Convolution identities and lacunary recurrences for Bernoulli numbers (Q877924)

The authors extend Euler's quadratic recurrence relation of Bernoulli numbers, which is \[ (B_0+ B_0)^n:= \sum^n_{j=0} {n\choose j} B_j B_{n-j}=- nB_{n-1}- (n- 1)B_n\quad\text{for }n\geq 1, \] to \[ \begin{multlined} (B_k+ B_m)^n:= \sum^n_{j=0} {n\choose j} B_{k+j} B_{m+ n-j}\\ =-{k!m!\over (k+m+1)!}(n+ \delta(k, m)(k+ m+ 1))B_{n+ k+ m}+ \sum^{k+m}_{r=1} (-1)^r{B_{k+ m+1-r}\over k+ m+ 1-r}\\ \times \Biggl\{(-1)^k{k+1\choose r}\Biggl({k+1-r\over k+1} n-{rm\over k+1}\Biggr)+ (-1)^m{m+1\choose r}\Biggl({m+1-r\over m+1}n- {rk\over m+1}\Biggr)\Biggr\} B_{n+r-1},\end{multlined} \] where \(\delta(k,m)= 0\) when \(k= 0\) or \(m= 0\), and \(\delta(k,m)= 1\) otherwise (Theorem 2.1). And from special cases of the above identity, they obtain the expression of \(B_{6k}\) in terms of \(B_{2k}, B_{2k+2},\dots, B_{4k}\) as \[ B_{6k}=-{4k+1\over 6k+1} \sum^k_{r=0} {2k\choose 2r} \Biggl({2k(2k- 4r- 1)\over (2r+ 1)(2k- r)}+ 1\Biggr) B_{2k+2r} B_{4k- 2r} \] and the expressions of \(B_{6k+2}\) and \(B_{6k+ 4}\) as well. The proof of Theorem 2.1 is given by considering the Stirling numbers of the second kind \(S(n,k)\), whose relations to Bernoulli numbers are \[ {d^m\over dx^m}{1\over e^x- 1}= (-1)^m \sum^{m+1}_{j=1} (j- 1)! {S(m+ 1,j)\over (e^x- 1)^j} \] and \[ {d^m\over dx^m}{x\over e^x- 1}= (-1)^m \sum^{m+1}_{j=1} (j-1)! {S(m+ 1,j)x- mS(m, j)\over (e^x- 1)^j}. \]