Higher-Order Generalized q-Difference Equations and Lucas-Type Series Solutions via Inverse q-Difference Operators

Abstract

In this paper, we introduce and analyze a novel class of t^th-order generalized q-difference equations associated with multi-parameter recurrence relations. By employing an inverse q-difference operator framework, explicit solution representations are derived in terms of generalized Lucas-type sequences. A higher-order Lucas series formula is established, providing closed-form summation identities for polynomial and logarithmic forcing functions. Several corollaries illustrate the effectiveness of the proposed method, including quadratic and logarithmic source terms. The obtained results extend classical Fibonacci-Lucas summation techniques to higher-order q-calculus and unify discrete summation identities within a single operator-theoretic setting. The proposed framework offers a systematic approach for solving higher-order q-difference equations and paves the way for further developments in fractional q-difference equations, discrete dynamical systems, and applications involving special sequences.