This paper generalizes a result of Gerdemann to show (with slight variations in some special cases) that, for any real number m and Horadam function Hn(A, B, P,Q), mHn(A, B, P, Q) = i=h k∑(t,Hn+i(A, B, P, Q)), for two consecutive values of n, if and only if, m= i=h k∑ (tiai)= i=h k∑ (t ibi) where a =(P+(P2-4Q) 1/2)/2 and b = (P-(P2-4Q) 1/2)/2. (Horadam functions are defined by: H 0(A, B, P, Q) = A, H1(A,B,P,Q) = B, Hn+1(A, B, P,Q) = PHn(A,B, P,Q)-QHn-1(A, B, P,Q).) Further generalizations to the solutions of arbitrary linear recurrence relations are also considered.
Bunder, M. W. (2012). Horadam functions and powers of irrationals. Fibonacci Quarterly, 50 (4), 304-312