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On determinants involving generalized Fibonacci numbers. (English) Zbl 0191.04501

Summary: The Generalized Fibonacci numbers are defined by the relation \(T_1 = a\), \(T_2 = b\), \(T_{n+2} = T_{n+1} + T_n\). On taking \(a = b = 1\), the Fibonacci numbers are obtained, while \(a=1\), \(b =3\), gives the Lucas numbers. In this paper we prove the relation: \[ T_{m+r}F_{n+r} + (-1)^{r+1} T_mF_n =T_{m+n+r}F_r, \] and evaluate some determinants and circulants whose elements are the Generalized Fibonacci numbers or their products. Several particular cases are obtained by specializing the parameters.
These results have now been extended for the Fibonacci polynomials, and have been submitted for publication.

MSC:

11C20 Matrices, determinants in number theory
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
15A15 Determinants, permanents, traces, other special matrix functions