ON QUADRAPELL NUMBERS AND QUADRAPELL POLYNOMIALS

Transcription

1 Hacettepe Journal of Mathematics and Statistics Volume 8() (009), ON QUADRAPELL NUMBERS AND QUADRAPELL POLYNOMIALS Dursun Tascı Received 09:0 :009 : Accepted 04 :05 :009 Abstract In this paper we define and deal with the quadrapell numbers,, in terms of a linear recurrence relation of order 4, and define the quadrapell polynomials in x. Further we give the generating function, a Binet-like formula and formulae for sums of these numbers. Keywords: Quadrapell numbers, Quadrapell polynomials. 000 AMS Classification: B9.. Introduction Recurrence relations and their generalizations have many interesting properties, and they have many applications to every field of science. Koshy s book [8] is a good source for these applications. One application of second order linear recurrences occurs in graph theory. In [0] Minc give the relationship between the permanent of a certain (0, ) superdiagonal matrix and the k-generalized Fibonacci numbers. Second order linear recurrences, especially the Fibonacci and Lucas numbers and their generalizations, are studied in [6, 7, 9, ] In [6] the authors consider the relationship between the generalized order k Fibonacci, Lucas numbers and their sums of bipartite graphs. In [7] the authors defined two tridiagonal matrices and they gave a relationship between the permanents and determinants of these matrices and second order linear recurrences. In [] the authors presented a sequence of progressively better upper bounds for the Perron root of a nonnegative matrix. They showed that each element in the sequence is a function of the Perron root of the arithmetic symmetrization of a power of the matrix. The authors in [] and [] defined and studied various fourth order linear recurrences and their polynomial analogous. Gazi University, Department of Mathematics, Teknikokullar, Ankara, Turkey. dtasci@gazi.edu.tr

2 66 D. Tascı In this paper we define and deal with the quadrapell numbers,, using a linear recurrence relation of order 4, and define the quadrapell polynomials in x. Then we give some identities and formulae among the terms of the Fibonacci sequence, the Lucas sequence, the sequence and the sequence C(n) all of whose elements are, 0 or. Also various summation formulae including the terms are given. Moreover a generating matrix (a companion matrix) of order 4 is given for generating the terms. The ratios of consecutive Fibonacci numbers and Lucas numbers converge to the Golden ratio α =.680, and we give evidence to support the conjecture that the ratio of consecutive terms of the Quadrapell numbers is also the Golden ratio.. Main results.. Quadrapell Numbers... Definition. The Quadrapell numbers are defined by the recurrence relation = D(n ) + D(n ) + D(n 4), n 4, where D(0) = D() = D() =, D() =. The first few Quadrapell numbers are,,,, 4, 5, 9, 5,, 8, 6, 99, 6, 6, 4, 68, 04, 785, 889, 4675, 756, 8,... Just as ratios of consecutive Fibonacci numbers and Lucas numbers converge to the the Golden ratio α =.680, so we conjecture that the Quadrapell ratios also D(n ) converge to α =.680 (Golden ratio). The characteristic equation of the Quadrapell recurrence relation is given by x 4 x x = 0. It can be shown that this equation has two real roots and two imaginary roots. The real roots are given by α = + 5 (Golden ratio), β = 5 and the imaginary roots are given by γ = + i and δ = i. Using the recursive definition, we can develop a generating function for the Quadrapell numbers. We require a real-valued function g(x) whose formal power series expansion at x = 0 has the form x n = D(0) + D()x + D()x + + x n +. If this is the case, the formal power series expansions of x g(x),x g(x) and x 4 g(x) are respectively D(0)x + D()x + D()x D(n )x n +, D(0)x + D()x 4 + D()x D(n )x n +,and D(0)x 4 + D()x 5 + D()x D(n 4)x n +,

3 On Quadrapell Numbers and Polynomials 67 so the expansion for g(x) x g(x) x g(x) x 4 g(x) is since and D(0) + D()x + (D() D(0))x + (D() D() D(0))x = + x x, D(0) = D() = D() =, D() = + (D(4) D() D() D(0))x 4 + = D(n ) + D(n ) + D(n 4). However, + x x is a finite series so we may write that is, ( x x x 4 )g(x) = + x x, + ( D(n ) D(n ) D(n 4))x n + + x x g(x) = x x x. 4 Now we present a theorem giving a Binet-like formula for Quadrapell numbers... Theorem. where = αn+ β n+ α β α = + 5, β = 5 are the roots of the equation x 4 x x = 0. ( α n β n α β ) + ( γ n δ n γ δ ),, γ = + i, δ = i Proof. Expressing the generating function of the Quadrapell sequence as a sum of partial fractions we obtain: g(x) = x x + x + x x + x + = x x x + x x + x x + x + = ( ) α n β n ( ) x n α n+ β n+ + x n + ( ) γ n δ n x n α β α β γ δ [ = ( ) ( ) α n β n α n+ β n+ + + ( )] γ n δ n x n α β α β γ δ We note from [] that and x x x = x x = ( α n β n α β ) x n ( α n+ β n+ α β ).

4 68 D. Tascı Thus, by the equality of generating functions, we get the stated Binet-like formula for the Quadrapell sequence... Corollary. i) C(n) = F(n + ) + F(n ), ii) C(n) = L(n), where F(n), L(n) and C(n) denote the Fibonacci sequence, the Lucas sequence, and C(n) = γn δ n γ δ, respectively. Now we present an important conjecture. It is of interest to consider the ratio of successive terms in this sequence. Initial values are given in Table. This ratio appears to converge to the Golden ratio, α =.680. For large n, this value is found to be D(n + ).680 (Golden Ratio) Table n D(n+) n D(n+) n D(n+) It appears from Table that D(n+) approaches α = , the magic number, as n. This encourages us to make the following conjecture:.4. Conjecture. D(n+) x 4 x x = 0. converges to the limit α, the positive root of We can extended the definition of to negative subscripts. If we apply the Quadrapell recurrence relation to the negative side, we obtain Table n D( n) We can generalize Table, for k N, n Z + : +, if n = 6k + D( n) =, if n = 6k + 4, if n = k + Now we give some theorems related to sums of Quadrapell numbers.

5 On Quadrapell Numbers and Polynomials Theorem. For k N k+ D(j) = D(k + ). Proof. We use the principle of mathematical induction. Since D(0) + D() = = D(), clearly the result is true for k = 0. Now assume it is true for an arbitrary positive integer k >, k+ Then, we have k+4 D(j) = D(k + ). D(j) = k+ D(j) + D(k + ) + D(k + ) + D(k + 4) = D(k + ) + D(k + ) + D(k + ) + D(k + 4) (by the induction hypothesis) = D(k + 4) + D(k + ) + D(k + ) = D(k + 6) (by the Quadrapell recurrence relation) So the formula works for k +. Thus, by the principle of mathematical induction the formula holds for every integer k Theorem. For k N, k+ D(j) = D(k + 4). Proof. We proceed by induction on k. Since D(0) + D() + D() = = D(4), the statement is true for k = 0. Assume it is true for k >, k+ D(j) = D(k + 4). Then, we show that the formula holds for k +. Indeed, k+5 D(j) = k+ D(j) + D(k + ) + D(k + 4) + D(k + 5) = D(k + 4) + D(k + ) Thus the formula works for k Lemma. + D(k + 4) + D(k + 5) (by the induction hypothesis) = D(k + 5) + D(k + 4) + D(k + ) = D(k + 7) (by the Quadrapell recurrence relation) + D(n + ) + D(n + ) + D(n + 5) = D(n + 6).

6 70 D. Tascı Proof. By the Quadrapell recurrence relation, we have D(n + 5) = D(n + ) + D(n + ) + D(n + ). Thus we obtain + D(n + ) + D(n + ) + D(n + 5) = + D(n + ) + D(n + ) + D(n + ) = D(n + ) + D(n + ) + + D(n + ) + D(n + ) = D(n + 4) + D(n + ) + D(n + ) (by the recurrence relation) = D(n + 6) (by the recurrence relation). So the lemma is proved..8. Theorem. For k N, k+ D(j) = D(6k + ). Proof. We use induction on k. Since D(0) + D() = = D(), the formula works for k = 0. Now we assume that it is true for k >. Then k+4 D(j) = k+ D(j) + D(6k + 4) + D(6k + 6) + D(6k + 8) = D(6k + ) + D(6k + 4) + D(6k + 6) + D(6k + 8) (by the induction hypothesis) = D(6k + 9) (by Lemma.7). Note that if we take n = 6k + in Lemma.7, then we obtain D(6k + ) + D(6k + 4) + D(6k + 6) + D(6k + 8) = D(6k + 9). Thus the formula is true for k +. So, the proof is complete..9. Theorem. For k N, k+ D(j) = D(6k + 5) +. Proof. We will use proof by induction. Since D(0) + D() + D(4) = 6 = D(5) +, clearly it is true for k = 0. We now suppose that the statement holds for k >, i.e. k+ D(j) = D(6k + 5) +.

7 On Quadrapell Numbers and Polynomials 7 Then k+5 D(j) = k+ D(j) + D(6k + 6) + D(6k + 8) + D(6k + 0) = D(6k + 5) + + D(6k + 6) + D(6k + 8) + D(6k + 0) = D(6k + ) + (by Lemma.7). Thus the formula works for k +. So, by the principle of mathematical induction, the given statement is true for every integer k Theorem. For k N, k D(j + ) = D(6k + ). Proof. We use the principle of mathematical induction on k. Since D() = = D(), the formula is true for k = 0. Now we assume that it is true for k >, that is, k Then we have k+ D(j + ) = D(6k + ). k D(j + ) = D(j + ) + D(6k + ) + D(6k + 5) + D(6k + 7) = D(6k + ) + D(6k + ) + D(6k + 5) + D(6k + 7) = D(6k + 8) (by Lemma.7). Thus, the formula works for k +. Consequently the given statement is true for every integer k 0... Theorem. For k N, k+ D(j + ) = D(6k + 4). Proof. We proceed by induction on k, Since D() + D() = = D(4), the formula is true for k = 0. Assume it is true for k >, that is k+ Then we have k+4 D(j + ) = D(6k + 4). D(j + ) = k+ D(j + ) + D(6k + 5) + D(6k + 7) + D(6k + 9) = D(6k + 4) + D(6k + 5) + D(6k + 7) + D(6k + 9) = D(6k + 0) (by Lemma.7).

8 7 D. Tascı Therefore the formula works for k +. So, by the principle of mathematical induction the formula holds for every integer k 0... Definition. The matrix of the Quadrapell recurrence relation is defined by 0 T = The characteristic polynomial of the matrix T is λ 4 λ λ. We note that the characteristic equation of the matrix T is same as that of the Quadrapell equation. Furthermore, the eigenvalues of the matrix T are λ = + 5 = α, λ = 5 = β, λ = + i = γ, λ4 = i = δ. Also the eigenvectors of T are X = = 5 + α α α + β β β, for λ = + 5 = α (Golden ratio), 5 X = 5 =, for λ = 5 = β, X = i + i = δ γ, for λ = + i = γ, X 4 = + i i = γ δ, for λ4 = i = δ. We note that det(t) =... Definition. [8] Let λ, λ,..., λ n be the (real or complex) eigenvalues of a matrix A. Then its spectral radius ρ(a) is defined as: ρ(a) = max i n ( λi )..4. Theorem. [] For any matrix norm, ρ(a) = lim n An n..5. Lemma. Let T be the matrix of the Quadrapell recurrence relation. Then for every integer n we have lim T n n = + 5 = α (Golden ratio). n

9 On Quadrapell Numbers and Polynomials 7 Proof. Since the spectral radius of the matrix T is ρ(t) = + 5, the desired result follows by Theorem Theorem. D(n + ) D(n + ) D(n + ) D(n + ) D(n + ) D(n ) D(n + ) D(n ) D(n ) = ( ) n 5 D(n ) D(n ) D(n ) Proof. We use the principle of mathematical induction on n. Since D(4) D() D() D() 4 D() D() D() D(0) D() D() D(0) D( ) = = 5 D() D(0) D( ) D( ) the result is true for n =. Now we assume that it is true for an arbitrary positive integer n >, that is D(n + ) D(n + ) D(n + ) D(n + ) D(n + ) D(n ) D(n + ) D(n ) D(n ) = ( ) n 5 D(n ) D(n ) D(n ) Then using the Quadrapell recurrence relation and properties of determinants, we have D(n + 4) D(n + ) D(n + ) D(n + ) D(n + ) D(n + ) D(n + ) D(n + ) D(n + ) D(n ) D(n + ) D(n ) D(n ) D(n + ) + D(n + ) + D(n + ) D(n + ) D(n + ) = D(n + ) + + D(n ) D(n + ) D(n + ) + D(n ) + D(n ) D(n + ) D(n ) D(n ) + D(n ) + D(n ) D(n ) D(n )

10 74 D. Tascı D(n + ) D(n + ) D(n + ) D(n + ) = D(n + ) D(n + ) D(n + ) D(n + ) D(n ) D(n ) D(n ) D(n ) }{{} 0 D(n + ) D(n + ) D(n + ) D(n + ) + D(n + ) D(n + ) D(n ) D(n + ) D(n ) D(n ) D(n ) D(n ) }{{} D(n + ) D(n + ) D(n + ) + D(n ) D(n + ) D(n + ) D(n ) D(n + ) D(n ) D(n ) D(n ) D(n ) D(n + ) D(n + ) D(n + ) = ( ) D(n + ) D(n + ) D(n ) D(n + ) D(n ) D(n ) D(n ) D(n ) D(n ) = ( )( ) n 5 (by the induction hypothesis) = ( ) n+ 5 Thus the statement works for n +. So, the equality is true for every integer n. 0.. Quadrapell Polynomials. Quadrapell numbers can be generalized to a set of polynomials, called the Quadrapell polynomials..7. Definition. The Quadrapell polynomials are defined by x, if n = 0,, if n =, D n(x) = x if n =, x, if n =, x D n (x) + xd n (x) + D n 4(x), if n 4. The first eight Quadrapell polynomials are: D 0(x) = x D (x) = D (x) = x D (x) = x D 4(x) = x 4 + x D 5(x) = x 5 + x + D 6(x) = x 6 + 4x 4 + x + x D 7(x) = x 7 + 4x 5 + x + 7x D 8(x) = x 8 + 8x 6 + x 5 + 6x 4 + 5x. Here we make an interesting observation: D i() = D(i), i =,,,..., n.

11 On Quadrapell Numbers and Polynomials 75 In fact, D n() =..8. Theorem. (x +x) n D i(x) = D n(x)+( x )D n+(x)+d n+(x)+d n+(x) x x x. Proof. Considering Definition 6, we have n n n n D i+(x) = x D i+(x) + x D i(x) + D i (x) or D n(x) + ( x )D n+(x) + D n+(x) + D n+(x) n = (x + x) D i(x) + D 0(x) + D (x) + D (x). On the other hand, since D 0(x) = x, D (x) = x and D (x) = x, it follows that n (x +x) D i(x) = D n(x)+( x )D n+(x)+d n+(x)+d n+(x) x x x. So the theorem is proved..9. Corollary. n D i = (Dn + Dn+ + Dn+ 4). Proof. Since D i() = D i, the proof is easily seen by Theorem.8. References [] Harary, F. Determinants, permanents and bipartite graphs, Math. Mag. 4, 46 48, 969. [] Harne, S. and Parihar, C. L. Some generalized Fibonacci polynomials, J. Indian Acad. Math. 8(), 5 5, 996. [] Harne, S. and Singh, B. Some properties of fourth-order recurrence relations, Vikram Math. J. 0, 79 84, 000. [4] Horadam, A. F. and Shannon, A. G. Irrational sequence-generated factors of integers, Fibonacci Quart. 9(), 40 50, 98. [5] Horn, R. and Johnson, C. Matrix Analysis (Cambridge University Press, 985). [6] Kilic, E. and Tasci, D. On families of bipartite graphs associated with sums of Fibonacci and Lucas numbers, Ars Combin. 89, 40, 008. [7] Kilic, E. and Tasci, D. On the second order linear recurrences by tridiagonal matrices, Ars Combin. (to appear). [8] Koshy, T. Fibonacci and Lucas Numbers with Applications (John Wiley & Sons Inc., 00). [9] Lee, G. -Y. k-lucas numbers and associated Bipartite graphs, Linear Algebra and its Appl. 0, 5 6, 000. [0] Minc, H. Permanents of (0,)-circulants, Canad. Math. Bull. 7 (), 5 6, 964. [] Sato, S. On matrix representations of generalized Fibonacci numbers and their applications, in Applications of Fibonacci Numbers, Vol. 5 (St. Andrews, 99) (Kluwer Acad. Publ., Dordrecht, 99), [] Tasci, D. and Kirkland, S. Sequence of upper bounds for the Perron root of nonnegative matrix, Linear Algebra and Its Appl. 7, 8, 998.