Complex Fibonacci Numbers and Fibonacci Quaternions

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  • Complex Fibonacci Numbers and Fibonacci QuaternionsAuthor(s): A. F. HoradamSource: The American Mathematical Monthly, Vol. 70, No. 3 (Mar., 1963), pp. 289-291Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2313129 .Accessed: 27/08/2013 10:22

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  • 1963] MATHEMATICAL NOTES 289

    COMPLEX FIBONACCI NUMBERS AND FIBONACCI QUATERNIONS A. F. HORADAM, University of New England, Armidale, Australia

    1. Introduction. In [1], [2] and [3], properties of the generalised Fibonacci sequence H1 H2 H3 H4 Hs H6 . . . Hn, . . . (1) H(_HXp): p p+q 2p+q 3p+2q 5p+3q 8p+5q (p-q)Fn+qFn+l*

    were discussed (p, q arbitrary integers), where F. is the nth term of the classical Fibonacci sequence F( H1o). The Lucas sequence is G-H12.

    Here, we extend the results of [1 ] and [2 ] to the complex case. Because of the various types of sequences dealt with, it is convenient to distinguish the cor- responding p, q, 1, m, e of [1] by an appropriate suffix. Thus, in (1), we must now write PH, * - - , eH for p, * * * ,e.

    2. Generalised complex Fibonacci sequence. Define the nth generalised com- plex Fibonacci number

    (2) D. = Hn + iHn+l = (PH - qH + iqH)Fn + (qH + ipH)Fn+l using (1), whence

    (3) PD = (1 + i)PH + iqH, D = qH + ipH

    i.e., the generalised complex Fibonacci sequence is D -H(1+i)pH+iqff,q+ipH. Special cases: (a) PH=1, qH=O: complex Fibonacci sequence C-H1+i,i

    (b) PH = 1, qH = 2: complex Lucas sequence H1+3i,2+i. From (a), we have

    (4) ec = 2pa-qc = 2( + i)-i=2 + i,

    and from [1], (3) and (4),

    eD = { (1 + i)PH + iqH}2 { (1 + i)PH + iqH} { qH + ipH} - { qH + iPH}2

    = (2 + i){pH - pHqH - qH} = eCeH.

    Another sequence, among many, for which ec =2 +i is H1,-i implying Hn =-iCn-1. The conjugate C,n of Cn = F,, +iFn+l leads to the sequence C-H_i,_ or, alternatively, H1,i for which Hn = iCn-,. Also Fc = ec.

    Parallelling [1], we have, for C, 2

    Cn-lCn+l - Cn = (-1)nec 2 2

    Cn, + Cn,+1 = iecF2n+2 2 2

    Cn+1-Cn-1 = iecF2n+l (5) ~ ~~2 32= 22 (2Cn+lCn+2) + (CnCn+3) = (2Cn+lCn+2 + Cn) (Pythagorean theorem)

    Cn,,+ + (_ 1)rC,_, = Fr+1 + Fr-1 = a, + bT (independent of n).

    C,,

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  • 290 MATHEMATICAL NOTES [March

    Also, for moduli, writing I Cn4 2 = cn, we find 2 2

    Jcn = Fn + Fn+l = F2n+l (6) {2f + 2~ Cn Cn-1 =Fn+l-Fn_l= F2n

    i.e., terms F1, F2, F3s, F4, *, F2n, F2n+l, * * * of F are expressible as co, c1-co, c1, c2 - c1, . . . i , cn-Cl C* *, respectively. Results (5) and (6) may be ex- tended to the general sequence without any difficulty, though some of the formulae begin to look a bit awkward. In the case of jDn| 2 =H + H = (2PH-qH)H2,+1-eHF2n+1, we see that there is no simple analogue to the moduli-sequence above for F.

    Whereas [3 ] the geometrical meaning of eH is the area of a certain parallelo- gram, in the complex cases ec and eD are merely complex numbers. Notice that arg Cn = tan-' (Fn+1/Fn) -> tan-' (1+V/5)/2 _ 58?17' and -r/4 < arg Cn

  • 1963] MATHEMATICAL NOTES 291

    Furthermore, we find 3

    (12) EI Qn+i = Gn+3 + iGn+4 + jGn+5 + kGn+6 i=O

    which is the quaternion Q,+3 for the Lucas sequence. Generalising, we define the nth generalised Fibonacci quaternion

    (13) Pn- =n + iHn+l + jHn?+2+ k1n+3

    yielding, in particular, PnPn = 3 { (2PH - qH) H2n+3-eHF2n+3 } It is not intended to carry the theory any further at this stage. Our definitions of complex Fibonacci sequences and quaternions do lead to

    some reasonably neat relationships. However, the question remains: are there other definitions which are more effective?

    References 1. A. F. Horadam, A generalized Fibonacci sequence, this MONTHLY, 68 (1961) 455-459. 2. , Fibonacci number triples, this MONTHLY, 68 (1961) 751-753. 3. , Fibonacci numbers and a geometrical paradox, Math, Mag., 35 (1962) 1-11.

    THE INVERSION OF A CONVOLUTION TRANSFORM WHOSE KERNEL IS A LAGUERRE POLYNOMIAL

    D. V. WIDDER, Harvard University

    Recently Ta Li [2] has shown that a certain convolution transform whose kernel involves a Tchebycheff polynomial can be inverted by another similar convolution. R. G. Buschman [1 ] has proved a similar result for Legendre poly- nomials. The present author has given new proofs of these two inversions by the method of the Laplace transform. That method reveals how similar formulas may be invented, and we apply it here to invert the transform

    (1) f(x) = Ln(- t)g(t)dt,

    where Ln(t) is the Laguerre polynomial

    1 Ln(X) = - Dn(e-xXn).

    ni

    We prove the following theorem.

    THEOREM. If g(x) C C1 for 0 ?x < oo, g(0) = 0, and f(x) is defined by equation (1), then f(x) E C2 for 0

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