Formulae related to the \(q\)-Dixon formula with applications to Fibonomial sums (Q276443)

Let \[ \binom{n}{k}_q=\frac{(q;q)_n}{(q;q)_k (q;q_{n-k}},\qquad {\text{where}}\qquad (x;q)_n=(1-x)(1-xq)\cdots (1-xq^{n-1}) \] be the \(q\)-binomial coefficient. In the paper under review, the authors present several formulae related to triple sums of \(q\)-binomial coefficients. For example, \[ \sum_{k=0}^{2n} \binom{2n}{k}_q^2 \binom{2n+1}{k}_q(-1)^k q^{k(3k-6n-1)/2}=(-1)^n q^{-n(3n+1)/2} \binom{2n}{n}_q \binom{3n+1}{n}_q. \] All proofs use the \(q\)-Dickson identity \[ \sum_k (-1)^k q^{k(3k+1)/2} \binom{a+b}{a+k}_q \binom{b+c}{b+k}_q \binom{c+a}{c+k}_q=\frac{[a+b+c]!}{[a]! [b]! [c]!},\quad {\text{where}}\quad [n]!=\frac{(q;q)_n}{(1-q)^n}. \] Some applications to Fibonomial sums are presented.