Convolution summations for Pell and Pell-Lucas numbers (Q2707845)

The \(m\)th convolution Pell resp. Pell-Lucas, polynomials \(P_n^{ (m)}(x)\) and \(Q_n^{(m)}(x)\) are defined by generating functions: NEWLINE\[NEWLINE \sum^\infty_{n=0} P_{n+1}^{(m)} (x)y^n=(1-2xy -y^2)^{-(m+1)},\quad \sum^\infty_{ n=0} Q_{n+1}^{(m)}(x) y^n=\left({2x+2y \over 1-2xy-y^2} \right)^{m+1}.NEWLINE\]NEWLINE Putting \(x=1\) yields the \(m\)th convolution Pell numbers resp. Pell-Lucas numbers \(P_n^{(m)}\) and \(Q_n^{(m)}\). With \(m=0\) we have the Pell numbers and the Pell-Lucas numbers. Then recurrence relations for \(P_n^{(m)}\) and \(Q_n^{(m)}\) are given \((m\geq 1\) in both cases).