Congruences for Lucas \(u\)-nomial coefficients modulo \(p^3\) (Q2478006)

Let \(\{u_n\}\) be the Lucas sequence \(u_{n+1}= Au_n- Bu_{n-1}\), \(n\geq 2\), \(u_0= 0\), \(u_1= 1\) and \([n]= \prod^n_{k=1} u_k\). Then the Lucas \(u\)-nomial coefficient \({n\brack k}\) is defined by \({n\brack k}= [n]/([k][n-k])\) if \(n\geq k\) and \(0\) otherwise. The main results of this paper are congruences for special Lucas \(u\)-nomial coefficients \(\text{mod\,}p^2\) and \(\text{mod\,}p^3\) where \(p\) is a prime \(> 3\) not dividing \(B\).