We extend Fibonacci numbers with arbitrary weights and generalize a dozen identities. As special case, we propose an elliptic extension which extends the q-Fibonacci polynomials appearing in Schur's work. The proofs of most identities are combinatorial, extending given by Benjamin Quinn, q-case, Garrett. Some proved telescoping.

Circulant matrix family occurs in various fields, applied in image processing, communications, signal processing, encoding and preconditioner. Meanwhile, the circulant matrices [1, 2] have been extended in many directions recently. The f(x)-circulant matrix is another natural extension of the research category, please refer to [3, 11]. Recently, some authors researched the circulant type matric...