/^creative ^commor ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 17 (2019) 125-139 https://doi.org/10.26493/1855-3974.1526.b8d (Also available at http://amc-journal.eu) Total positivity of Toeplitz matrices of recursive hypersequences* Tomislav Doslic Faculty of Civil Engineering, University of Zagreb, Zagreb, Croatia Ivica Martinjak Faculty ofScience, University ofZagreb, Zagreb, Croatia Riste Skrekovski FMF, University ofLjubljana, Ljubljana, Slovenia Faculty of Information Studies, Novo Mesto, Slovenia FAMNIT, University of Primorska, Koper, Slovenia Received 9 November 2017, accepted 12 March 2019, published online 6 September 2019 Abstract We present a new class of totally positive Toeplitz matrices composed of recently introduced hyperfibonacci numbers of the r-th generation. As a consequence, we obtain that (r) all sequences Fn of hyperfibonacci numbers of r-th generation are log-concave for r > 1 and large enough n. Keywords: Total positivity, totally positive matrix, Toeplitz matrix, Hankel matrix, hyperfibonacci sequence, log-concavity. Math. Subj. Class.: 15B36, 15A45 1 Introduction and preliminary results A matrix M is totally positive if all its minors are positive real numbers. When it is allowed that minors are zero, then M is said to be totally non-negative. Such matrices appears in many areas having numerous applications including, among other topics, graph theory, * Partial support of the Croatian Science Foundation through project BioAmpMode (Grant no. 8481) is gratefully acknowledged by the first author. All authors gratefully acknowledge partial support of Croatian-Slovenian bilateral project "Modeling adsorption on nanostructures: A graph-theoretical approach" BI-HR/16-17-046. The third author is partially supported by Slovenian research agency ARRS, program no. P1-0383. E-mail addresses: doslic@grad.hr (Tomislav Doslic), imartinjak@phy.hr (Ivica Martinjak), skrekovski@gmail.com (Riste Skrekovski) ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ 126 Ars Math. Contemp. 17 (2019) 291-310 Polya frequency sequences, oscillatory motion, symmetric functions and quantum groups among these areas [1, 2, 12, 13, 18]. The notion of total positivity is closely related with log-concavity and more on this one can find in a paper by Stanley [21]. A classical result by Whitney, Loewer and Cryer [8] says that any totally non-negative matrix M can be factored as a product of totally non-negative matrices M = L1 ■ ■ ■ LmDU1 ■ ■ ■ Um, where D is a diagonal matrix with non-negative elements, Li is a matrix of the form I + cEj+1j, Ui is a matrix of the form I + cEjij+i and Ek,i is the matrix which has a 1 on the k, l position and zeros elsewhere. There is also a connection between totally non-negative matrices and planar networks proved by Karlin and McGregor [15], and Lindstrom [16]. The famous Lindstrom lemma gives combinatorial interpretation of a minor through the weights of collections of vertex-disjoint paths in a planar network. An important notion when testing a matrix on total positivity is initial minor. We let I, J denote column set and row set, respectively. A minor A/j where both I and J consist of several consecutive indices and where I U J contain 1, is called initial. Thus, each matrix entry is the lower-right corner of exactly one initial minor. In this work we use Theorem 1.1, which is proved by Gasca and Peiia [14]. Theorem 1.1. A square matrix is totally positive if and only if all its initial minors are positive. The notion of total positivity can be refined as follows. A matrix M is said to be totally positive of order p (or TPp, in short) if all its minors of all orders < p are positive. The concept of total positivity extends in a straightforward manner also to (semi)infinite matrices. It turns out that many such triangular matrices appearing in combinatorics are indeed TP [3]. Recently, Wang and Wang proved total positivity of Catalan triangle via Aissen-Schonberg-Whitney theorem [22]. Further general results on triangular matrices and Riordan array have been obtained by Chen, Liang and Wang [5, 6] as well as Zhao and Yan [23], while Pan and Zeng give combinatorial interpretation of results on total positivity of Catalan-Stieltjes matrices [20]. A Toeplitz matrix T = [ti,j] is a (finite or infinite) matrix whose entries satisfy tijj = tj+1,j+1. In finite case, T= ( to ti t-1 to \tn-1 tn-2 t-n+l\ t-n+2 to In words, elements of a Toeplitz matrix are constant along diagonals descending from left to right. If the elements of a matrix are constant along diagonals ascending from left to right, the matrix is called a Hankel matrix. An example is given here, H to t1 t n-1 t1 t2 tn-2 tn 1 tn 2 t2n-2 Obviously, each Toeplitz (or Hankel) matrix of order n gives rise to a unique sequence (of length 2n - 1 in the finite case) of its elements. The connection also works the other way: T. Doslic et al.: Total positivity of Toeplitz matrices of recursive hypersequences 127 Given an (infinite) sequence (an) and given integers n0 and m, we can construct a Toeplitz (or a Hankel) matrix of order m having an0 in the upper left corner. In what follows we present a class of totally positive Toeplitz matrices whose entries are hyperfibonacci numbers [4, 17, 24]. These sequences of numbers were recently introduced by Dil and Mezo in a study of a symmetric algorithm for hyperharmonic and some other integer sequences [9]. ? ice of the i -ui generation (F quence arising from the recurrence relation Definition 1.2. The hyperfibonacci sequence of the r-th generation (Fnr))n>° is a se- n Fir) = EF(0) = Fn, F°r) = 0, F(r) — 1, (1.1) k=0 where r e N and Fn is the n-th term of the Fibonacci sequence, Fn = Fn-1 + Fn-2, F0 = 0,Fi = 1. Proposition 1.3 gives some basic identities for hyperfibonacci sequences [7]. Proposition 1.3. For hyperfibonacci sequence (F,lr) )n>° we have (i) (ii) (iii) (iv) F(r) — F^i + F,(r-1) (1.2) Fi1)2 - FiVi+i = F(% + 1 + (-1)n +1 FiM+i - F^F^ — Fi-2 + 1 - (-1)n +1 r— 1 ' n + r + k F—F.+=r-Z r+;+k . (.3) fc=0 Explicit formula for determinant of the Hankel matrix of hyperfibonacci sequence of r-th generation A — -^r,? — ( F(r) F F(r) F n +1 F(r) F n+1 • F(+ F n+2 F (r) \ • F n+r+1 F (r) F n+r+2 F(r) n+r + 1 F(r) F n+r+2 • F(r) • F n+2r+2/ has been obtained in [19] and here we state it in Theorem 1.4. We will find it useful in establishing our main result, the total positivity of the Toeplitz matrix of the same sequence with odd-indexed hyperfibonacci number in the upper left corner. Theorem 1.4. For the sequence (Fkr))k>0, r e N and n e N a determinant of a matrix Ar,n takes values ±1, det(Ar,n ) = (-1)n +L ^ J. 128 Ars Math. Contemp. 17 (2019) 291-310 The TP2 property of Toeplitz and Hankel matrices is closely related to log-concavity and log-convexity, respectively, of the associated sequences. Recall that a sequence (an) of positive numbers is log-concave if a?n > an-1an+1 holds for all n > n0 for some n0 G N. If the inequality is reversed, the sequence is log-convex. The literature on log-concavity and log-convexity is vast. Besides already mentioned classical papers by Stanley [21] and Brenti [3], we refer the reader also to [10, 11, 20, 22] for some recently developed techniques. In particular, the log-concavity of hyperfibonacci numbers of all generations r > 1 has been established in [24] by using recurrence relations. Here we proceed to prove more general claims that will imply the log-concavity results of reference [24]. 2 Positivity of hyperfibonacci determinant We let B^n = ] denote the matrix of order m consisting of hyperfibonacci numbers of the r-th generation, B(r) •= m,n ( F(r) F n (r) F (r) F n+1 V F(r) \Fn+m-1 Fn n-1 (r) F (r) n—m+1 F(r) Fn-m+2 F (r) F(r) Fn / n+m-2 with the constraint r > m - 1. In what follows we will show that there exist q(r) G N such that det(Bm,n) is positive for n > q(r). From the elementary properties of the Fibonacci sequence known as Cassini identity we immediately have that the matrix / F2n+1 F2n+2 | \F2n+2 F2n+3 J is positive for n G N0 and the matrix M M ' = F2n+1 F2n+2 F2n F2n+1 is positive for n G N. In Proposition 2.1 we extend the property of positivity to matrices of order 2 consisting from first generation of hyperfibonacci numbers while a general result, involving r-th generation of hyperfibonacci numbers is given in Theorem 3.5. Proposition 2.1. For n, r G N determinant of the matrix B(in is positive, det(B(m ) = det Fn (1) F (1) n+1 ev Fn1) , > 0. Proof. We apply relations presented in Proposition 1.3 to get F(1) - Fn- = Fn. Now, by the properties of determinant (column subtraction and then row subtraction) we obtain (1) det Fn(1) Fn F (1) F (1) n+1 Fnn(-1)1 = det det Fn Fn-A Fn+1 F^J Fn Fn+1 - Fn-1 Fn = det Fn F (1) n-1 Fn > 0. □ Fn l T. Doslic et al.: Total positivity of Toeplitz matrices of recursive hypersequences 129 Theorem 2.2. Let m e N. Then there is nm e N such that det (B^n > 0 for all n > nm. Proof. Employing elementary transformation on matrices and using relation (1.2) we get ( F det (B^) = det F n+1 VF ( F = det F n-1 VF F (1) F Fn n-1 (-1) F (1) F (1) 1 Fn F F (2) F( 2 ) Fn 1 n+m-1 F n+m-2 F n+m-3 F (2) F(2) F( -Fn 1 FF F n-m+2 F n-m+3 Fn n-m+1 F n-m+2 F n-m+3 F n-m+4 F(m-1) 1 n-m+1 F(m-1) F n-m+2 Fnm-1)/ F(m-1) \ F n-m+1 F(m-2) F n-m+2 F (1) 1 F (2.1) Having in mind relation (1.3) we immediately obtain r-1 F (r) _ F+ n + k n-r _ Fn+r U - 1 - k fc=Q v and furthermore F(r) _ F _ S Fn—r Fn+r Sr7 (2.2) where S _ r-1 f " + k Sr U-1 -kj- k=Q v Thus, S1 _ 1, S2 _ n +1, S3 _ ) + n + 2, S _ + n + 3, etc. Now, nln- '2 = n +1, S3 = ——" + n + 2, S4 substitute entries in (2.1) according to (2.2) to get we / Fn Fn+1 - S1 Fn+2 - S2 • • • Fn+m-1 - Sm-A det (B^) _ det F n-1 F Fn+1 - S1 • • • Fn+m-2 - Sm-2 VFn-m+1 Fn-m+2 Fn-m+3 F . (2.3) In the following steps of this proof we let Ai, A2, A3 denote matrices we deal with. We will show that determinants of these matrices are equal to each other. In order to make the proof more readable, the elements of the last two columns of Ai, A2, A3 are denoted by c-ij, cij, , respectively. On the other hand, the elements of the first m - 2 columns of these matrices are denoted by 6ijj and they do not change their values under performed transformation. 130 Ars Math. Contemp. 17 (2019) 291-310 When performing elementary transformations on matrix columns of (2.3) we obtain det (Bmn 1}) = det S2 — S1 S3 — S2 — S1 • • Fn+m- 2 Sm- 2 Fn+m- 1 Sm- S1 S2 — S1 • • Fn+m- 3 Sm- 3 Fn+m- 2 Sm- 2 0 S1 • • Fn+m- 4 S ^ m- 4 Fn+m- 3 S - m- 3 \ Fn Fn-1 Fn+1 — S1 Fn = det(A1 ) where we get A1 = [b j ] by similar transformation on rows, /$2 — 2S1 S3 — 2S2 — S1 S1 S2 — 2S1 A1 V S1 0 0 0 — Sm-1 + Sm-2 + Sm-3\ — Sm-2 + Sm-3 + Sm-4 — Sm-3 + Sm-4 + Sm-5 S2 — S1 Fn+1 — S1 Fn bj,j = bj+1,j+1, i = 1, .. ., m — 1, j = 1, .. ., m — 3, bj,j = cj,j, i = 1, .. ., m, j = m — 1, m, ci,m-1 c¿+1îm, i -1, . . . , m 3 and where entries bjj get values &1,1 = S2 — 2S1 b1,2 = S3 — 2S2 — S1 b1,3 = S4 — 2S3 — S2 + 2S1 b1,4 = S5 — 2S4 — S3 + 2S2 + S1 b1,5 = S6 — 2S5 — S4 + 2S3 + S2 b = Sm-1 — 2Sm-2 — Sm-3 + 2Sm-4 + S; 1,m-2 Sm-1 ~m-5 , while for entries Cj j we have c1,m-1 C2,m-1 = — Sm-2 + Sm-3 + Sm-4 — Sm-3 + Sm-4 + Sm-5 Cm-3,m-1 = —S2 + S1 cm-2,m-1 = —S1 cm-1,m-1 Fn cm,m-1 Fn+1, T. Doslic et al.: Total positivity of Toeplitz matrices of recursive hypersequences 131 and cm—1,m Fn+1 S1 cm,m Fn. Furthermore, we form matrix A2 = [b^-] with b^- = cij, i = 1,..., m, j = m — 1, m, by performing row transformations m—3 Ci,m-1 = Ci,m— 1 + £ , i = 1, . . . , m j=1 m—2 ci,m ci,m + ^ ^ bi,j, i -1, . . . , m. j = 1 As a consequence of these two operations for the last two columns of A2 we obtain / —Sm-4 + Sm-6 + Sm-7 + • • • + S2 — Sm-3 + Sm-5 + Sm-6 • + S2\ \ S4 + S2 — S5 + S3 + S; —S3 — S4 + S2 —S2 — S3 —S1 — S2 0 — S1 0 0 Fn Fn+1 Fn—1 Fn / (while the other entries of A2 are equal to those of A1). Clearly, det(A1) = det(A2). Furthermore, we perform row transformations c i,m— 1 ci,m— 1 I ci,m— 1 + bi,m—5 + 2bi,m—6 + 4bi,m—7 + • • • + (Fm—3 — 1)&i,1 + bi,m—4 + 2bi,m—5 + 4bi,m—6 + • • • + (Fm—2 — 1)bi,1 to get matrix A3 = [bi,j ] where bi,j = cij, i = 1,..., m, j = m — 1, m. Then, the last two columns of A3 are / —Fm—2 — Fm —A 0 0 0 Fn 0 Fn+1 \ Fn—1 Fn J Namely, a straighforward but tedious algebraic manipulation give us a nice value for c 1 ,m— 1 c1I,m—1 = (Fm—6 — 1)S1 + (Fm—5 — 1)2^1 — (Fm—4 — 1)^1 — (Fm—3 — 1)2^1 = Fm 2 . 2 132 ArsMath. Contemp. 17 (2019) 125-139 In the same fashion one can prove that c'1/,m = —Fm_1 and cij =0, i = 2,..., m - 2, j = m — 1, m. Again, determinant is not affected under these transformations, det(A3) = det(A2). We shall now separately treat the matrix A3, for even and odd n. Using the Fibonacci recurrence relation, for even n we immediately obtain det (Bm^) = det -det (Ki bi,2 • • bi;m-2 -Fm-2 Fm-1 b2,i &2,2 • • b2,m-2 0 0 0 &3,2 • • b3,m-2 0 0 0 0 • bm-2,m-2 0 0 0 0 • F„_i 0 1 V 0 0 • Fn-2 1 1 /bi.i bi,2 • y Fm-3 -Fm-2 b2,i b2,2 • • b2,m-2 0 0 0 &3,2 • • b3,m-2 0 0 0 0 • bm-2,m-2 0 0 0 0 0 1 0 0 0 0 0 1 where b[ n_2 = b1jm_2 + Fm_3Fn_1 — Fm_2Fn_2. This determinant can be represented as the sum of the upper triangular determinants. Now we use the fact that there is q e N such that the Fibonacci number Fq is bigger that the value P(q), Fq > P(q), where P(n) is a polynomial of any degree. The only element in the matrix above containing Fibonacci numbers is b'1m_2. The fact that the term Fn_1Fm_3 has a positive contribution in the determinant completes the proof for case when n is even. When n is odd we have det (Bm-15) = det (Ki bi,2 • • b i,m-2 -Fm-2 Fm-1 b2,i b2,2 • • b2,m-2 0 0 0 b3,2 • • b3,m-2 0 0 0 0 • bm-2,m-2 0 0 0 0 • Fn-2 1 1 0 0 • —F„-i 0 1 Now, analogue arguments as when n is even completes the proof. □ In particular, when m = 4 we have det (B43 (3) ) 4,n) det /S — 2 S3 — 2S2 — 1 —1 —2 1 S2 — 2 0 0 0 1 F„ Fn+i 0 0 F„-i F„ / T. Doslic et al.: Total positivity of Toeplitz matrices of recursive hypersequences 133 When n is even then det (Sj (3) A 4,nJ /S - 2 S3 - 2S2 - 1 -1 -2 det ( 1 S2 - 2 0 0 0 Fn-1 0 1I 0 - Fn-2 1 1 /S2 - 2 S3 - 2S2 - 1 - Fn- -2 + Fn-1 -1 -1\ det ( 1 S2 - 2 0 0 0 0 0 1I 0 0 1 0 /S2 - 2 0 0\ -(S2 - 2) ( 0 1 0I 0/01 /S3 - 2S2 -1 - Fn -2 + Fn-1 -1 + ( 3 2 0 1 V 0 0 The inequality -(n - 1)2 + - n - 1 + Fn-3. Fn-3 > (n - 1)2 - + n + 1 holds true for n > 15 and consequently det (b4 n) > 0 for n > 15 when n is even. Similarly, when n is odd ^2 - 2 0 0\ /S3 - 2S2 - 1 - Fn-3 -1 -1' det (S43,^) = (S2 - 2) ( 0 1 0 I - I 0 10 0 0 1 V 0 01 = (n - 1)2 - i(n - 1) + n +1 + Fn-3. Thus, it follows from these two cases that det (B^) > 0 for n > 15. Note that the proof of Theorem 2.2 can be used to efficient calculation of determinants (m— 1) of matrices Bm,n ). We will illustrate this on the example for m = 4 and n = 5. In that case, when applying the proof of Theorem 2.2 we have det (B45) det 51 25 11 4 6 11 -1 -1 97 51 25 11 = det ( 1 4 0 0 176 97 51 25 0 0 1 0I \309 176 97 51 0 0 0 1 = 24 - 11 = 13. Corollary 2.3. Let m, n, r G N and r > m - 1. Then there is q G N such that determinant of the matrix Bm,n is positive for all n > q, det (Bm)n) > 0. 134 Ars Math. Contemp. 17 (2019) 291-310 Proof. We proceed by induction on r. The base case, r = m - 1, is provided by Theorem 2.2. Let us now assume that the claim is true for m - 1 < p < r - 1. Our task is to show that the determinant i F,r) det(Bm) ) = det F (r) VF n+1 (r) n+m-1 F F (r) n-1 (r) F (r) n+m-2 F(r) n-m+1 F (r) n-m+2 F 0, and hence /(c,..., c) = c • det(Bm,n1)) > 0, ±v 2 (r-1) for any positive constant c. In particular, /(^2,..., ^2) > 0, where — Since / is continuous, there must exist a neighborhood W = (^2 - ¿1 + ¿1) x ■■■ x (^2 - ¿m,^2 + ¿m) such that f is positive on W. Now we use the explicit expression F (r) = F V1 (n + r + k F,) = Fn+2r ^ _ 1 _ k k=0 v from Proposition 1.3. By dividing it through by analogous expression for F, passing to limit when n ^ œ, one readily obtains ,r 1) and lim Fn(r) Fn n (r-1) Fn n — i+1 T. Doslic et al.: Total positivity of Toeplitz matrices of recursive hypersequences 135 That further implies that, for large enough n, the coefficient xi - Si, 4>2 + Si) for all i, and hence F (r) _ Fn — i+1 = P (r 1) falls into That completes the proof. F (r) n—m+1 F (r— I)'"' ' F (r —1) Fn—m+1, = det(Bm)n) > 0. □ 3 Main results We let Tr,n denote the matrix of order r + 2 consisting of hyperfibonacci numbers of the r-th generation, T •= r,n / F(r) F2n+1 F(r) F2n+2 VF2n+r+2 F (r) F2n F (r) F2n+1 F (r) \ F2n—r F (r) F (r) 2n+r+1 Lemma 3.1. For n G N and the hyperfibonacci sequence (F„')n>0 the matrix 2n—r+1 F(r) F2n+1 / Tl- F(1) F2n+1 F(1) F2n+2 \F2n+3 F(1) F 2n F (1) F2n+1 F(1) F2n+2 F(1) F 2n— 1 F2n F (1) F2 n)1 is totally positive. Proof. According to Proposition 2.1 the three initial minors of order 2 of Ti,n are positive. It is immediately seen from Theorem 1.4 that determinant det(Ti,n) is positive. These facts complete the proof. □ Note that the matrix T1n = [tijj] is a Toeplitz matrix, with the element ¿1,1 being hyperfibonacci number of the first generation having odd index. If we allow both even and odd indices for t1,1 then the property of total positivity is lost. Such determinant of order 3 in not positive for even indices (by Theorem 1.4), while it keeps the positivity of minors of order 2. We express this fact, that follows from the proof of Lemma 3.1, in Corollary 3.2. Corollary 3.2. For n G N and the hyperfibonacci sequence (Fn^)n>0 the matrix T 1,n Fn F(1) Fn)1 F(1) Fn-1 Fn F (1) n)1 F(1) Fn 2 n — 1 F ( 1)1 F is TP 2. Lemma 3.3. For n > 4 and the hyperfibonacci sequence (F, (2h 1>0 the matrix T2,n = F(2) F2n)1 F(2) F2n+2 F(2) F2n+3 XFw F(2) F2n F (2) F(2) F(2) F2n)3 F(2) F2n 1 F2n( )1 F2n (22 n) F(2) F2n 2 F2(n2) 2 F2n 1 F2n( )1 F2n F(2) I F2 n)1 F 2n)1 F (2) 2n)2 n — i+1 136 Ars Math. Contemp. 17 (2019) 291-310 is totally positive. Proof. According to Proposition 2.1 the five initial minors of order 2 of T2,n are positive. Furthermore, the three initial minors of order 3 are positive when n > 3 by Corollary 2.3. However, when n = 3 determinant det(T2,n) is negative (by Theorem 1.4) so the matrix T2,n is totally positive for n > 4. □ (2) Having in mind Proposition 2.1 and the fact that the matrix B3 „ has positive determinant for n > 7 we immediately derive Corollary 3.4. Corollary 3.4. For n > 8 and the hyperfibonacci sequence (F, (2h ^>0 the matrix t Fn(2) F(2) T' F (2) Fn+1 Fn+2 \Fn+3 -1 F(2) Fn F(2) F2n+1 F(2 + F2n+2 F(2) Fn-2 F(2) Fn-1 Fnn(-2)1 Fn F(2) F2n+1 F(2)3\ n—3 F(2) Fn-2 F<2>/ is TP3. Furthermore, it holds true that det(Bg)) > 0, n > 15 det(Bg)) > 0, n > 5. When r > 5 there is no constraint on the value of n when asking for positivity of de^B^). Theorem 3.5. For the hyperfibonacci sequence (Fnr))n>0 there is q G N such that the matrix Tr,n of order r + 2 Tr, is totally positive for n > q. i F(r) F2n+1 F(r) F2n+2 F(r) \Fn+r+2 F(r) F2n F (r) F2n+1 F2(r) \ 2n—r F (r) F (r) 2n+r+1 n—r+1 F(r) F2n+1 / Proof. First we prove that 2n +1 initial minors of order 2 are positive. These submatrices (r) are of the form B( ^ where m2 > 2n - r, so there they have positive determinant for r > 1 and n > 1, according to Corollary 2.3. Obviously, another initial minors are of the form R(r) R(r) R(r) B3,m3 , B4,m4, . . . , Br+1,mr+i . According to Corollary 2.3 there exist numbers q3, q4,..., qr)1 G N such that det(s3ri3 ) > 0, m3 > q3 det(s4ri4 ) > 0, m4 > q4 det(Br+)1,mr+i ) > 0, mr + 1 > qr+1. T. Doslic et al.: Total positivity of Toeplitz matrices of recursive hypersequences 137 It remains to show that det(Tr,n) is itself positive. We start by noticing that Tr,n can be obtained from Ar,2n_r by reversing the order of columns. That corresponds to right multiplication of Ar,2n_r by Ur+2, where Ur+2 is a square matrix of order r + 2 whose elements are (Ur+2)ijj = 1 if i + j = r + 3 and zero otherwise. It is immediately seen that det(Ur+2) = (_1)L(r+2)/2J. Now we have det(Tr,„) = det(Ar,2„-r)det(Ur+2), and Theorem 1.4 implies det(Tr,n) = (_i)2n-r+L(r+3)/2J + L(r+2)/2J = (_i)2 = i, for all r. That completes the proof. □ We conclude the section with another result that follows directly from Corollary 3.4. Corollary 3.6. For the hyperfibonacci sequence (Fnr)) 0 there is q G N such that the matrix Tr',n of order r + 2 Fn Fn-1 ••• Fn-r-1 F(r) F(r) . . . F(r) t, = Fn+1 Fn Tr,n . . . F(r) F(r) F(r) . \Fn+r + 1 Fn+r • • • Fn / is TPr+1 for n > q. 4 Concluding remarks In this paper we have considered several classes of Toeplitz matrices associated to sequences of hyperfibonacci numbers of given generation. We have established various pos-itivity results for such matrices. In particular, we showed that such matrices with odd-indexed hyperfibonacci numbers on the main diagonal are totally positive for large enough values of index n. When the restriction to odd-valued indices is omitted, the total positivity is not preserved, but we established that those matrices are TPr+1 for a given generation r and large enough n. That implies (at least asymptotical) log-concavity of hyperfibonacci numbers of all generations r > 1. Our results thus extend and strengthen results of reference [24] established by a different approach. It would be interesting to have combinatorial proofs of log-concavity of Fn ) for r > 1; at the moment, we are not aware of any. We have also tried to explore the form of dependence of qr on r. The numerical evidence, collected in Table 1, suggests that 2qr + 1, the index in the upper left corner, behaves as 7r _ 5 for even r and 7r _ 4 for r odd. It would be interesting to examine whether the Table 1: Some values of parameter qr in Theorem 3.5. r 1 2 3 4 5 6 7 8 9 10 11 2qr + 1 5 9 17 23 31 37 45 51 59 65 73 pattern (or at least a linear dependence) persists for larger r, and if it does, to find some explanation. We are fairly confident that the methods and results presented here could be extended so as to encompass also other sequences defined by two-term recurrences and their iterated 138 Ars Math. Contemp. 17 (2019) 291-310 partial sums. It would be worthwhile to explore whether the same approach could be applicable to the sequences defined by longer linear recurrences with constant coefficients, such as the sequence of tribonacci numbers. References [1] T. 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