qq-derivative operator proof for a conjecture of Melham

Melham conjectured two identities between Fibonacci and Lucas numbers in 1999. Subsequently, Kilic et al. showed them in 2010 by contour integration. In this paper, we present a new proof using the qq-derivative operator. By examining two rational functions, we further derive several identities concerning Fibonomial coefficients. Finally, two well-known summation formulae on qq-series are transformed into convolution formulae on Fibonomial coefficients.