Higher-order balancing numbers: new sequences, recurrence relations, generating functions and identities

In this article, we study a novel extension of the classic balancing numbers, referred to as the higher-order balancing numbers and denoted by. This sequence is analogous to the higher-order Fibonacci numbers and follows the same recurrence relation as the balancing sequence itself. The case k=1 gives the classic balancing numbers (A001109) and for k=2 gives the sequence A029547, thus establishing a direct link to existing number sequences. Here, we first establish the Binet-like formula and then, with its help, present various algebraic properties of this newly introduced sequence, such as recurrence relations, generating functions (both ordinary and exponential), partial sums, binomial sums, combined identities, and more. We also obtain the limiting ratio and establish several well-known identities, including Catalan’s identity, d’Ocagane’s identity, Vajda’s identity, Honsberger’s identity, using the Binet-like formula. Finally, we give some mixed identity and series sum formulae. In this study, the obtained identities and algebraic properties are expressed in terms of the existing balancing and Lucas-balancing numbers.