Some Generalizations and Properties of Balancing Numbers

Abstract

The sequence of balancing numbers admits generalization in two different ways. The first way is through altering coefficients occurring in its binary recurrence sequence and the second way involves modification of its defining equation, thereby allowing more than one gap. The former generalization results in balancing-like numbers that enjoy all important properties of balancing numbers. The second generalization gives rise to gap balancing
numbers and for each particular gap, these numbers are realized in multiple sequences. The definition of gap balancing numbers allow further generalization resulting in higher order gap balancing numbers but unlike gap balancing numbers, these numbers are scarce, the existence of these numbers are often doubtful. The balancing zeta function—a variant of Riemann zeta function—permits analytic continuation to the entire complex plane, while the series converges to irrational numbers at odd negative integers. The periods of balancing numbers modulo positive integers exhibits many wonderful properties. It coincides with the modulus of congruence if calculated modulo any power of two. There are three known primes such that the period modulo any one of these primes is equal to the period modulo its square. The sequence of balancing numbers remains stable modulo half of the primes, while modulo other half, the sequence is unstable.