Morgan-Voyce polynomial derivative sequences (Q2719872)

There are two sequences of Morgan-Voyce polynomials, denoted \(B_n(x)\), \(C_n(x)\) respectively. Both satisfy the binary linear recurrence: NEWLINE\[NEWLINEX_n(x)= (x+2) X_{n-1}(x)- X_{n-2}(x)NEWLINE\]NEWLINE with initial conditions \(X_0(x)= a_0\), \(X_1(x)= a_1\), where \(a_0= 0\), \(a_1=1\) if \(X_n(x)= B_n(x)\); \(a_0= 2\), \(a_1= 2+x\) if \(X_n(x)= C_n(x)\). NEWLINENEWLINENEWLINEIn this paper, the authors investigate the two derivative sequences defined by: \(R_n= B_n'(1)\), \(S_n= C_n'(1)\). They obtain various identities for \(R_n\) and \(S_n\), some of which link these sequences to the better-known Fibonacci and Lucas sequences. They also present theorems regarding the divisibility of \(R_n\) and of \(S_n\) by 2, 3, and 5.