Sums of products of two \(q\)-Bernoulli numbers
Let \(q\in \mathbb C\), \(| q| <1\). The sequence of \(q\)-Bernoulli numbers \(\beta _n\) (\(n\geq 0\)) is defined by the relations \[ \beta _0=1,\quad \sum \limits_{j=0}^nq^{j+1}\binom nj\beta_j- \beta_n=\begin{cases} 1, &\text{for \(n=1\)}\\ 0, &\text{for \(n>1\)}.\end{cases} \] Let \(G_q\) be the generating function of the sequence \(\{ \beta_n\}\). It is proved that \[ G_q*\varphi (G_q)=G_q-(G_q+qG_q')*\varphi \] where \(*\) is a \(q\)-analogue of the product of formal power series [\textit{J. Satoh}, Nagoya Math. J. 127, 129-143 (1992; Zbl 0761.11023)], while \(\varphi\) is the differential operator \(I+(q-1)\frac{d}{dt}\).
scientific article; zbMATH DE number 1271088