q-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials - Kachurik, I. I. (Ukrains’kyi Matematychnyi Zhurnal, 1998)
We obtain algebraic relations (identities) for q-numbers that do not contain q^α-factors. We derive a formula that expresses any q-number [x] in terms of the q-number [2]. We establish the relationship between the q-numbers [n] and the Fibonacci numbers, Chebyshev polynomials, and other special functions. The sums of combinations of q-numbers, in particular, the sums of their powers, are calculated. Linear and bilinear generating functions are found for “natural” q-numbers.