Balancing polynomials and their derivatives

Abstract

We study the generalization of balancing numbers with a new sequence of numbers called $k$-balancing numbers. Moreover, by using the Binet formula for $k$-balancing numbers, we obtain the identities including the generating function of these numbers. In addition, the properties of divisibility of these numbers are investigated. Further, balancing polynomials that are natural extensions of the $k$-balancing numbers are introduced and some relations for the derivatives of these polynomials in the form of convolution are also proved.