A generalization of André-Jeannin’s symmetric identity

In 1997, Richard André-Jeannin obtained a symmetric identity involving the reciprocal of the Horadam numbers W_n, defined by a three-term recurrence W_n+2 = P W_n+1− QW_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+1a_n+1 + q_n+1a_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.