On The Second Order Linear Recurrences By Generalized Doubly Stochastic Matrices

In this paper, we consider the relationships between the second order linear recurrences, and the generalized doubly stochastic permanents and determinants. 1. Introduction The Fibonacci sequence, fFng ; is de…ned by the recurrence relation, for n 1 Fn+1 = Fn + Fn 1 (1.1) where F0 = 0; F1 = 1: The Lucas Sequence, fLng ; is de…ned by the recurrence relation, for n 1 Ln+1 = Ln + Ln 1 (1.2) where L0 = 2; L1 = 1: The well-known Fibonacci, Lucas and Pell numbers can be generalized as follows: Let A and B be nonzero, relatively prime integers such that D = A 4B 6= 0: De…ne sequences fung and fvng by, for all n 2 (see [14]), un = Aun 1 Bun 2 (1.3) vn = Avn 1 Bvn 2 (1.4) where u0 = 0; u1 = 1 and v0 = 2; v1 = A: If A = 1 and B = 1; then un = Fn (the nth Fibonacci number) and vn = Ln (the nth Lucas number). If A = 2 and B = 1; then un = Pn ( the nth Pell number). An alternative is to let the roots of the equation t At+B = 0 be, for n 0 un = n n and vn = n + : (1.5) There are many connections between permanents or determinants of tridiagonal matrices and the Fibonacci and Lucas numbers. For example, 2000 Mathematics Subject Classi…cation. 11B37, 15A15, 15A51. Key words and phrases. Second order linear recurrences, Generalized doubly stochastic matrix, Permanent, Determinant. 1 2 E. KILIC AND D. TASCI Minc [12] de…ne a n n super diagonal (0; 1) matrix F (n; k) for n > k 2; and show that the permanent of F (n; k) equals to the generalized order-k Fibonacci numbers. Also he give some relations involving the permanents of some (0; 1) Circulant matrices and the usual Fibonacci numbers. In [8], the authors present a nice result involving the permanent of an ( 1; 0; 1)-matrix and the Fibonacci Number Fn+1: The authors then explore similar directions involving the positive subscripted Fibonacci and Lucas Numbers as well as their uncommon negatively subscripted counterparts. Finally the authors explore the generalized order-k Lucas numbers, (see [19] and [7] for more detail the generalized Fibonacci and Lucas numbers), and their permanents. In [9] and [10], the authors gave the relations involving the generalized Fibonacci and Lucas numbers and the permanent of the (0; 1) matrices. The results of Minc, [12], and the result of Lee, [9], on the generalized Fibonacci numbers are the same because they use the same matrix. However, Lee proved the same result by a di¤erent method, contraction method for the permanent (for more detail of the contraction method see [1]). In [11], Lehmer proves a very general result on permanents of tridiagonal matrices whose main diagonal and super-diagonal elements are ones and whose subdiagonal entries are somewhat arbitrary. Also in [16] and [17], the authors de…ne a family of tridiagonal matrices M (n) and show that the determinants ofM (n) are the Fibonacci numbers F2n+2. In [4] and[3], the family of tridiagonal matrices H (n) and the authors show that the determinants of H (n) are the Fibonacci numbers Fn: In a similar family of matrices, the (1; 1) element of H (n) is replaced with a 3. The determinants, [2], now generate the Lucas sequence Ln: In [5], the authors …nd the families of (0; 1) matrices such that permanents of the matrices, equal to the sums of Fibonacci and Lucas numbers. Recently, in [6], the authors de…ne two tridiagonal matrices and then give the relationships the permanents and determinants of these matrices and the second order linear recurrences. The permanent of an n-square matrix A = (aij) is de…ned by perA = X 2Sn n Y i=1 ai (i) where the summation extends over all permutations of the symmetric group Sn: Also one can …nd more applications of permanents in [13]. Let A = [aij ] be an m n real matrix row vectors 1; 2; : : : ; m: We say A is contractible on column (resp. row:) k if column (resp. row:) k contains exactly two nonzero entries. Suppose A is contractible on column k with aik 6= 0 6= ajk and i 6= j: Then the (m 1) (n 1) matrix Aij:k obtained from A by replacing row i with ajk i + aik j and deleting row ON THE SECOND ORDER LINEAR RECURRENCES 3 j and column k is called the contraction of A on column k relative to rows i and j: If A is contractible on row k with aki 6= 0 6= akj and i 6= j; then the matrix Ak:ij = h Aij:k iT is called the contraction of A on row k relative to columns i and j: Every contraction used in this paper will be on the …rst column using the …rst and second rows. We say that A can be contracted to a matrix B if either B = A or exist matrices A0; A1; : : : At (t 1) such that A0 = A; At = B; and Ar is a contraction of Ar 1 for r = 1; 2; : : : ; t: One can …nd the following fact in [1]: let A be a nonnegative integral matrix of order n > 1 and let B be a contraction of A. Then perA = perB: (1.6) We also recall the following de…nitions : De…nition 1. A matrix A = (aij) of order n is said to be nonnegative if aij 0; i; j = 1; 2; : : : ; n: De…nition 2. A nonnegative n n matrix A is called row stochastic, or simply stochastic, if all its rows sum 1. De…nition 3. A nonnegative n n matrix A is called row stochastic, if all its rows and colums sum 1. We give the following de…nitions (see [15] and [18], respectively). De…nition 4. A matrix A = (akj) of order n is said to be generalized stochastic if n X j=1 akj = s; k = 1; 2; : : : ; n where s is a complex number. De…nition 5. If A = (akj) is such that n X j=1 akj = s; k = 1; 2; : : : ; n and n X k=1 akj = s; j = 1; 2; : : : ; n then A is said to be generalized doubly stochastic matrix. Note that a generalized stochastic or generalized doubly stochastic matrix need to be nonnegative. In this paper, we give the relationships between the permanents of some generalized symmetric doubly stochastic matrices and the second order linear recurrences. 4 E. KILIC AND D. TASCI 2. The Main Results In this section, we de…ne a n n generalized symmetric doubly stochastic matrix Dn and then show that its permanent equals to the nth term of the sequence fvng : We de…ne a n n generalized symmetric doubly stochastic matrix Dn with d11 = + ; dii = 0 for 2 i n 1; let n be an even number, d2k;2k+1 = + for 1 k n 2 2 ; d2k 1;2k = + for 1 k n 2 and dnn = + ; and, let n be an odd number, d2k;2k+1 = + and d2k 1;2k = + for 1 k n 1 2 ; and dnn = + : Clearly, if n is an even , then Dn = 266666666666664 + + 0