A formula for the generating functions of powers of Horadam's sequence

The second-order linear recurrence sequence (wn(a, b; p, q))n≥0, or briefly (wn)n≥0, is defined by wn+2 = pwn+1 + qwn, (1) with w0 = a, w1 = b and n ≥ 0. This sequence was introduced in 1965 by Horadam [3, 4], and it generalizes many sequences (see [1, 5]). Examples of such sequences are Fibonacci number sequences (Fn)n≥0, Lucas number sequences (Ln)n≥0, and Pell number sequences (Pn)n≥0, when one has p = q = b = 1, a = 0; p = q = b = 1, a = 2; and p = 2, q = b = 1, a = 0; respectively. In this paper we are interested in studying the generating function for powers of Horadam’s sequence, that is, Hk(x; a, b, p, q) = Hk(x) = ∑ n≥0 w k nx . In 1962, Riordan [7] found the generating function for powers of Fibonacci numbers. He proved that the generating function Fk(x) = ∑ n≥0 F k nx n satisfies the recurrence relation