces. I sectio 3, we discuss their log-cocavity ad log-covexity. We ivestigate also the q-log-cocavity of some polyomials related to hyperpell ad hyperpell-lucas umbers.
6 M. Ahmia, H. Belbachir, A. Belhir 2. The geeratig fuctios The geeratig fuctio of Pell umbers ad Pell-Lucas umbers deoted G P t ad G Q t, respectively, are ad G P t := G Q t := + =0 + =0 P t = Q t = t 1 2t t 2, 2.1 2 2t 1 2t t 2. 2.2 So, we establish the geeratig fuctio of hyperpell ad hyperpell-lucas umbers usig respectively = 1 + P [r 1] ad = 1 + Q[r 1]. 2.3 The geeratig fuctios of hyperpell umbers ad hyperlucas umbers are G [r] P t = =0 P [r] t t = 1 2t t 2 1 t r, 2.4 ad G [r] Q t = =0 Q [r] t 2 2t = 1 2t t 2 1 t r. 2.5 3. The log-cocavity ad log-covexity properties We start the sectio by some useful lemmas. Lemma 3.1. [12] If the sequeces x ad y are log-cocave, the so is their ordiary covolutio z = x y, = 0, 1,... Lemma 3.2. [12] If the sequece x is log-cocave, the so is the biomial covolutio z = x, = 0, 1,... Lemma 3.3. [8] If the sequece x is log-covex, the so is the biomial covolutio z = x, = 0, 1,... The followig result deals with the log-cocavity of hyperpell umbers ad hyperlucas sequeces. Theorem 3.4. The sequeces ad are log-cocave for r 1 ad r 2 respectively. 0 0
The log-cocavity ad log-covexity properties... 7 Proof. We have Whe = 1, ad 1.8 that The P [1] = 1 4 Q +1 2 ad Q [1] = 2P +1. 3.1 2 [1] P 1 +1 = 1 > 0. Whe 2, it follows from 3.1 2 [1] P 1 +1 = 1 [ ] Q +1 2 2 Q 2 Q +2 2 16 = 1 Q 2 +1 Q Q +2 4Q +1 + 2Q + 2Q +2 0 16 = 1 2 1 1 + Q +1 0. 4 is log-cocave. r 1 is log-cocave. It follows from 3.1 ad 1.7 that Hece Q [1] Q [1] By Lemma 3.1, we ow that 0 2 Q [1] 1 Q[1] +1 = 4 P 2 +1 P P +2 = 4 1 = ±4 3.2 0 Oe ca verify that The 0 is ot log-cocave. = 1 2 Q +2 2 = 2 +1. 3.3 is log-cocave. By Lemma 3.1, we ow that r 2 is log-cocave. This completes the proof of Theorem 3.4. The we have the followig corollary. Corollary 3.5. The sequeces log-cocave for r 1 ad r 2 respectively. [r] P 0 ad Proof. Use Lemma 3.2. Now we establish the log-cocavity of order two of the sequeces for some special sub-sequeces. 0 Theorem 3.6. Let be for 1 T := 2 [1] P 1 +1 ad R := [r] Q 2 Q [2] 1 Q[2] +1. The T 2 1, R 2+1 0 are log-cocave, ad T 2+1 0, R 2 1 are logcovex. 0 0 0 are ad
8 M. Ahmia, H. Belbachir, A. Belhir Proof. Usig respectively 3.3 ad 1.8, we get 2 Q [2] 1 Q[2] +1 = 2 1 + Q +1, ad thus, for 1, T = 1 4 2 1 1 + Q ad R = 2 1 + Q +1. 3.4 By applyig 3.4 ad 1.8, for 1 we get Q 2 2 Q 2 2 Q 2+2 = 32 ad Q 2 2+1 Q 2 1 Q 2+3 = 32. 3.5 The T2 2 T 2 1 T 2+1 = 1 Q 2 16 2 Q 2 2 Q 2+2 4Q 2 + 2Q 2 2 + 2Q 2+2 = 4Q 2 4 > 0. ad R 2 2+1 R 2 1 R 2+3 = Q 2 2+2 Q 2 Q 2+2 4Q 2+2 + 2Q 2 + 2Q 2+4 = 64Q 2+2 4 > 0. The T 2 1 ad R 2+1 0 are log-cocave. ad Similarly by applyig 3.4 ad 3.5, we have T 2 2+1 T 2 1 T 2+3 = 1 2 Q 2+1 < 0, R 2 2 R 2 1 R 2+1 = 8Q 2+1 < 0. The T 2+1 0 ad R 2 1 are log-covex. This completes the proof. Corollary 3.7. The sequeces T2 0 ad R2+1 log-cocave. Proof. Use Lemma 3.2. Corollary 3.8. The sequeces T2+1 1 ad log-covex. Proof. Use Lemma 3.3. R2 0 are 1 are Lemma 3.9. Let a := P+1, where P 0 is the Pell sequece. The a 0 satisfy the followig recurrece relatios 2 a = 3a 1 + a ad a = 4a 1 2a 2.
The log-cocavity ad log-covexity properties... 9 Proof. Let be b := P, where P 1 is the Pell sequece exteded to P 1 = 1. Usig Pascal formula ad the recurrece relatio of Pell sequece together ito the developmet P+1 we get a = 3a 1 + b 1, the by b = b 1 + a 1. By iterated use of this relatio with the precedet oe, we get a = 3a 1 + 2 a with b 0 = 0 ad a 0 = 1, thus a = 4a 1 2a 2. Theorem 3.10. The sequeces are logcocave ad log-covex, respectively. Proof. Let be Q [1] 0 ad 2 S := 2 Q [1] 2 1Q [1] 1 Q[1] +1 ad K := with the covetio that K <0 = 0. From 3.2, we have S = 4 2 1 1 + The Q [1] 2 [1] Q 0 Q [1], = 4 [ 2 1 1 + P+1] 2 [ 4 2 1 1 + + 1 2] > 0. is log-cocave. Q [1] 0 Usig Lemma 3.9, we ca verify that K = 4K 1 2K 2. 3.6 The associated Biet-formula is 1 + 2 α 1 2 β K =, with α, β = 2 ± 2, α β which provides The [1] Q 0 K 2 K 1 K +1 = 2 +1 < 0. is log-covex. Remar 3.11. The terms of the sequece K satisfy K = 2 +2/2 P +1 if is eve, ad K = 2 1/2 Q +1 if is odd. Theorem 3.12. The sequeces ad are log-balaced.! 0! Proof. By Theorem 3.4, i order to prove the log-balaced property of!p [1] 0 ad we oly eed to show that they are log-covex. It follows from! 0 the proof of Theorem 3.4 that 2 P [1] 1 +1 = 1 4 0 2 1 1 + Q +1, 3.7
10 M. Ahmia, H. Belbachir, A. Belhir ad from the proof of Theorem 3.6 that 2 Q [2] 1 Q[2] +1 = 2 1 + Q +1. 3.8 Let M := B := 2 + 1P [1] 1 +1, 2 + 1Q [2] 1 Q[2] +1, from 3.3, 3.7 ad 3.8, we get M = + 1 4 2 1 1 + Q +1 1 4 Q +1 2 2, B = + 1 2 1 + Q +1 1 4 Q +2 2 2. Clearly B 0 for = 0, 1, 2. We have by iductio that for 1, Q + 1. This gives B Q +1 1 2 1 + Q +1 1 4 2Q +1 + Q 2 2 < 0. Also, M 0 for = 2 ad for 3, Q + 6. This gives + 1 Q +1 6, ad M 1 [ Q +1 6 2 1 1 + Q +1 Q +1 2 2] 4 = 1 [ 2 + 2 1 1 Q +1 4 12 1 1] < 0. 4 Hece!P [1] 0 ad! 0 are log-covex. As the sequeces P [1] 0 ad 0 are log-cocave, so the sequeces!p [1] 0 ad! 0 are log-balaced. Theorem 3.13. Defie, for r 1, the polyomials P,r q := q ad Q,r q := q. The polyomials P,r q r 1 ad Q,r q r 2 are q-log-cocave. Proof. Whe 1, r 1, P,rq 2 P 1,r qp +1,r q 2 1 = q q +1 q
The log-cocavity ad log-covexity properties... 11 = = = =1 q 2 q +1 q+1 1 +1 q q q + P [r] +1 q2+1 q +. Whe 1, r 2, through computatio, we get Q 2,rq Q 1,r qq +1,r q = As ad =1 r 1 ad Q,r q r 2 are q-log-cocave. q + +1 q+1 Q[r] 1 Q[r] +1 q + + q. r 2 are log-cocave, the the polyomials P,r q Acowledgemets. We would lie to tha the referee for useful suggestios ad several commets witch ivolve the quality of the paper. Refereces [1] Belbachir, H., Belhir, A., Combiatorial Expressios Ivolvig Fiboacci, Hyperfiboacci, ad Icomplete Fiboacci Numbers, J. Iteger Seq., Vol. 17 2014, Article 14.4.3. [2] Belbachir, H., Becherif, F., Uimodality of sequeces associated to Pell umbers, Ars Combi., 102 2011, 305 311. [3] Belbachir, H., Becherif, F., Szalay, L., Uimodality of certai sequeces coected with biomial coefficiets, J. Iteger Seq., 10 2007, Article 07. 2. 3. [4] Belbachir, H., Szalay, L., Uimodal rays i the regular ad geeralized Pascal triagles, J. Iteger Seq., 11 2008, Article. 08.2.4. [5] Breti, F., Uimodal, log-cocave ad Pólya frequecy sequeces i combiatorics, Mem. Amer. Math. Soc., o. 413 1989. [6] Cao, N. N., Zhao, F. Z, Some Properties of Hyperfiboacci ad Hyperlucas Numbers, J. Iteger Seq., 13 8 2010, Article 10.8.8. [7] Che, W. Y. C., Wag, L. X. W., Yag, A. L. B., Schur positivity ad the q-log-covexity of the Narayaa polyomials, J. Algebr. Comb., 32 2010, 303 338. [8] Daveport, H., Pólya, G., O the product of two power series, Caadia J. Math., 1 1949, 1 5. [9] Dil, A., Mező, I., A symmetric algorithm for hyperharmoic ad Fiboacci umbers, Appl. Math. Comput., 206 2008, 942 951. [10] Liu, L., Wag, Y., O the log-covexity of combiatorial sequeces, Advaces i Applied Mathematics, vol. 39, Issue 4, 2007, 453 476.
12 M. Ahmia, H. Belbachir, A. Belhir [11] Sloae, N. J. A., O-lie Ecyclopedia of Iteger Sequeces, http://oeis.org, 2014. [12] Wag, Y., Yeh, Y. N., Log-cocavity ad LC-positivity, Combi. Theory Ser. A, 114 2007, 195 210. [13] Zheg, L. N., Liu, R., O the Log-Cocavity of the Hyperfiboacci Numbers ad the Hyperlucas Numbers, J. Iteger Seq., Vol. 17 2014, Article 14.1.4.