A series whose terms are products of two \(q\)-Bernoulli numbers in the \(p\)-adic case (Q2771471)

If \(\{ B_n\}\) is the sequence of Bernoulli numbers, and NEWLINE\[NEWLINE A_{m,n}=\frac{1}n\sum\limits_{j=1}^n(-1)^j\binom njB_{m+j}B_{n-j}, NEWLINE\]NEWLINE then \(A_{m,n}=A_{n-1,m+1}\) [\textit{C. F. Woodcock}, J. Lond. Math. Soc. (2) 20, 101-108 (1979; Zbl 0406.12009)]. NEWLINENEWLINENEWLINEThe authors construct analogs of the above convolution and symmetry property for two \(p\)-adic \(q\)-analogs of the Bernoulli numbers introduced by \textit{H. Tsumura} [J. Number Theory 39, 251-256 (1991; Zbl 0735.11009)] and \textit{T. Kim} [J. Number Theory 76, 320-329 (1999; Zbl 0941.11048)]. As an application, they obtain some analogs of the Euler identity for Bernoulli numbers.