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Horadam functions and powers of irrationals

Journal Article


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Abstract


  • 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.

Publication Date


  • 2012

Citation


  • Bunder, M. W. (2012). Horadam functions and powers of irrationals. The Fibonacci Quarterly: a journal devoted to the study of integers with special properties, 50 (4), 304-312.

Scopus Eid


  • 2-s2.0-84870490515

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1172&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/167

Has Global Citation Frequency


Number Of Pages


  • 8

Start Page


  • 304

End Page


  • 312

Volume


  • 50

Issue


  • 4

Abstract


  • 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.

Publication Date


  • 2012

Citation


  • Bunder, M. W. (2012). Horadam functions and powers of irrationals. The Fibonacci Quarterly: a journal devoted to the study of integers with special properties, 50 (4), 304-312.

Scopus Eid


  • 2-s2.0-84870490515

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1172&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/167

Has Global Citation Frequency


Number Of Pages


  • 8

Start Page


  • 304

End Page


  • 312

Volume


  • 50

Issue


  • 4