On the Generalised Second-Order Linear Recurrences Relations and Identities

In this article, we study the sequence {Vn}, which is generated by the (p, q)-Generalised linear recurrence relation of second order Vn(p, q, a, b) = pVn-1 + qVn-2, n ≥ 2, with the initial terms V0 = a, V1 = b, where p, q, a and b (ab ≠ 0, pq ≠ 0) are arbitrary real numbers. Addition, subtraction formulas, Binet formula and some new results are obtained and studied in the generalised form. Some existing and new identities are also explored, employing this generalized definition of the sequence {Vn} and becoming the special cases, on substituting the coefficients p, q of the recurrence relation and the initial terms V0, V1.