Some formulate for the Fibonacci and Lucas numbers

This paper mainly study some identities involving Fibonacci and Lucas numbers of interest. By using the properties of Chebyshev polynomials and combining the elementary and combinatorial method, we establish identities that Sigma(d-1)(j=0) F(n+j)F(m+j)x(j) , Sigma(d-1)(j=0) Ln+j Lm+j x(j) , Sigma F-d-1(j=0)n1+j Fn2+j Fn3+j, and Sigma L-d-1(j=0)n1+j Ln2+j Ln3+j. These identities are extensions of Sigma(d-1)(j=0) Fn+jFm+j and Sigma(d-1)(j=0) Ln+j Lm+j which have been proved by Brian Curtin before.