k-pell, k-pell-lucas and Modified k-pell Numbers: Some Identities and Norms of Hankel Matrices

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1 International Journal of Mathematical Analysis Vol. 9, 05, no., 3-37 HIKARI Ltd, k-pell, k-pell-lucas and Modified k-pell Numbers: Some Identities and Norms of Hankel Matrices P. Vasco, P. Catarino, H. Campos, A. P. Aires 3 and A. Borges Universidade de Trás-os-Montes e Alto Douro, UTAD Quinta de Prados, Vila Real, Portugal Copyright c 04 P. Vasco, P. Catarino, H. Campos, A. P. Aires and A. Borges. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we present some identities involving terms of k-pell, k-pell-lucas and Modified k-pell sequences. We also give some results on the column and row norms of Hankel matrices which entries are numbers of these sequences. Mathematics Subject Classification: B37, B83, 5A60 Keywords: k-pell numbers, k-pell-lucas numbers, Modified k-pell numbers, Hankel matrices, Hadamard product, matrix norms. Introduction Hankel matrices play an important role in many science fields namely signal processing in engineering or times series in statistics. An Hankel matrix is a matrix which entries are the same along the anti-diagonals. This type of matrices has been subject of study with respect to its spectrum (collection of eigenvalues) and other interesting results were derived. Member of CM-UTAD, Portuguese Research Centre Member of CM-UTAD, Collaborator of CIDTFF, Portuguese Research Centres 3 Member of CIDTFF, Collaborator of CM-UTAD,Portuguese Research Centres

2 3 P. Vasco, P. Catarino, H. Campos, A. P. Aires and A. Borges Some norms of Hankel matrices involving Fibonacci and Lucas numbers were presented in [] and, in what concerns spectral norms of these matrices their work was continued in [9]. In this paper we are interested in the special cases of Hankel matrices involving k-pell, k-pell-lucas and Modified k-pell numbers, denoted by P k,n, Q k,n and q k,n, with k a real number and n a non negative integer, respectively. These sequences are particular cases of the sequences W n (a, b; p, q) defined by the general recurrence relation W n pw n qw n, n () with W 0 a, W b, and a, b, p, q, integers, with p > 0, q 0, by Horadam (see [5], [6] and [7]). In fact, in the Horadam notation we have and The corresponding characteristic equation of () is P k,n W n (0, ;, k) () Q k,n W n (, ;, k) (3) q k,n W n (, ;, k). (4) x px q, and its roots are r p+ p 4q and r p p 4q. For (), (3) and (4), their roots are and verify r + + k and r + k, (5) r + r, r r + k and r r k. (6) Their Binet s formulas are well known and given by P k,n r n r n r r (7) Q k,n r n r n q k,n r n n + r, (9) respectively. The main purpose of this paper is the study of some norms of Hankel matrices involving k-pell, k-pell-lucas and Modified k-pell numbers. With this purpose, after this introductory section, we present, in section two, a set of identities involving the terms of these sequences. These identities are an attempt to generalize the study started in [4]. In section 3 we present some results for the norms... and... of Hankel matrices involving k-pell, k-pell-lucas and Modified k-pell numbers. (8)

3 Some identities and norms of Hankel matrices 33. Some Identities Let P k,n, Q k,n and q k,n, with k a real number and n a nonnegative integer, be defined as above. The following proposition establishes several identities between the terms of the referred sequences. Proposition.. For any real number k and a nonnegative integer n (different from zero in items i), iii), vii) and viii)), i) Pk,n + P [ k,n+p k,n (+k) Q k,n ( k) n ( k) ] ; ii) (P k,n+ P k,n ) ( + k)pk,n + ( k)n ; iii) P k,n P k,n + Pk,n P k,n ( k)p k,n + ( k)n ; iv) Pk,n [Q 4(+k) k,n ( k) n ] ; v) P k,n+ Pk,n+ + kp k,n ; vi) P k,n+ P k,n q k,n ; vii) ( + k)p k,n Q k,n + kq k,n ; viii) P k,n + kp k,n q k,n. Proof. In order to prove i), ii), iv) and v) we use the Binet s formulas (7), (8), (9), the expressions of the roots (5) and the relations involving the roots (6). To prove iii) we also use ii) in the form (P k,n P k,n ) ( + k)p k,n + ( k) n. The fact that q k,n Q k,n, n 0, is used in the proof of vi) and vii). Proposition 3 of [] is required to obtain vi) and Proposition 4 of the same article to obtain vii). We note that for k these identities generalize the relationships (9), (0), (), (), (4) and (5) established by Halici in [4]. 3. Norms of Hankel matrices involving k-pell, k-pell-lucas and Modified k-pell numbers An Hankel matrix is an n n matrix H n (h ij ) where h ij h i+j, that is a matrix of the form h h h 3 h n h H n h 3 h 4 h n h n h n+ h n+ h n H n is symmetric and consequently its column norm H n is equal to its row norm H n, H n max j n i h ij max i n h ij H n. i

4 34 P. Vasco, P. Catarino, H. Campos, A. P. Aires and A. Borges For this reason, we will calculate, from now on, one of them in each case. We start by defining the following Hankel matrices and A (a ij ), a ij P k,i+j (0) B (b ij ), b ij Q k,i+j () C (c ij ), c ij q k,i+j. () We shall give the column and row norms of the Hankel matrices involving k-pell, k-pell-lucas and Modified k-pell numbers and note that these results generalize the corresponding results for Pell, Pell-Lucas and Modified Pell numbers given in [4]. Theorem 3.. If A is an n n matrix with a ij P k,i+j then A A (k + ) (Q k,n Q k,n ) k + (q k,n q k,n ). Proof. From the definition of the matrix A, we can write A max a ij j n i max { a j + a j + + a nj } j n P k,n + P k,n+ + + P k,n n P k,i n P k,i. Now, using Proposition of [3], this last equality can be written as k + ( + P k,n + kp k,n ) k + ( + P k,n + kp k,n ) and by Proposition 4 of [] as consequently which is equal to since Q k,j q k,j. k + A A ( Q k,n ) Q k,n, (k + ) (Q k,n Q k,n ), k + (q k,n q k,n ), Theorem 3.. If B is an n n matrix with b ij Q k,i+j then B B k + (Q k,n+ + Q k,n Q k,n Q k,n+ ) (P k,n P k,n ).

5 Some identities and norms of Hankel matrices 35 Proof. Similarly to the proof of the previous theorem and using now Proposition 3 of [0], Q k,i (Q k+ k,n+ + kq k,n ), and Propositions 3 and 4 of [] we obtain B Q k,n + Q k,n+ + + Q k,n n Q k,i n Q k,i k+ (Q k,n + kq k,n Q k,n kq k,n ) [(Q k+ k,n + kq k,n ) Q k,n (Q k,n kq k,n ) + Q k,n ] (Q k+ k,n+ Q k,n Q k,n+ + Q k,n ) ((P k+ k,n+ + kp k,n ) (P k,n+ P k,n ) (P k,n+ + kp k,n )+ (P k,n+ P k,n )) [(k + )P k+ k,n (k + )P k,n ] (P k,n P k,n ). Theorem 3.3. If C is an n n matrix with c ij q k,i+j then C C (k + ) (Q k,n+ + Q k,n Q k,n Q k,n+ ) P k,n P k,n. Proof. Since Q k,j q k,j we have B C and B C C, and the result follows by the previous theorem. In addition if we consider the Hankel matrices D A + B (d ij ) where E A + C (e ij ) where and F B + C (f ij ) where d ij P k,i+j + Q k,i+j, e ij P k,i+j + q k,i+j f ij Q k,i+j + q k,i+j 3 Q k,i+j and the definitions of the used norms, we get and D D E A E A (3) F F 3 B 3 B. (4) To obtain E we can use Theorem 3. and Theorem 3.3 since, in this case, E A + C. So we have E (k + ) (Q k,n+ Q k,n+ ) k + (q k,n+ q k,n+ ). (5)

6 36 P. Vasco, P. Catarino, H. Campos, A. P. Aires and A. Borges The same result could be obtained if we follow the same steps of Theorem 3. and use Proposition 4 item of [0] (P k,i + q k,i ) k + ( + kp k,n + (k + )P k,n+ ). In this case we obtain E (k + ) (k(p k,n P k,n ) (k + )(P k,n P k,n ) and using the definition of k-pell numbers this is equivalent to E (k + ) (P k,n+ + kp k,n P k,n+ kp k,n ). Now using Proposition 4 of [] we have exactly the same as in (5). Now we will consider the Hadamard product, denoted by, of the previously defined matrices, that is the entrywise product, see [8]. Consider the Hankel matrices G A B (g ij ) where H A C (h ij ) where and L B C (l ij ) where Since Q k,j q k,j we have that g ij P k,i+j Q k,i+j, h ij P k,i+j q k,i+j l ij Q k,i+j q k,i+j H G. Now following the same steps of Theorem 3. and using Proposition 6 item. of [0], (P k,i Q k,i ) [ ] 3 k k + ( + P k,n+ + kp k,n ) kp k,n, we obtain [ G (P 3 k k+ k,4n + kp k,4n P k,n kp k,n ) kp k,4n + kp k,n ] and using Proposition 6 item 3. of [0], (Q k,i q k,i ) ( k) (q k, + q k, q k,3 + + q k,n q k,n ) + q k,n q k,n+ k, we obtain L ( k) (q k,n q k,n + q k,n q k,n+ + + q k,n q k,n ) + q k,n q k,n q k,n q k,n. (6) (7)

7 Some identities and norms of Hankel matrices 37 Acknowledgements. This work has been supported by the Portuguese Government through the FCT - Fundação para a Ciência e a Tecnologia - under the project PEst-OE/MAT/ UI4080/04. References [] M. Akbulak and D. Bozkurt, On the norms of Hankel matrices involving Fibonacci and Lucas numbers. Selçuk J. Appl. Math. 9 (008),, [] P. Catarino, A Note Involving Two by Two Matrices of the k-pell and k-pell-lucas Sequences. Int. Math. Forum 8 (03), 3, [3] P. Catarino and P. Vasco, Some basic properties and a two-by-two matrix involving the k-pell numbers. Int. J. Math. Anal. 7 (03), 45, [4] S. Halici, On some inequalities and Hankel matrices involving Pell, Pell-Lucas numbers. Math. Rep. (Bucur.) 5 (03), 65, -0. [5] A. F. Horadam, Generating functions for powers of a certain generalized sequences of numbers. Duke Math. J. 3 (965), 3, [6] A. F. Horadam, Basic properties of a certain generalized sequence of numbers. Fibonacci Quart. 3 (965), 3, [7] A. F. Horadam, Special properties of the sequence w n (a, b; p, q). Fibonacci Quart. 5 (967), 4, [8] R. A. Horn and C. Johnson, Topics in matrix Analysis. University Press, Cambridge [9] S. Solak and M. Bahsi, On the Spectral Norms of Hankel Matrices with Fibonacci and Lucas Numbers. Selçuk J. Appl. Math. (0),, [0] Vasco P. and Catarino P. (04). Sums and products involving terms of k-pell, k-pell- Lucas and Modified k-pell sequences. JP Journal of Algebra, Number Theory and Applications, 3(), Received: December 4, 04; Published: January 9, 05