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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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