## 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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