Representations for the first associated \(q\)-classical orthogonal polynomials (Q1863294)

In the present paper the author considers the expansion of the first associated \(q\)-classical polynomials \(p_n^{(1)}(x;q)\) in term of the basis \(x^k\) or \((x;q)_k\) , \(k\geq 0\), and also in term of the \(q\)-classical polynomials \(p_n(x;q)\) theirselves, i.e. the problem \[ p_n^{(1)}(x;q)=\sum_{k=0}^n c_{n,k} r(x;q), \] where \(r(x;q)=x^k\), \((x;q)_k\) or \(p_n(x;q)\). In the paper the explicit formulas for the coefficients \(c_{n;k}\) and also the recurrence relations involving these connection coefficients. The case of little and big \(q\)-Jacobi polynomials are considered in details.