Decompositions of the Cauchy and Ferrers-Jackson Polynomials

Abstract

Recently, Witula and Slota have given decompositions of the Cauchy and Ferrers-Jackson polynomials [Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences, Central Europan J. Math., 2006]. Our main purpose is to derive a different decomposition of the Cauchy and Ferrers-Jackson polynomials. Our approach is to use the Waring formula and the Saalschutz identity to prove the claimed results. Also, we obtain generalizations of the results of Carlitz, Hunter and Koshy as corollaries of our results about sums and differences of powers of the Fibonacci and Lucas numbers.