Generalized k-Balancing and k-Lucas Balancing Numbers and Associated Polynomials

Kalika Prasad; Munesh Kumari*; Jagmohan Tanti

doi:10.5666/KMJ.2023.63.4.539

Article

Kyungpook Mathematical Journal 2023; 63(4): 539-550

Published online December 31, 2023 https://doi.org/10.5666/KMJ.2023.63.4.539

Generalized k-Balancing and k-Lucas Balancing Numbers and Associated Polynomials

Kalika Prasad and Munesh Kumari*, Jagmohan Tanti

Department of Mathematics, Central University of Jharkhand, Ranchi, India
e-mails: klkaprsd@gmail.com and muneshnasir94@gmail.com

Department of Mathematics, Babasaheb Bhimrao Ambedkar University, Lucknow, India
e-mail: jagmohan.t@gmail.com

Received: February 7, 2023; Revised: August 2, 2023; Accepted: August 9, 2023

Abstract

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Abstract

In this paper, we define the generalized k-balancing numbers {Bn(k)} and k-Lucas balancing numbers {Cn(k)} and associated polynomials, where n is of the form sk+r, 0≤r<k. We give several formulas for these new sequences in terms of classic balancing and Lucas balancing numbers and study their properties. Moreover, we give a Binet style formula, Cassini's identity, and binomial sums of these sequences.

Keywords: k-balancing numbers, k-Lucas balancing numbers, k-balancing and k-Lucas balancing polynomials, Partial sums