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

Abstract

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.