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A generalization of a congruential property of Lucas. (English) Zbl 0755.11001
For a prime \(p\) and integers \(n,k\) let \(n=n_ 0+n_ 1p+\dots+n_ r p^ r\) (\(0\leq n_ i\leq p-1\)) and \(k=k_ 0+k_ 1p+\dots+k_ r p^ r\) (\(0\leq k_ i\leq p-1\)). If the functions \(F(n)\) and \(L(n,k)\) satisfy the congruences \[ \begin{aligned} F(n) & \equiv F(n_ 0)F(n_ 1)\dots F(n_ r)\quad \mod p\qquad\text{and}\\ L(n,k) &\equiv L(n_ 0,k_ 0)L(n_ 1,k_ 1)\dots L(n_ r,k_ r) \quad \mod p\end{aligned} \] for every prime \(p\), then we say that \(F\) has the Lucas property (LP) and \(L\) has the double Lucas property (DLP). In 1878 Lucas proved that the binomial coefficient function \(L(n,k)={n\choose k}\) is a DLP function. The author presents various properties and connections on these functions. A typical result: If \(L(n,k)\) is a DLP function, then \(F(n)=\sum_{k=0}^ n L(n,k)\) is an LP function.
Reviewer: P.Kiss (Eger)

MSC:
11A07 Congruences; primitive roots; residue systems
11B65 Binomial coefficients; factorials; \(q\)-identities
11A25 Arithmetic functions; related numbers; inversion formulas
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