Download Full text

MORE FIBONACCI FUNCTIONS
M.W. BONDER
Wollongong University College, The University of New South Wales, Wollongong, N. S. W., Australia
Recently there have appeared in this Quarterly a number of generalizations of the Fibonacci number Fn to functions Fix), defined for all real*, and, in general, continuous everywhere.
For such a generalization two properties are particularly desirable:
(A)
Fix) = Fn for x = n a natural number
and
(B)
Fix+2) = F(x) + F(x+1).
Spickerman [6] proved some general properties of functions satisfying (B).
Of the various generalizations Halsey's [1] does not generally satisfy (B) (see [7]) and even if defined for all real*,
is not continuous dXx= 1.
Heimer's function [2] satisfies (A) and (B) but is quasilinear. Elmore's function [3] is not a generalization in the
above sense, it is a function of a natural number variable and a real variable.
Parker's [4] and Scott's [5] functions which are identical are "smooth curves," satisfy both (A) and (B) but can
be generalized further.
Both take
FM
=
Re
(\*-(-V*\-*\
=
\«-X«co»7ar
where
It seems, however, that a lot is lost in taking only the real part of
Clearly this complex function itself (we will call it Fx) satisfies (A), and also (B) for any complex number*. Also
as the real part of Fx satisfies (B) so does the imaginary part and any linear combination of these.
If we let
Ft(x) = Re(Fx),
F2(x) = l(Fx) = ^
™ nx
,
f o r x real, then Fi (x) +aF2(x) satisfies (A) and (B) for each real numbers.
Scott gives a number of identities concerning F^ix) and also concerning the corresponding Lucas function which
we will call
Li(x) = He(Lx) = ne(\x + (-7)x\~x) = Xx + X~x cos nx .
Of course l(Lx) =
-F2(x)s/J.
We now list some easily derivable properties of F2(x) some of which relate it to F± (x):
p2(x + 1) - F2(x -1) = F22(x),
F2M - F2(-x) = zmlMf
F2(x + %) • F2(x - %) = F2(2x) ™±?M
2y/(S) '
97
Fix +1/2)• F2(x - %) = -F2Jx) cot2nx,
2
98
APRIL 1978
MORE FIBONACCI FUNCTIONS
Ft(x) = i
^ + ^Mcotir*,
2M
5F
F2(nx) = s i ™
sin*™
S^^fiM
(-VnH
Another possible generalization of Fn forx = n \s\Fx\, which we will call Gi(x).
Thus
G
r X\ = y/F\(x)
lM*/ ~
= \\F
+ FH'x) "* -!—
sj\2x - 2 cosnx + X-2*
x\ " v r l ' A ' T^IX/
-
yft5)
Another such function is
1
G2(x) = s/F\(x) - F\(x) _
= -7
— sfk2x - 2 cos nx + X' 2 * cos lux .
Clearly
kGl(x) + (1-k)G2M
\Nhenx = n for all real k.
The following are some properties of these functions:
G2(x + 1) - G2(x) = G\(x +1/2>- 2/5 sin irx + 4/5 cos -nx
G2(2x) = 5G41(X)+4COSTTXG21(X)
G2VJx) =
'2
2
(1/5)(Ll(2x)-2cos7rx)
G (x)-G2(x)
= 2F2(x).
REFERENCES
1. Eric Halsey, "The Fibonacci Number Fu where u is not an Integer/' The Fibonacci Quarterly 3 (1965), pp. 147152.
2. Richard L. Heimer, "A General Fibonacci Function/' The Fibonacci Quarterly 4 (1967), pp. 481-483.
3. M. Elmore, "Fibonacci Functions," The Fibonacci Quarterly b (1967), pp. 371-382.
4. Francis D. Parker, "A Fibonacci Function," The Fibonacci Quarterly 6 (1968), pp. 1-2.
5. A. Scott, "Continuous Extensions of Fibonacci Identities," 77?e Fibonacci Quarterly 6 (1968), pp. 245-250.
6. W. R. Spickerman, "A Note on Fibonacci Functions," The Fibonacci Quarterly 8 (1970), pp. 397-401.
7. M. W. Bunder, "On Halsey's Fibonacci Function," The Fibonacci Quarterly n (1975), pp. 209-210.
*******