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<p>In this article, we introduce a three-parameter family of k-Horadam sequences with arbitrary initial values and real-valued coefficient functions, extending the classical Horadam and Fibonacci sequences to third-order recurrence relations. We present several algebraic properties, including a Binet-type formula, partial sum identities, and closed-form ordinary and exponential generating functions. Moreover, we derive Catalan, d’Ocagne, and Vajda-type identities for this family of sequences. We also provide illustrative examples involving generalized Tribonacci, Perrin/Padovan, and third-order Jacobsthal sequences. The proposed results unify and generalize several known identities and properties from the literature.</p>

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On the three parameter family of the generalized k-Horadam sequences

  • 期刊论文

  • 发表日期：

    2026年1月13日

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  • Kalika Prasad,
  • Munesh Kumari,
  • Ritanjali Mohanty,
  • Hrishikesh Mahato

摘要

In this article, we introduce a three-parameter family of k-Horadam sequences with arbitrary initial values and real-valued coefficient functions, extending the classical Horadam and Fibonacci sequences to third-order recurrence relations. We present several algebraic properties, including a Binet-type formula, partial sum identities, and closed-form ordinary and exponential generating functions. Moreover, we derive Catalan, d’Ocagne, and Vajda-type identities for this family of sequences. We also provide illustrative examples involving generalized Tribonacci, Perrin/Padovan, and third-order Jacobsthal sequences. The proposed results unify and generalize several known identities and properties from the literature.