A generalization of André-Jeannin's symmetric identity

In 1997, Richard André-Jeannin obtained a symmetric identity involving the reciprocal of the Horadam numbers Wn, defined by a three-term recurrence W_{n+2} = P W_{n+1} − Q W_n with constant coefficients. In this paper, we extend this identity to sequences {a_n}_{n∈ℕ} satisfying a three-term recurrence a_{n+2} = p_{n+1} a_{n+1} + q_{n+1} a_n with arbitrary coefficients. Then, we specialize such an identity to several q-polynomials of combinatorial interest, such as the q-Fibonacci, q-Lucas, q-Pell, q-Jacobsthal, q-Chebyshev and q-Morgan-Voyce polynomials.