On unrestricted dual-generalized complex Horadam numbers

Abstract

This research introduces a novel category of dual-generalized complex numbers, with components represented by unrestricted Horadam numbers. We present various recurrence relations, summation formulas, the Binet formula, and the generating function associated with these numbers. Additionally, a comprehensive bilinear index-reduction formula is derived, which encompasses Vajda’s, Catalan’s, Cassini’s, D’Ocagne’s, and Halton’s identities as specific cases.