# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] BEEBEE J.: Bernoulli numbers and exact covering systems. Amer. Math. Monthly 99 (1992), 946-948. · Zbl 0776.11008 [2] DEEBA E. Y., RODRIGUES D. M.: Stirling’s series and Bernoulli numbers. Amer. Math. Monthly 98 (1991), 423-426. · Zbl 0743.11012 [3] ERDÖS P.: On integers of the form 2k + p and some related problems. Summa Brasil. Math. 2 (1950), 113-123. [4] FRAENKEL A. S.: A characterization of exactly covering systems. Discrete Math. 4 (1973), 359-366. · Zbl 0257.10003 [5] NAMIAS V.: A simple derivation of Stirling’s asymptotic series. Amer. Math. Monthly 93 (1986), 25- 29. · Zbl 0615.05010 [6] PORUBSKÝ Š.: Natural exactly covering systems of congruences. Czechoslovak Math. J. 24(99) (1974), 598-606. · Zbl 0327.10005 [7] PORUBSKÝ Š.: Covering systems and generating functions. Acta Arith. 26 (1975), 223-231. [8] PORUBSKÝ Š.: On m times covering systems of congruences. Acta Arith. 29 (1976), 159-169. [9] PORUBSKÝ Š.: Results and Problems on Covering Systems of Residue Classes. Mitt. Math. Sem. Giessen. Heft 150, Univ. Giessen, Giessen, 1981. · Zbl 0479.10032 [10] RAABE J. L.: Die Jacob Bernoulli’sche Function. Zürich, 1848. [11] STERN J.: Beiträge zur Theorie der Bernoullischen und Eulerschen Zahlen. Abh. Geselschaft Wiss. Göttingen 23 (1878). 1-44. [12] ZNÁM Š.: Vector-covering systems of arithmetical sequences. Czechoslovak Math. J. 24(99) (1974), 455-461. · Zbl 0311.10003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.