THE DIFFERENCE MATRICES OF GENERALIZED FIBONACCI AND LUCAS SEQUENCES
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The difference sequence ∆ = (∆a0, ∆a1, ∆a2, . . .)T of a sequence a = (a0, a1, a2, . . .)T is defined by ∆ai = ai+1 - ai for each i = 0, 1, 2, . . .. Let ∆ia = (∆ia0, ∆ia1, ∆ia2, . . .)T , i = 0, 1, 2, . . ., be the ith difference sequence defined inductively by ∆ia = ∆(∆-1a) where ∆0a = a. Let Ma = [mij ], (i, j = 0, 1, 2, . . .) denote the difference matrix of a whose ith row is the transpose of the ith difference sequence and let Ha denote an infinite Hankel matrix with aT as its first row. The entries in Ma satisfy the Γlaw defined by the recurrence relation mi,j+1 = mi+1,j+mij. In this paper, the Γ-law is applied to examine the recurrence relations of difference sequences of Fk,l and Lk,l, where Fk,l and Lk,l are (k, l)-Fibonacci and (k, l)-Lucas sequences, respectively, for positive real numbers k and l. Finally, we show that MFk,l = P-1HFk,l and MLk,l = P-1HLk,l , where P is the Pascal matrix.
Abstract
The difference sequence ∆ = (∆a0, ∆a1, ∆a2, . . .)T of a sequence a = (a0, a1, a2, . . .)T is defined by ∆ai = ai+1 - ai for each i = 0, 1, 2, . . .. Let ∆ia = (∆ia0, ∆ia1, ∆ia2, . . .)T , i = 0, 1, 2, . . ., be the ith difference sequence defined inductively by ∆ia = ∆(∆-1a) where ∆0a = a. Let Ma = [mij ], (i, j = 0, 1, 2, . . .) denote the difference matrix of a whose ith row is the transpose of the ith difference sequence and let Ha denote an infinite Hankel matrix with aT as its first row. The entries in Ma satisfy the Γlaw defined by the recurrence relation mi,j+1 = mi+1,j+mij. In this paper, the Γ-law is applied to examine the recurrence relations of difference sequences of Fk,l and Lk,l, where Fk,l and Lk,l are (k, l)-Fibonacci and (k, l)-Lucas sequences, respectively, for positive real numbers k and l. Finally, we show that MFk,l = P-1HFk,l and MLk,l = P-1HLk,l , where P is the Pascal matrix.
참고문헌 (11)
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