The Lucas property of a number array (Q1598821)

Let \(p\) be a prime and \(\alpha\), \(\beta\), \(\gamma\), \(\delta\) be nonnegative integers such that \(0\leq \beta< p\) and \(0\leq\delta< p\). A double integer number array \(N(i,j)\), where \(i\) and \(j\) are nonnegative integers, is said to satisfy the Lucas property, if \(N(\alpha p+\beta, \gamma p+\delta)\equiv N(\alpha, \gamma)N(\beta,\delta)\pmod p\). The binomial coefficients \(C(i,j)= {i\choose j}\) are one of the double sequences satisfying the Lucas property. For all nonnegative integers \(i\), \(j\) let \(w(i,j\mid a,b,c)\) denote the number of all paths in the plane from \((0,0)\) to \((i,j)\) with steps \((1,0)\), \((0,1)\), \((1,1)\) and positive integer weights \(a\), \(b\), \(c\), respectively. The author proves the following theorem: \(w(\alpha p+\beta, \gamma p+\delta\mid a,b,c)\equiv w(\alpha,\gamma\mid a,b,c)w(\beta,\delta\mid a,b,c)\pmod p\).