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## Combinatorial formula for generalized Fibonacci sequence and Chebyshev's polynomials of two variables. (English)

Authors: Kolesnik Alexander

### Abstract

This paper concerns the explicit combinatorial formula for calculating the general term of the generalized Fibonacci sequence $c_0 = {\beta}$, \ $c_1 = {\alpha},$ $c_n = ac_{n-1} + bc_{n-2}, \quad n{\ge 2},$ where ${\alpha},{\beta},a,b$ are the elements of an arbitrary commutative algebra over the field of complex numbers. The connection between this formula and Chebyshev's polynomials of two variables is also established.

Institutul de Matematică Academia de Ştiinţe a Moldovei
str. Academiei 5, Chişinău MD-2028, Moldova