Some congruences for generalized binomial coefficients
For a sequence of integers \((u_n)\), the generalized binomial coefficients are defined by \([\begin{smallmatrix} n\\ k\end{smallmatrix}]= (u_n u_{n- 1}\cdots u_1)/(u_k u_{k- 1}\cdots u_1)\cdot (u_{n- k}\cdots u_1)\). Let \((u_n)\) be a second order recurrence sequence satisfying \(u_n= au_{n- 1}+ bu_{n- 2}\) \((n\geq 2)\), \(u_0:= 0\), \(u_1:= 1\). Let \((a, b)= 1\), and put \(D= a^2+ 4b\neq 0\), \(c= (a+ \sqrt D)/2\), \(d= (a- \sqrt D)/2\), \(v_n= c^n+ d^n\) \((n\geq 0)\). The authors prove three theorems, with consequences, relating to certain congruence properties of \([\begin{smallmatrix} n\\ k\end{smallmatrix}]\) for the above defined sequences. As a sample, we quote the following result: Let \(p\) be a prime, \(p\nmid b\), \(p\nmid D\). Then for \(n\geq m\geq 0\), \([\begin{smallmatrix} nr\\ mr\end{smallmatrix}]\equiv (v_r/2)^{(n- m) mr}\cdot (\begin{smallmatrix} n\\ m\end{smallmatrix})(\text{mod } p^2)\), where \(r\) denotes the index of the first element of \((u_n)\) divisible by \(p\).
scientific article; zbMATH DE number 878072