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Identities involving covering systems. I. (English) Zbl 0809.11012
Applications of the concept of covering systems of congruences are given for proving some identities containing Bernoulli numbers. Let (1) \(a_ t \pmod {n_ t}\), \(0\leq a_ t< n_ t\), \(n_ t\in \mathbb{N}\) \((t=1,2,\dots, k)\) be given residue classes \(\pmod {n_ t}\). Assume that for (1) a real valued function \(\mu(t)= \mu_ t\) is given. Put \({\mathfrak m}(n)= \sum_{t=1}^ k \mu_ t \chi_ t(n)\) \((n\in \mathbb{Z})\), where \(\chi_ t\) is the indicator function of the class \(a_ t\pmod {n_ t}\) \((1\leq t\leq k)\). Then (1) is said to be a \((\mu,{\mathfrak m})\)-covering. It is called a covering system provided that \(\mu\equiv 1\) and \({\mathfrak m}(n) =1\) for each \(n\in \mathbb{Z}\). There is a relationship between \((\mu,{\mathfrak m})\)-coverings and some identities containing numbers \(a_ t\). The author illustrates this fact by proving three theorems with many corollaries. E.g. he proves that if (1) is a \((\mu,{\mathfrak m})\)-covering then for every real \(x\) we have \[ \begin{split} {\mathfrak m}(0) \Biggl[ {x\over {n_ 0}} \Biggr]+ {\mathfrak m}(1) \Biggl[ {{x+1} \over {n_ 0}} \Biggr]+ \cdots+ {\mathfrak m}(n_ 0 -1) \Biggl[ {{x+n_ 0 -1} \over {n_ 0}} \Biggr]=\\ \mu_ 1 \Biggl[ {{x+a_ 1} \over {n_ 1}} \Biggr]+ \mu_ 2 \Biggl[ {{x+a_ 2} \over {n_ 2}} \Biggr]+ \cdots+ \mu_ k \Biggl[ {{x+a_ k} \over {n_ k}} \Biggr],\end{split} \] where \(n_ 0\) is the least positive period of the function \({\mathfrak m}\).

MSC:
11B68 Bernoulli and Euler numbers and polynomials
11A07 Congruences; primitive roots; residue systems
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References:
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