The generalized order-k Fibonacci Pell sequence by matrix methods

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1 Journal of Computational and Applied Mathematics 09 (007) wwwelseviercom/locate/cam The generalized order- Fibonacci Pell sequence by matrix methods Emrah Kilic Mathematics Department, TOBB University of Economics and Technology, Sogutozu, Anara, Turey Received 4 July 006; received in revised form October 006 Abstract In this paper, we consider the usual and generalized order- Fibonacci and Pell recurrences, then we define a new recurrence, which we call generalized order- F P sequence Also we present a systematic investigation of the generalized order- F P sequence We give the generalized Binet formula, some identities and an explicit formula for sums of the generalized order- F P sequence by matrix methods Further, we give the generating function and combinatorial representations of these numbers Also we present an algorithm for computing the sums of the generalized order- Pell numbers, as well as the Pell numbers themselves 006 Elsevier BV All rights reserved MSC: B37; C0; 5A8; 05A5 Keywords: Generalized Fibonacci and Pell number; Sums; Generating function; Matrix methods Introduction The Fibonacci and Pell sequences can be generalized as follows: let r be a nonnegative integer such that r 0 and 4 r + 0 Define the generalized Fibonacci sequence as shown x n+ = r x n + x n, () where x 0 = 0 and x = When r = 0, then x n = F n (nth Fibonacci number) and when r =, then x n = P n (nth Pell number) Miles [9] define the generalized -Fibonacci numbers as shown for n> f n = f n + f n + +f n, where f = f = =f = 0 and f = f = Then the author have studied some properties of the sequence {f n } Further, in [5], the authors consider the generalized -Fibonacci numbers, then they give the Binet formula of the generalized Fibonacci sequence {f n } address: eilic@etuedutr /$ - see front matter 006 Elsevier BV All rights reserved doi:006/jcam006007

2 34 E Kilic / Journal of Computational and Applied Mathematics 09 (007) Er [8] give the definition of the sequences of the generalized order- Fibonacci numbers as follows: for n>0 and i gn i = gn j i, () j= with initial conditions { gn i if i = n = 0 otherwise for n 0, where gn i is the nth term of the ith sequence Also Er give the generating matrix for the generalized order- Fibonacci sequence {gn i } Also in [], the authors give the relationships between the generalized order- Fibonacci numbers gn i and the generalized order- Lucas numbers (see for more details about the generalized Lucas numbers [3]), and give the some useful identities and Binet formulas of the generalized order- Fibonacci sequence {gn i } and Lucas sequence {li n } In [3], the authors define the sequences of the generalized order- Pell numbers as follows: for n>0 and i P i n = P i n + P i n + +P i n, (3) with initial conditions { Pn i if i = n = 0 otherwise for n 0, where Pn i is the nth term of the ith generalized order- Pell sequence Also authors give the generating matrix, Binet formula, sums and combinatorial representations of the terms of generalized order- Pell sequence {Pn i} The above sequences are a special case of a sequence which is defined recursively as a linear combination of the preceding terms a n+ = c a n+ + c a n+ + +c a n, where c,c,,c are real constants In [], Kalman derived a number of closed-form formulas for the generalized sequence by companion matrix method as follows: c c c c A = (4) Then by an inductive argument, he obtains a 0 a n A n a = a n+ a a n+ Linear recurrence relations are of great interest and have been a central part of number theory These recursions appear almost everywhere in mathematics and computer science (for more details see [7,5,4,0,9]) For example, many families of orthogonal polynomials, including the Tchebychev polynomials and the Dicson polynomials, satisfy recurrence relations Linear recurrence relations are of importance in approximation theory and cryptography and they have arisen in computer graphics and time series analysis Furthermore, higher order linear recurrences have leaderships to the new ideas for the generalization of orthogonal polynomials satisfying second order linear recurrences to the order In a recently published article [], the authors define the orthogonal polynomial satisfying a generalized order- linear recurrence

3 E Kilic / Journal of Computational and Applied Mathematics 09 (007) Also matrix methods are of importance in recurrence relations For example, the generating matrices are useful tools for the number sequences satisfying a recurrence relation Further, the combinatorial matrix theory is very important tool to obtain results for number theory [] In[5,,3], the authors define certain generalizations of the usual Fibonacci, Pell and Lucas numbers by matrix methods and then obtain the Binet formulas and combinatorial representations of the generalizations of these number sequence Furthermore, using matrix methods for computing of properties of recurrence relations are very convenient to parallel algorithm in computer science (see [4,6,7,8,,,5]) Now we define and study properties common to Fibonacci and Pell numbers by investigating a number sequence that satisfy both Fibonacci and Pell recurrences of second order and th order in matrix representation Then extending the matrix representation, we give sums of the generalized Fibonacci and Pell numbers subscripted from to n could be derived directly using this representation Furthermore, using matrix methods, we obtain the generalized Binet formula and combinatorial representation of the new sequence A generalization of the Fibonacci and Pell numbers In this section, we define a new order- generalization of the Fibonacci and Pell numbers, we call generalized order- F P numbers Then we obtain the generating matrix of the generalized order- F P sequence Also using the generating matrix, we derive some interesting identities for the generalized order- F P numbers which is the well-nown formula for the usual or generalized order- Fibonacci and Pell numbers We start with the definition of the generalized order- F P sequence Define sequences of the generalized order- F P numbers as shown: for n>0, m 0 and i u i n = m u i n + ui n + +ui n, with initial conditions for n 0 { u i if i = n, n = 0 otherwise, where u i n is the nth term of the ith generalized F P sequence For example, when = i = and m =, the sequence {u i n } is reduced to the usual Pell sequence {P n} Also when m = 0, the sequence {u i n } is reduced to the generalized order- Fibonacci sequence {gn i } By the definition of generalized F P numbers, we can write the following vector recurrence relation: u i n+ u i u i n u n u i i n n = T u i n, (5) u i n + u i n + where T is the companion matrix of order as follows: m T = (6) The matrix T is said to be generalized order- F P matrix To deal with the sequences of the generalized order- F P numbers, we define an ( ) matrix D n =[d ij ] as follows: u n u n u n u n u n u n D n = u n + u n + u n + (7)

4 36 E Kilic / Journal of Computational and Applied Mathematics 09 (007) If we expand Eq (5) to the columns, then we can obtain the following matrix equation: D n = TD n (8) Then we have the following lemma Lemma Let the matrices D n and T have the forms (7) and (6), respectively Then for n D n = T n Proof We now that D n = TD n Then, by an inductive argument, we can write D n = T n D Since the definition of generalized order- F P numbers, we obtain D = T and so D n = T n which is desired Then we can give the following theorem Theorem Let the matrix D n have the form (7) Then for n { if is odd, det D n = ( ) n if is even Proof From Lemma, we have D n = T n Thus, det D n = det(t n ) = (det T) n where det T = ( ) Thus, { if is even, det D n = ( ) n if is odd So the proof is complete Now we give some relations involving the generalized order- F P numbers Theorem 3 Let u i n be the generalized order- F P number, for i Then, for m 0 and n u i n+m = u j mu i n j+ j= Proof By Lemma, we now that D n = T n ; so we rewrite it as D n = D n D = D D n In other words, D is commutative under matrix multiplication Hence, more generalizing, we can write D n+m = D n D m = D m D n (9) Consequently, an element of D n+m is the product of a row D n and a column of D m ; that is u i n+m = u j mu i n j+ j= Thus, the proof is complete For example, if we tae = and m = 0 in Theorem 3, the sequence {u i n } is reduced to the usual Fibonacci sequence and we have Fn+m = FmF j n j+ j= = F m F n + F m F n and, since Fn = F n+ for all positive n and =, we obtain F n+m = F m+ F n + F m F n,

5 E Kilic / Journal of Computational and Applied Mathematics 09 (007) where Fn is the usual Fibonacci number Indeed, we generalize the following relation involving the usual Fibonacci numbers [4] F n+m = F m+ F n + F m F n For later use, we give the following lemma Lemma 4 Let u i n be the generalized order- F P number Then u i n+ = u n + ui+ n for i, (0) u n+ = m u n + u n, () u n+ = u n () Proof From (9), we have D n+ = D n D Then by using a property of matrix multiplication, the proof is readily seen Generalizing Theorem 3, we can give the following Corollary without proof since D n =T n and so D n+t =D n+r D t r for all positive integers t and r such that t>r Corollary 5 Let u i n be the generalized order- F P number Then for n, t such that t>r u i n+t = u j n+r ui t r j+ j= 3 Generalized Binet formula In 843, Binet derived the following formula for the Fibonacci numbers: F n = αn β n α β, where α and β are given by ( 5)/ Also in [6], using the generating function, the author give the Binet formula for generalized Fibonacci numbers In this section, we derive the generalized Binet formula for generalized order- F P numbers by using matrix methods Before this, we give some results From the companion matrices, it is well nown that the characteristic equation of the matrix T given by (6) is x m x x x = 0 which is also the characteristic equation of generalized order- F P numbers Lemma 6 The equation x + ( m + )x + ( m )x + = 0 does not have multiple roots for and m 0 Proof Let f(x)=x m x x x and h(x)=(x )f (x)=x + ( m +)x +( m )x + Thus, is a root but not a multiple root of h(x), since and f() Suppose that α is a multiple root of h(x) for all integers and m such that and m 0 Note that α 0 and α Since α is a multiple root, h(α) = α + ( m + )α + ( m )α + = 0 and h (α) = ( + )α ( m + )α + ( m )( )α = α [( + )α ( m + )α + ( m )( )]=0 Thus, α, = (( m + ) Δ)/( + ) where Δ = ( m + ) 4( m ) ( ) and hence, for α 0 = h(α ) = α ( α + (m + )α ( m ))

6 38 E Kilic / Journal of Computational and Applied Mathematics 09 (007) Let a,m = α ( α + (m + )α ( m )) Then we write the above equation as follows: for and m 0 0 = a,m However, if we choose = and m =, a, = 39, a,, a contradiction Similarly, hence, for α 0 = h(α ) = α ( α + (m + )α ( m )) Let b,m = α ( α + (m + )α ( m )) Then we write the above equation as follows: for and m 0 0 = b,m For = and m =, b, = 3 7, b,, a contradiction because we suppose that α is a multiple root for any integers and m such that and m 0 Therefore, the equation h(x) = 0 does not have multiple roots Consequently, by Lemma 6, the equation x m x x x = 0 does not have multiple roots for and m 0 Let f(λ) be the characteristic polynomial of the generalized order- F P matrix T Then f(λ) = λ m λ λ λ, which is mentioned above Let λ, λ,,λ be the eigenvalues of matrix T Then, by Lemma 6, λ, λ,,λ are distinct Let V be a Vandermonde matrix as follows: V = λ λ λ λ λ λ λ λ λ Let e i be a matrix as follows: e i = λ n+ i λ n+ i λ n+ i and V (i) j be a matrix obtained from V by replacing the jth column of V by e i Denote V T by Λ Then we obtain the generalized Binet formula for the generalized order- F P numbers with following theorem Theorem 7 Let u i n be the nth term of ith F P sequence, for i Then u j n i+ = det(v (i) j ) det(v ) Proof Since the eigenvalues of T are distinct, T is diagonalizable Since Λ is invertible Λ T Λ = E where E = diag(λ, λ,,λ ) Hence, T is similar to E So we obtain T n Λ = ΛE n By Lemma, T n = D n Then we have the

7 following linear system of equations: d i λ + d i λ + +d i = λ n+ i, d i λ + d i λ + +d i = λ n+ i, d i λ + d i λ E Kilic / Journal of Computational and Applied Mathematics 09 (007) d i = λ n+ i, where D n =[d ij ] Thus, for each j =,,,, we obtain d ij = det(v (i) j ) det(v ) Note that d ij = u j n i+ So we have the conclusion 4 Sums of the generalized F P numbers Now we extend the matrix representation of generalized order- F P numbers in Section and derive a generating matrix for the sums of the generalized order- F P numbers subscripted from to n To calculate the sums S n of the generalized order- F P numbers, defined by n S n = i=0 u i since u n+ = u n which given in Lemma 4, we rewrite it as n S n = u i i= Let A and W n be ( + ) ( + ) square matrix, such that A = 0 T 0 and W n = D n, (4) S n S n S n + where T and D n are the matrices as in (6) and (7), respectively Using formula (), we obtain the following equation: S n = u n + S n and so we derive the following matrix recurrence equation: W n = W n A Inductively, we also have W n = W A n (3)

8 40 E Kilic / Journal of Computational and Applied Mathematics 09 (007) Since S i = 0, i and by the definition of the generalized order- F P numbers, we thus infer W = A, and in general, W n = A n So we obtain the generating matrix for the sums of the generalized order- F P numbers S n Since W n = A n, we write W n+ = W n W = W W n (5) which shows that W is commutative as well under matrix multiplication By an application of Eq (5), the sums of generalized order- F P numbers satisfy the recurrence relation: S n = + m S n + S n i (6) i= Substituting S n = u n + S n into the Eq (6) and by (), we express u n in terms of the sums of the generalized order- F P numbers u n = + (m )S n + S n i (7) i= For example, when = and m = 0, the sequence {u i n } is reduced to the Fibonacci sequence {F n} and so the above equation is reduced to F n = + S n So we derive the well-nown result [4] n F i = F n i= 5 An explicit formula for the sums of generalized F P numbers In this section, we derive an explicit formula for the sums of generalized order- F P numbers subscripted from to n by matrix methods Recall that the ( + ) ( + ) matrix A be as in (3) We consider the characteristic equation of matrix A, then the characteristic polynomial of A is K A (λ) = λ T λi 0, where the matrix T given by (6) and I is the identity matrix of order Computing the above determinant by the Laplace expansion of determinant with respect to the first row gives as K A (λ) = ( λ) T λi (8) From Section, we have the characteristic polynomial of matrix T, then we obtain the characteristic equation of matrix A as follows: x + ( m + )x + ( m )x + = 0 From Lemma 6, we now that the above equation does not have multiple roots Thus, we can diagonalize the matrix A Recall that the eigenvalues of the matrix T are λ, λ,,λ By (8) and previous results, we have that the eigenvalues of matrix A are, λ, λ,,λ and all of them are distinct

9 E Kilic / Journal of Computational and Applied Mathematics 09 (007) We define an ( + ) ( + ) matrix G as follows: ( m λ + ) λ λ ( m λ + ) λ λ G =, (9) ( m λ λ λ + ) ( m + ) where, λ, λ,,λ are the eigenvalues of A Then we give an explicit formula for the sums of generalized order- F P numbers with the following theorem Theorem 8 Let S n denote the sums of the terms of the sequence {u n } subscripted from to n Then ( )/ S n = ( m + ), i= u i n where u i n is the nth term of ith generalized order- F P sequence Proof It is easily verified that AG = GM, (0) where M = diag(, λ, λ,,λ ) and the matrices A and G given by (3) and (9), respectively If we compute the det G by the Laplace expansion according to the first row, then, by the definition of Vandermonde matrix V,we easily obtain det G = det V We now that det V 0 since the all λ i s are distinct for i Thus, det G 0 and so the matrix G is invertible Therefore, we write (0) as G AG = M Then the matrix A is similar to the matrix M and so we obtain the matrix equation G A n G = M n or A n G = GM n Since A n = W n and S n = (W n ),, W n G = GM n Thus, by a matrix multiplication, the conclusion is easily obtained Taing by = and m =, then the sequence {u i n } is reduced to the usual Pell sequence and by Theorem 8, we obtain n P i = P n + P n and since P n = P n+, n P i = P n+ + P n which is well-nown fact from [0] 6 Generating function and combinatorial representation In this section we give the generating function and combinatorial representation of generalized order- F P sequence {u n } We start with the generating function of the sequence {u n }

10 4 E Kilic / Journal of Computational and Applied Mathematics 09 (007) Let Then G (x) = u + u x + u 3 x + u x + u + x + +u n+ xn + G (x) m xg (x) x G (x) x G (x) = ( m x x x )G (x) = u + x(u m u ) + x (u 3 m u u ) + +x (u + m u u u u ) + +x n (u n+ m u n u n u n + u n + ) + By the definition of generalized order- F P numbers, we obtain G (x) m xg (x) x G (x) x G (x) = u and since u =, G (x) = ( m x x x ) for 0 m x + x + +x < Let f (x) = m x + x + +x Then 0 f (x) < and we give exponential representation for generalized order- F P numbers ln G (x) = ln[ ( m x + x + +x )] = ln[ ( m x + x + +x )] = [ ( m x + x + +x ) (m x + x + +x ) n (m x + x + +x ) n ] = x[( m + x + x + +x ) + (m + x + x + +x ) + + n (m + x + x + +x ) n + ] = x n (m + x + x + +x ) n n=0 Thus, ( ) G (x) = exp x n (m + x + x + +x ) n n=0 Now we give combinatorial representation of the generalized order- F P numbers In [3], the authors obtained an explicit formula for the elements in the nth power of the companion matrix and gave some interesting applications The companion matrix A be as in (4), then we find the following theorem in [3] Theorem 9 The (i, j) entry a (n) ij (c,c,,c ) in the matrix A n (c,c,,c ) is given by the following formula: a (n) ij (c,c,,c ) = ( ) t j + t j+ + +t t + t + +t c t t + t + +t t,t,,t c t, () (t,t,,t ) where the summation is over nonnegative integers satisfying t + t + +t = n i + j, and the coefficients in () is defined to be if n = i j Then we have the following Corollary

11 E Kilic / Journal of Computational and Applied Mathematics 09 (007) Fig Illustrates the MATLAB code concerning the algorithm

12 44 E Kilic / Journal of Computational and Applied Mathematics 09 (007) Corollary 0 Let u i n be the generalized order- F P number for i Then u i n = ( ) r r + r + +r mr, r + r + +r r,r,,r (r,r,,r ) where the summation is over nonnegative integers satisfying r + r + +r = n i + Proof In Theorem 9, if j = and c = m, then the proof is immediately seen from (6) 7 An algorithm for the sum In this section, we are going to give a computational algorithm for the sums of the generalized order- F P numbers The sums of generalized order- F P numbers may be estimated via the equation S n = u n + S n mentioned above But this estimation based on the matrix multiplication is not useful for the large values of n, m and Consequently, to obtain the matrix W n generating sums of the generalized order- F P numbers, we give an algorithm that is entirely based on the sums of the series rather than the one based on matrix multiplication The following considerations have played important results in forming such an algorithm First, whatever values n, m and tae, the first line of the resulting matrix is fixed as follows: for all i and j, W i (,j) = for j = i and W i (,j) = 0 for j>i Thus, there is no need for further processes for the first line Second, when i =, the elements of the (i + )th line of D i can be obtained by using the identity matrix I of order Following the second step, when i>, the operation to do is to erase the th line of I and for j =, 3,, tomovethejth row into (j + )th line and the (i + )th line of D i into the first line of I This operation is repeated by using the second step until i = n Finally, after obtaining the (n + )th line of the matrix D n, it is easy to compute the other element of W n by using the equation S n =u n +S n and the Eq () (Fig ) Acnowledgment The author would lie to than to the anonymous referee for helpful suggestions References [] LD Antzoulaos, MV Koutras, On a class of polynomials related to classical orthogonal and Fibonacci polynomials with probability applications, J Statist Plann Inference 35 () (005) 8 39 [] RA Brualdi, HJ Ryser, Combinatorial Matrix Theory, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 99 [3] WYC Chen, JD Louc, The combinatorial power of the companion matrix, Linear Algebra Appl 3 (996) 6 78 [4] P Cull, JL Holloway, Computing Fibonacci numbers quicly, Inform Process Lett 3 (3) (989) [5] T Cusic, C Ding, A Renvall, Stream Ciphers and Number Theory, North-Holland, Amsterdam, 998 [6] MC Er, A fast algorithm for computing order- Fibonacci numbers, Comput J 6 (983) 4 7 [7] MC Er, Computing sums of order- Fibonacci numbers in log time, Inform Process Lett 7 () (983) 5 [8] MC Er, Sums of Fibonacci numbers by matrix methods, The Fibonacci Quarterly (3) (984) [9] G Everest, T Ward, An Introduction to Number Theory, Graduate Texts in Mathematics, Springer, London, 005 [0] AF Horadam, Pell identities, Fibonacci Quart 9 (3) (97) 45 5, 63 [] D Kalman, Generalized Fibonacci numbers by matrix methods, Fibonacci Quart 0 () (98) [] E Kilic, D Tasci, On the generalized order- Fibonacci and Lucas numbers, Rocy Mountain J Math, to appear [3] E Kilic, D Tasci, The generalized Binet formula, representation and sums of the generalized order- Pell Numbers, Taiwanese J Math, to appear [4] V Kumar, T Helleseth, in: VS Pless, W Huffman (Eds), Handboo of Coding Theory, North-Holland, Amsterdam, 998 [5] G-Y Lee, S-G Lee, J-S Kim, HK Shin, The Binet formula and representations of -generalized Fibonacci numbers, Fibonacci Quart 39 () (00) [6] C Levesque, On the mth-order linear recurrences, The Fibonacci Quarterly 3 (4) (985) [7] F MacWilliams, N Sloane, The Theory of Error Correcting Codes, North-Holland, Amsterdam, 977 [8] AJ Martin, M Rem, A presentation of the Fibonacci algorithm, Inform Process Lett 9 () (984) [9] EP Miles Jr, Generalized Fibonacci numbers and associated matrices, Amer Math Monthly 67 (960) [0] H Niederreiter, J Spanier (Eds), Monte Carlo and Quasi-Monte Carlo Methods, vol 998, Springer, Berlin, 000

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