A difference of sums of finite products of lucas-balancing polynomials

Abstract

The Lucas-balancing numbers arise naturally from the balancing numbers which were introduced by Behera and Panda about twenty years ago and have been undergone intensive studies by many researchers. Natural extensions of the Lucas-balancing numbers are the Lucas-balancing polynomials. In this paper, we will consider a difference of sums of finite products of Lucas-balancing polynomials and represent them in terms of nine orthogonal polynomials in two different ways each. In particular, this gives us an expression of such a difference of sums of finite products in terms of Lucas-balancing polynomials. Our proof is based on the recent observation by Frontczak as to a fundamental relation between Chebyshev polynomials of the first kind and Lucas-balancing polynomials. ? 2020 Jangjeon Mathematical Society. All rights reserved.