Concerning an identity for differences of arbitrary order (Q1282541)

The authors prove the identity \[ \sum^{2m-1}_{i=0}a_{im}b_i=\sum^{m-1}_{k=0}\lambda_{km}\Delta^mb_k,\quad m\in\mathbb{N}, \] in which the numbers \(b_k\), \(k\in\mathbb{Z}\), are arbitrary and the numbers \(a_{im}\) and \(\lambda_{km}\) are given by \[ a_{im}=\sum^m_{j=i+1}{(-1)^{j+1}\over j}\cdot{m!\over j!(m-j)!}+\sum_{(i+1)/2\leq j\leq i}{(-1)^j\over j}{m!\over j!(m-j)!},\quad i=0,\dots,2m-1, \] \[ \lambda_{km}=(-1)^ma_{km}+(-1)^{k+1}\sum^{k-1}_{j=0}(-1)^j {m!\over(k-j)!(m-k+j)!}\lambda_{jm},\quad k=0,\dots,m-1, \] and \(\Delta^mb_k\) is an \(m\)-th order difference \[ \Delta^mb_k=\sum^m_{j=0}(-1)^{m+j}{m!\over j!(m-j)!}b_{k+j}. \]