Shortened recurrence relations for Bernoulli numbers
The authors start with two little known results of {Saalschütz} giving recurrence relations for the Bernoulli numbers \(B_{2n}\). The first of these results actually is contained in the second one which reads as \[ \sum_{k=m}^n (-1)^{k+1} (2^{2k}-2) C(n-m,k-m) B_{2k}=\sum_{j=0}^m (-1)^j\frac{4^{n-j}(n-j)!}{2(n-j)+1} C^{(n)}(m,j) \] with certain numbers \(C(n,m)\) and \(C^{(n)}(m,j)\) which may be given explicitly and where \(0\leq j\leq m\leq n\). The case \(m=0\) gives the first identity. Then the authors express these numbers \(C(n,m)\), \(C^{(n)}(m,j)\) in terms of Stirling numbers of both kinds. The also discuss results by \textit{P. G. Todorov} [J. Math. Anal. Appl. 104, 309--350 (1984; Zbl 0552.10007)] of a similar taste involving Stirling numbers \(s(k,l)\) and \(S(m,j)\) of the first and second kind respectively. Finally, using generating functions for the numbers \(S(n,k)\) they get for \(1\leq m\leq n\) and \(k\geq 0\) the formula \[ \sum_{j=m}^n \binom{n+k}{j-m} S(n-j+k+m,k+m) B_j=\frac{(-1)^m}{k+m} \sum_{j=1}^{m+1} \frac{N(n,m,k,j)}{\binom{k+m-1}{j-1}}, \] where \(N(n,m,k,j)=(n+k) S(m+1,j) S(n+k-1,k+m-j)-m S(m,j)S(n+k,k+m-j)\). From this they also derive some other formulas.
scientific article; zbMATH DE number 5542383
scientific article