A note on sums of finite products of Lucas-balancing polynomials
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Behera and Panda introduced balancing numbers about twenty years ago. Since then, these numbers have been intensively studied by many researchers and lots of interesting properties of them have been unveiled. Lucas-balancing numbers have close connection with balancing numbers and their natural extensions are Lucas-balancing polynomials. In this pa- per, we will study sums of finite products of Lucas-balancing polynomials and represent them in terms of nine orthogonal polynomials in two differ- ent ways each. In particular, we obtain an expression of such sums of finite products in terms of Lucas-balancing polynomials. Our proof is based on a fundamental relation between Lucas-balancing polynomials and Chevby- shev polynomials of the first kind observed by Frontczak.
Abstract
Behera and Panda introduced balancing numbers about twenty years ago. Since then, these numbers have been intensively studied by many researchers and lots of interesting properties of them have been unveiled. Lucas-balancing numbers have close connection with balancing numbers and their natural extensions are Lucas-balancing polynomials. In this pa- per, we will study sums of finite products of Lucas-balancing polynomials and represent them in terms of nine orthogonal polynomials in two differ- ent ways each. In particular, we obtain an expression of such sums of finite products in terms of Lucas-balancing polynomials. Our proof is based on a fundamental relation between Lucas-balancing polynomials and Chevby- shev polynomials of the first kind observed by Frontczak.
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