$q$-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials

Abstract

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.