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SYMMETRIC RATIONAL EXPRESSIONS IN THE FIBONACCI
NUMBERS
MARTIN GRIFFITHS
Abstract. In this paper we consider the evaluation of numerical expressions obtained by
specializing the n variables of a symmetric rational function to the Fibonacci numbers.
In particular, we derive both exact and asymptotic formulas for elementary symmetric
expressions in the Fibonacci numbers, and go on to demonstrate some applications of these
results. The asymptotic formula is then generalized to all sequences sharing a particular
mathematical property with the Fibonacci sequence.
1. Introduction
A symmetric rational function in the variables x1 , x2 , . . . , xn is left unchanged by any
permutation of these variables (unchanged, that is, other than in the order of the terms and
factors). To take an example,
x21 x2 x1 x22 x21 x3 x1 x23 x22 x3 x2 x23
+
+
+
+
+
x3
x3
x2
x2
x1
x1
is a symmetric rational function in x1 , x2 , and x3 . The elementary symmetric polynomial ek,n is defined as the sum of all possible products of k distinct elements from the set
{x1 , x2 , . . . , xn }. The elementary symmetric polynomials in three variables are thus given by
e1,3 = x1 + x2 + x3 , e2,3 = x1 x2 + x1 x3 + x2 x3 and e3,3 = x1 x2 x3 .
A well-known result is that any symmetric rational function in n variables can always be
expressed as a rational function in e1,n , e2,n , . . . , en,n (for a proof of this see Theorem 13.5.1
in [2]). For example,
f (x1 , x2 , x3 ) =
e1,3 e22,3 − e2,3, e3,3 − 2e21,3 e3,3
f (x1 , x2 , x3 ) =
.
e3,3
We may therefore think of the elementary symmetric polynomials as basic building blocks
for symmetric rational functions.
In this article the variables x1 , x2 , . . . , xn are first specialized to the Fibonacci numbers by
setting xk = Fk , k = 1, 2, . . . , n. We use Sk,n to denote the elementary symmetric Fibonacci
expression consisting of the sum of all possible products of k elements from {F1 , F2 , . . . , Fn }
having distinct indices (if k > n then Sk,n is defined to be zero). We obtain exact formulas
for some elementary symmetric Fibonacci expressions and, for any fixed k ∈ N, a general
asymptotic formula for Sk,n . We then go on to illustrate some applications of our results.
Finally, our asymptotic formula is generalized further to cater for all Fibonacci-like sequences.
2. Exact formulas
It is not hard to prove the recursive formula
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SYMMETRIC RATIONAL EXPRESSIONS IN THE FIBONACCI NUMBERS
ek,n = ek,n−1 + xn ek−1,n−1 ,
which, for the Fibonacci numbers, specializes to Sk,n = Sk,n−1 + Fn Sk−1,n−1 . Thus, noting
that
n
X
S1,n =
Fk = Fn+2 − 1,
k=1
we have, for n ≥ 3,
S2,n = S2,n−1 + Fn S1,n−1 = S2,n−1 + Fn (Fn+1 − 1),
from which it can be seen that
à n
! Ã n
!
X
X
S2,n =
Fk Fk+1 − F2 F3 −
Fk − F1 − F2
k=1
=
n
X
k=1
Fk (Fk+1 − 1)
k=1
1
(1 + (−1)n )
2
¢
1¡ 2
=
Fn+1 + Fn Fn+2 − 2Fn+2 + 1 .
2
= Fn Fn+2 − Fn+2 +
Next, for n ≥ 4,
S3,n = S3,n−1 + Fn S2,n−1
¢
Fn ¡ 2
= S3,n−1 +
Fn + Fn−1 Fn+1 − 2Fn+1 + 1 .
2
From this it follows that
¢
1¡ 2
1
(F3n+2 − (−1)n Fn−1 ) −
Fn+1 + Fn Fn+2 − Fn+2
S3,n =
10
2
¢
1
1¡ 2
=
(F3n+2 + Fn F2n − Fn+1 F2n−1 ) −
Fn+1 + Fn Fn+2 − Fn+2 .
10
2
In a similar manner it is possible to derive formulas for S4,n , S5,n , and so on.
3. Asymptotic formulas
It can be verified that
φ3n+2
φn+2
φ2n+2
φ4n+2
√ ,...,
, S3,n ∼ √ , S4,n ∼
S1,n ∼ √ , S2,n ∼
5
5
10 5
25(5 + 5)
where
√
1
5+1
=
2
φ−1
is the golden ratio. More generally we find, for fixed k, that
φ=
Sk,n ∼
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(3.1)
φkn+2
,
dk
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THE FIBONACCI QUARTERLY
where dk satisfies the recurrence relation
√ ¡ k
¢
5 φ −1
dk =
dk−1
φ
√
for k ≥ 2, with d1 = 5.
On using (3.1) and (3.2) we have:
S1,n
∼√
φn+1
,
5(φ − 1)
S2,n
(3.2)
φ2n+2
∼
,
5(φ − 1)(φ2 − 1)
...,
leading to the following explicit asymptotic formula for any fixed k ∈ N:
φk(n+1)
Sk,n ∼ ¡√ ¢k
5 (φ − 1)(φ2 − 1) . . . (φk − 1)
k
φ 2 (2n−k+1)
= ¡√ ¢k
5 (1 − φ1 )(1 − φ12 ) . . . (1 −
µ ¶
k
φ 2 (2n−k+1)
1
= ¡√ ¢k Pk
,
φ
5
1
)
φk
(3.3)
where
Pk (x) =
k
Y
1
1 − xm
m=1
is the generating function for pl (k), the number of partitions of l into parts not exceeding k
(see [1], for example).
4. Some applications
We can use the results from Sections 2 and 3 to derive formulas for symmetric rational
expressions in the Fibonacci numbers, and to tackle related problems:
(a) As a first example, let us consider
X
Fk2 Fm ,
where the sum is taken over all distinct ordered pairs (k, m) from the set {1, 2, . . . , n}.
It may be noted that of the 12 n2 (n−1) terms in the expansion of S1,n S2,n , 12 n(n−1)(n−
2) are of the form Fk Fl Fm , where k, l and m are distinct elements from {1, 2, . . . , n},
while the remaining n(n − 1) terms are of the form Fk2 Fm . For a particular choice
of three distinct integers, k, l and m say, from {1, 2, . . . , n}, there are exactly three
appearances of the term Fk Fl Fm in the expansion of S1,n S2,n . On the other hand,
each term of the form Fk2 Fm appears in the expansion exactly once. From this we
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SYMMETRIC RATIONAL EXPRESSIONS IN THE FIBONACCI NUMBERS
see that
X
Fk2 Fm = S1,n S2,n − 3S3,n
¡ 2
¢
1
= (Fn+2 − 1) Fn+1
+ Fn Fn+2 − 2Fn+2 + 1
2
3
−
(F3n+2 + Fn F2n − Fn+1 F2n−1 )
10
¢
3¡ 2
+
Fn+1 + Fn Fn+2 − Fn+2
2
¢
1¡ 2
2
=
Fn+1 Fn+2 + Fn Fn+2
− 2Fn Fn+1 − 1
2
3
−
(F3n+2 + Fn F2n − Fn+1 F2n−1 ) .
10
(b) Next we obtain an asymptotic formula for
µX
¶−1
1
,
Fk1 Fk2 . . . Fkn−3
where the sum is taken over all possible sets of n−3 distinct elements, {k1 , k2 , . . . , kn−3 },
from {1, 2, . . . , n}. This expression may be rewritten as
µX
¶−1
Fk Fl Fm
F1 F2 . . . Fn
,
=
F1 F2 . . . Fn
S3,n
where the sum on the left now ranges over all distinct trios, k, l and m, from
{1, 2, . . . , n}. An asymptotic formula for the numerator is given by
n(n+1)
Cφ 2
¡√ ¢n ,
5
where C is the Fibonacci factorial constant defined as
Ã
µ
¶k !
∞
Y
1
1− − 2
C=
= 1.226742 . . .
φ
k=1
(see, for example, [3] and [4]). We therefore have
√
µX
¶−1
n(n+1)
1
10 5 Cφ 2
∼ 3n+2 × ¡√ ¢n
Fk1 Fk2 . . . Fkn−3
φ
5
n2 −5n−4
2Cφ 2
= ¡√ ¢n−3 .
5
(c) With gn (xn ) = (1 + F1 xn )(1 + F2 xn ) . . . (1 + Fn xn ), our results can be used to obtain
good numerical approximations to limn→∞ gn (xn ) for various sequences {xn }. Let us,
for example, consider the evaluation of
µ ¶
1
.
lim gn
n→∞
Fn
In order to facilitate this calculation we may note both that
gn (xn ) = 1 + S1,n xn + S2,n x2n + . . . + Sn,n xnn
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THE FIBONACCI QUARTERLY
and, for any k ∈ N,
µ ¶ ¡√ ¢k
k
5
1
Sk,n
φ 2 (2n−k+1)
lim
= ¡√ ¢k Pk
×
k
kn
n→∞ Fn
φ
φ
5
µ ¶
k(1−k)
1
= φ 2 Pk
.
φ
The resulting series, given by
µ ¶
µ ¶
µ ¶
1
1
1
1
1
1 + P1
+ P2
+ 3 P3
+ ...,
φ
φ
φ
φ
φ
³ ´
converges quite quickly to limn→∞ gn F1n . Indeed, by the tenth term the relative
error is less than 1 in 2 × 109 .
³ ´
(d) It is also interesting to note that the constants C and Pk φ1 appearing in this and
the previous section are related in the sense that they may both be expressed in terms
of generating functions of partition functions whose arguments are simple functions
of φ. We have
µ
¶
¶
µ
1
1
1
= P − 2 = lim Pk − 2 ,
k→∞
C
φ
φ
where Pk (x) is the function defined previously, and P (x) is the generating function
for the unrestricted partition function. Although it does need to be borne in mind
³ ´
that our asymptotic formulas are only valid for fixed k, we note that limk→∞ Pk φ1
also exists. Indeed,
µ ¶
µ ¶
1
1
=P
= 8.278013 . . . ,
lim Pk
k→∞
φ
φ
providing us with the incidental result that, within an order of magnitude, Sk,n is
given by
k
φ 2 (2n−k+1)
¡√ ¢k−2 .
5
5. Generalizing
Let us define a sequence of positive integers, {Gn } as follows. With a, b ∈ N, set G1 = a
and G2 = b. Now, for n ≥ 3, we define Gn recursively as Gn = Gn−2 + Gn−1 . Thus {Gn } is
a Fibonacci-like sequence with, as is easily verified,
Gn = aFn−2 + bFn−1
Ã
µ
¶n−2 µ
¶!
1
1
b
n−2
=√
φ (a + bφ) − −
a−
.
φ
φ
5
Using Sk,n (a, b) to denote the sum of all possible products of k elements from {G1 , G2 , . . . , Gn }
having distinct indices, we may obtain, in a manner analogous to that used to derive (3.3),
the asymptotic formula
µ ¶
µ
¶k
k
1
a + bφ
(2n−k−3)
√
φ2
Pk
Sk,n (a, b) ∼
φ
5
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SYMMETRIC RATIONAL EXPRESSIONS IN THE FIBONACCI NUMBERS
for fixed k.
It is possible to take the generalization of (3.3) still further. Let c, d ∈ R with c > 0 and
d > 1. Suppose the sequence {Hn } has the following property:
|Hn − cdn | → 0 as n → ∞.
Then, for fixed k,
µ ¶
1
,
Tk,n (c, d) ∼ cd
Pk
d
where Tk,n (c, d) represents the sum of all possible products of k elements from {H1 , H2 , . . . , Hn }
with distinct indices.
³
1
(2n−k+1)
2
´k
References
[1] T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976.
[2] P. J. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1994.
[3] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences,
http://www.research.att.com/∼njas/sequences/A003266
[4] E. W. Weisstein, “Fibonacci Factorial Constant,” From MathWorld–A Wolfram Web Resource,
http://mathworld.wolfram.com/FibonacciFactorialConstant.html
MSC2000: 05A15, 05A16, 05E05, 11B37, 11B39
Department of Mathematical Sciences, University of Essex, Colchester, CO4 3SQ, United
Kingdom
E-mail address: [email protected]
AUGUST 2008/2009
267