Recently Witula and Slota give decompositions of the Cauchy andFerrers-Jackson polynomials [Cauchy, Ferrers-Jackson and Chebyshevpolynomials and identities for the powers of elements of some conjugaterecurrence sequences, Central Europan J. Math., 2006]. Our main purpose isto derive different decomposition of the Cauchy and Ferrers-Jacksonpolynomials. Our approach is to use the Waring formula and Saalsch\"{u}tz'sidentity to prove claimed results. Also we obtain generalizations of theresults of Carlitz, Hunter and Koshy as corollaries of our results aboutsums and differences of powers of the Fibonacci and Lucas numbers.