Generalized Pascal matrix and recurrence sequences. (English) Zbl 1067.15502
Summary: This paper gives a product formula of the generalized Pascal matrix \(\phi_n[x,y]\), from this, getting the striking simplicity of the powers of \(\phi_n[x,y]\). Furthermore, this paper defines the IEGP-matrix \(\Omega_n[x,y].\) It is shown that not only can \(\Omega_n[x,y]\) be factorized by special summation matrices, but also has the closely relation with generalized Fibonacci sequences \(W_n(x,y)\). An explicit diagonalization of \(\Omega_n[x,y]\) is given.
MSC:
| 15A23 | Factorization of matrices |
| 11C20 | Matrices, determinants in number theory |
| 15B57 | Hermitian, skew-Hermitian, and related matrices |
References:
| [1] | Zhang, Z. Z.; Liu, M. X., An extension of generalized Pascal matrix and its algebraic properties, Linear Algebra Appl., 271, 169-177 (1998) · Zbl 0892.15018 |
| [2] | Call, G. S.; Velleman, D. J., Pascal’s matrices, Amer. Math. Monthly, 100, 372-376 (1993) · Zbl 0788.05011 |
| [3] | Zhang, Z. Z., The linear algebra of generalized Pascal matrix, Linear Algebra Appl., 250, 51-60 (1997) · Zbl 0873.15014 |
| [4] | Liverance, E.; Pitsenberger, J., Diagonalization of the binomial matrix, The Fibonacci Quarterly, 34, 1, 55-67 (1996) · Zbl 0853.11015 |
| [5] | Horadam, A. F., Basic properties of a certain generalized sequences of numbers, The Fibonacci Quarterly, 3, 2, 161-176 (1965) · Zbl 0131.04103 |
| [6] | Mercier, A., Identities containing Gauss binomial coefficients, Discrete Math., 76, 67-73 (1989) · Zbl 0672.05014 |
| [7] | Chen, W. Y.C.; Louck, J. D., The combinatorial power of the companion matrix, Linear Algebra Appl., 232, 261-278 (1996) · Zbl 0838.15015 |
| [8] | Lancaster, P.; Tismenesky, M., The Theory of Matrices (1985), Academic Press: Academic Press New York · Zbl 0558.15001 |
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