A note on the \(q\)-integral and \(q\)-series (Q2729683)

The authors calculate the integral NEWLINE\[NEWLINE \int\limits_0^1\int\limits_0^1\frac{(xy)^n(qx;q)_n(q^{- n}y;q)_\infty }{(qxy;q)_{n+1}(y;q)_\infty }d_qx\cdot d_qy NEWLINE\]NEWLINE understood in the sense of Jackson. Here \(q\in (0,1)\), \((x;q)_n=(1-x)(1-qx)\cdots (1-q^{n-1}x)\), and \((x;q)_\infty =\prod_{i=0}^\infty (1-xq^i)\). The result is used to find the value of \(\sum_{n=1}^\infty \frac{1}{F_nF_{n+1}}\) where \(\{F_n\}\) is the sequence of Fibonacci numbers. It is shown that the number \(\sum_{n=1}^\infty \frac{1}{F_nF_{n+1}}\) is irrational.