SUMS OF n-th POWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION
N.A. Draim, Ventura, C a l i f . , and Marjorie B i c k n e l l
Wilcox High School , Santa Clara, C a l i f .
The q u a d r a t i c equation whose r o o t s a r e the sum or difference
of the n - t h p o w e r s of the r o o t s of a given q u a d r a t i c equation is of
i n t e r e s t to the a l g o r i t h m a t i s t both for the n u m b e r p a t t e r n s to be seen
and for the implied connection between q u a d r a t i c equations and the
factorability and p r i m e n e s s of n u m b e r s . In this p a p e r , an e l e m e n t a r y
a p p r o a c h which points out r e l a t i o n s h i p s which might o t h e r w i s e be
m i s s e d is used to d e r i v e g e n e r a l e x p r e s s i o n s for the subject equation
and to show how Lucas and Fibonacci n u m b e r s a r i s e as s p e c i a l c a s e s .
Sums of like p o w e r s of r o o t s of a given equation have been
studied in d e t a i l .
F o r a few e x a m p l e s , in this Q u a r t e r l y S. L. Basin
[l~j h a s used a development of W a r i n g ' s formula [ 2 ] for s u m s of like
p o w e r s of roots to w r i t e a generating function for Lucas n u m b e r s and
to w r i t e the kth Lucas n u m b e r as a sum of binomial coefficients.
R. G. B u s c h m a n [ 3 J has used a l i n e a r combination of like powers of
r o o t s of a q u a d r a t i c to study the difference equation u
a r r i v i n g at a r e s u l t which e x p r e s s e s
u
, = au
+ bu
in t e r m s of a and b.
,
Paul
F . Byrd [4"j, [ 5 ] h a s studied the coefficients in expansions of a n alytic functions in polynomials a s s o c i a t e d with Fibonacci n u m b e r s .
A n u m b e r of the r e s u l t s in the foregoing r e f e r e n c e s a p p e a r as s p e c i a l
c a s e s of those of the p r e s e n t p a p e r .
A.
THE SUMS OF THE n - t h POWERS
Given the p r i m i t i v e equation f(x) = x
r,
- px -q = 0, with r o o t s
and r ? , the f i r s t p r o b l e m c o n s i d e r e d h e r e is to w r i t e the equa-
tion
£i \n
f(x)
with r o o t s
r]
and
2
=x
, n
n*
.
*n
- (r1 + r2)x - (rxr2)
n
= 0
,
r?.
Solving /the p r i m i t i v e equation for
formula,
170
r,
and r
by the q u a d r a t i c
1966
SUMS O F n - t h POWERS OF ROOTS OF A GIVEN
r x = (P + J?Z + 4q)/2 ,
Since
r
and r
r 2 = (p - JpZ + 4q)/2
.
satisfy the equation (x - r )(x - r ) = 0,
2
x
2
- (r
identically, so r , + r
f(x) , having r o o t s
+ r )x + r r
=[(p -
. r,
V/P
=x
= p and r r~ = -q.
2
+ 4q)/2]
2
2
2
Adding, r , + r ? = p + 2q.
f(x)
2
- px - q = 0
Next, w r i t e the equation
and r „ Squaring r o o t s ,
2
r J = [(p + \/p2 + 4q)/2]
r\
171
= (2p 2 + 4q + 2p / p 2 + 4q)/4
= (2p 2 + 4q - 2p / p 2 + 4 q ) / 4 .
The d e s i r e d equation is then
=x
2
- (p 2 + 2q)x + q 2 = 0 .
3
the coefficient of x for the equation f(x) = 0, by m u l t i 3
3
3
plying and adding, is r + r = p
+ 3pq. Continuing in this m a n n e r ,
the coefficients for x for the equations whose r o o t s a r e higher p o w e r s
of r , and r
can be tabulated as follows:
Similarly,
The sequence for
r, + r
ends when, for a given n, the last
t e r m in the sequence b e c o m e s equal to z e r o .
a s s u m e s the form
2q '
The penultimate t e r m
when n is e v e n a n d npq
'
when n is
odd.
Writing the n u m e r i c a l coefficients from Table 1 in the form of an
a r r a y of m columns and n rows and letting
C(m, n) denote the c o -
efficient in the m t h column and nth row leads to Table 2.
It is ob-
vious that .the i t e r a t i v e o p e r a t i o n in Table 2 to g e n e r a t e additional num e r i c a l coefficients is
C(m, n) = C ( m - 1 , n-2) + C(m, n-1)
,
which s u g g e s t s a l i n e a r combination of binomial coefficients.
By d i r e c t
expansion,
n ( n - m ) ( n - m - l ) . . . (n-2m+3) _
(m-l)I
"
?
,n+l-m,
,n-m
m-1 } " (m-l
(
172
QUADRATIC EQUATION
TABLE 1:
r
l
+ r
=
2
2 , 2
1
2
April
r* + r£
P
P
2
+ 2q
3 , 3 = 3
T r
P + 3pq
1
2
4
4 , 4
+ 4p q + 2q
r
l + r2 = P
r
5 , 5
5
-t- r
=
P
=
P
+ 5p q + 5:pq'
1
2
6
4
2 2
3
6
6
r
l + r 2 = P + 6p q + 9p q + 2q
r
7 , 7
l + r 2
r
8,8
l + r2
r
n .
l
7
=
n
2
P
= p
8
5
3 2
3
+ 7p q + 14p q + 7pq
+ 8p q + 20p q
n ,
P
n-Z
q
+ ... +
+ l6p q
, n(n - 3) n - 4 2 , n(n - 4)(n - 5) n - 6 3
P
q
p
q
~T!
3T
n(n - m)(n - m - 1). . . (n - 2m + 3) n-2m+2 m - 1
( m - 1)!
P
q
TABLE 2:
m
+ 2q
3
4
5
C(m, n)
SC(m, n)
L
2
L
0
1
L
2
3
L
3
0
4
L
4
2
7
L
5
5
0
11
L
6
9
2
18
L
7
14
7
0
29
L
8
20
16
2
47
1966
SUMS OF n - t h POWERS OF ROOTS OF A GIVEN
173
F r o m the above t a b l e s ,
where
[a/2]
z^: 1 ) - ( n - i - 1 )
1
1
1
'
n . n
r
l +r2
(A. 1)
[x]
p
s
i=0
n-2i i
q
is the g r e a t e s t i n t e g e r l e s s than or equal to x,
and
(n)
is the binomial coefficient
(m-n)l n!
Proof:
By a l g e b r a ,
k
k+1
k+1
, k+1 A k+1
k , k x ,
k t k .
r ,1
+ rn
= (r,
+ r~
+ r n1r29 + Ar l^ 2'
) - ( ^1r 2, + r 1, ^2 )'
= (r^ + r ^ ) ( r i + r 2 ) - r ^ r * "
= p(r
'I
+ r ) + q(r
F r o m Table 1, (A. 1) holds for
1
+ r
1
+
)
r
k
"l)
.
n = 1, 2, . . . , 8. To p r o v e (A. 1) by m a t h -
e m a t i c a l induction, a s s u m e that (A0 1) holds when n = k and n = k - 1 .
Using the r e s u l t j u s t given and the inductive h y p o t h e s i s ,
[k/2]
k+1 , k+1
r ,1
+ r
=p
2( k :S
. ( k_ii" 1 ) P k-2i q i
i
v
2
i=0
[(k-l)/2]
+q
1
k-l-i
_
I
k-i-2
P
I
i=0
Multiplying a s indicated and using the index substitution i-1
k-l-2i i
q
for
the second s e r i e s yields
k+1 , k+1
r
+ r
l
2
[V2]
1
2(k:i) 1 '
(k"i_1)
1
P
k+l-2i i
q
i=0
0+l)/2]
+
i
i=0
[2( k :j> - (k;1;11) 1 P
k+l-2i i
q
i in
174
QUADRATIC EQUATION
April
Since the r e c u r s i o n formula for binomial coefficients is
l
m
n'
m
l
n+l;
{
m+1
n+l'
'
combining the s e r i e s above will yield
[2(k+i-i}
for the coefficient of p
and for
_
q .
(k-i}]
Considering the s e r i e s for
k even
k odd leads to
[(k+l)/2]
k+1 ,
rx
k+1
+r2
v
=
'
2(k+l-ij
_ (k-i} '
I
p
k+l-2i i
q
i=0
so that (A. 1) follows by m a t h e m a t i c a l induction.
An e x p r e s s i o n eq-
uivalent to (A. 1) was given by Basin in [ ! ] •
Using formula (A. 1), we can w r i t e the d e s i r e d equation
,. n
2
n
n.
. vn n n
f(x) = x - (r x + r 2 ) x + (-1) q = 0 .
2
Example:
Given the equation x - 5x + 6 = 0, w r i t e the equation whose r o o t s a r e the fourth powers of the r o o t s of the given equation, without solving the given equation. In the given equation, p = 5,
q = 6. F r o m Table 1 or formula (A. 1),
r^ + rjj = (5) 4 + 4(5) 2 (-6) + 2 ( - 6 ) 2 = 97 ,
4
2
= 0 . As a check, by factoring x -5x + 6 = 0,
4
4
4
2
4
r ] = 2 and r_ = 3, giving us f(x) = (x - 2 )(x - 3 ) = x - 97x + 6 .
R e t u r n i n g to Table 2, the s u m m a t i o n of the coefficients by rows
yields the s e r i e s 1, 3, 4, 7, 11, 18, 29, 47, . . . , the s u c c e s s i v e
so f(x)
2
=x
4
- 97x + 6
Lucas n u m b e r s defined by L, = 1, L n = 3, and L = L , + L n .
7
1
2
n
n-1
n-2
The g e n e r a l equation for r, + r ? r e d u c e s to the n u m e r i c a l values of
2
C(m, n) of Table 2 for p = q = 1 in the p r i m i t i v e equation x - px -q =
c
•p
0, which b e c o m e s x - x - 1 = 0 with r o o t s a = (1 + /fT)/2 and
(3 = (1 - \/~5)/2. This yields an e x p r e s s i o n for the nth Lucas n u m b e r , since
L = a •+ 6 ;
n
1966
SUMS OF n - t h POWERS OF ROOTS OF A GIVEN
(A. 2)
L^ = 1 + n + n ( N - 3 ) / 2 ! + n ( n - 4 ) ( n - 5 ) / 3 ! + n ( n - 5 ) ( n - 6 ) ( n - 7 ) / 4 !
[n/2]
1
2(n:i) -
175
(n'Yly
i=0
F o r e x a m p l e , L, = 1 + 6 + 6(3)/2 + 6(2)(l)/6 = 18.
2
Also, the equation whose r o o t s a r e the nth roots of x - x - 1 = 0
is x - L x + (-1) = 0. ( E s s e n t i a l l y , equation (A. 2), a s w e l l a s (A. 3),
(B. 2), (B 0 3), a p p e a r s in [ 3 ] by B u s c h m a n . )
An a l t e r n a t e e x p r e s s i o n for L is obtained by e x p r e s s i n g
r
+r
in t e r m s of p and A , w h e r e A = \/p
+ rq e
Now,
n
n
T i _LA\/-?1 n
o-n/n.
~l
. n ( n - l ) n~2 2 . n ( n - l ) ( n - 2 ) n - 3
r x = L(P +A )/2J = 2 (p + np
A +
3/
'p
A + - ^ T " P
n r,
\/oin
r 2 = [(P - A )/2J
which, for
(A. 3)
= 2o~
n
/ n n n _ lA.
(P - P
p = q = 1, A = v5,
L
+.
n ( n - l ) n-2 2 , n ( nn - l ) ( n - 2 ) n - 3
"^T~- P
A +-^r
P
i m p l i e s , on adding,
= 2 1 ~ n ( l - 5 n ( n - l ) / 2 I + 5 2 n(n~ l ) ( n - 2 ) ( n - 3 ) / 4 ! + . . . )
[n/2]
1
—
—T
1
?n-l
Z
B.
_i, n,
5^;.)
x
2r
i=0
THE D I F F E R E N C E S OF THE nth POWERS
Now let us t u r n to the equation whose r o o t s a r e
where r
1
and r
a r e the r o o t s of the p r i m i t i v e equation x
Ct
r
2
and
-r
-px -q = 0
Since
r
l
•"
r
2
=
( p
+
^
p 2
+
r q ) / / 2
"
( p
" Vpl
+
4 q ) / / 2
the equation whose r o o t s a r e r , and ~r„ is x
2
and f(x) = x - A x + q = 0. S i m i l a r l y ,
*\-x\=
O
+
/P2
+ 4(
=
^
- yp
1)/ 2 ] 2 - [(P - V/P 2
p 2
+
4 q
=
A
+ rq x - r , r
+ 4
<1>/2] 2 =
'
= 0,
3,
+
3,
+
176
April
QUADRATIC EQUATION
Continuing in this m a n n e r ,
and a s s e m b l i n g r e s u l t s , we have
TABLE 3
n
l "
n
2
n
r
1
A.
2
PA
3
(p
4
( p 3 + 2pq)A
5
(p4 + 3p2q + q2)A
6
(p5 + 4p3q + 3pq2)A
7
( p 6 + 5 p 4 q + 6 p 2 q 2 + q 3 )A.
n
, n-1
. _. n - 3 ^ ( n - 3 ) ( n - 4 ) n - 5 2
(p
+ (n-2)p
q+-i
JJ
' p
q + ...
r
+q)A
( n - m - l )Li( n - m - 2 )' , . , ( in - 2 m ) L p n - 2 m - l q m .) A
.
+, 1_
mi
The sequence for
r1 - r ?
as given in Table 3 ends, for a given n,
on a r r i v i n g at t e r m z e r o .
Writing the n u m e r i c a l coefficients from Table 3 in the form of
an a r r a y of m columns and n r o w s , and letting C (m, n)
the coefficient in the m t h column and nth
TABLE 4:
m
2
3
4
5
row,
C (m, n)
A
6
% C A ( m , n)
1
L
0
1
2
L
0
1
3
L
1
0
2
4
L
2
0
3
5
L
3
1
0
6
L
4
3
0
7
L
5
6
1
0
13
L
6
10
4
0
21
L
7
15
10
1
5
8
0
34
denote
1966
SUMS OF n - t h POWERS OF ROOTS OF A GIVEN
177
It is obvious that the i t e r a t i v e p r o c e s s to g e n e r a t e additional n u m e r i c a l
coefficients r e s t s on the r e l a t i o n
C A ( m ? n ) = C A ( m - l , n-2) + C A ( m , n-1)
the s a m e a s for Table 2„
,
The coefficients g e n e r a t e d a r e a l s o the c o -
efficients a p p e a r i n g in the F i b o n a c c i polynomials defined by
f n + 1 (x) = x f j x ) + f n _ 1 ( x ) ,
f 1 (x) = l ,
f 2 (x)
( F o r further r e f e r e n c e s to Fibonacci p o l y n o m i a l s ,
= x
.
see Byrd, [ 4 ] ,
p.17.)
Tables 3 and 4 lead to
(B.l)
rj-rj
= A
l>/2]
(""j"1) P ^ ' V
2
.
j=0
which is proved s i m i l a r l y to (A. 1)
The equation whose r o o t s a r e
4,-u
£
a r e the r o o t s of
r\
2
x
- px - q = 0,
r ' and - r ? , w h e r e
n
•
£i
is
f(x) A = x
\
2
/
n
- (r
r
n
\
and
J_
n
- r )x + q
i
L
r
n
=0.
2
We use ( B . l ) to solve the following: Given the equation x - lOx +
21 = 0 with unsolved r o o t s r , and r ? , w r i t e the q u a d r a t i c equation
3
3
whose r o o t s a r e r , and - r ? „ F r o m the given equation, p = 10, q = - 2 1 ,
r
and A = / 1 0
- 4(21) = 4,
r
l " r2
Using ( B . l ) or Table 3,
= (p2 + q)
= (1
°2 " 21)4
= 3l6
e
3
2
3
Hence f (x) A = x - 3 l 6 x + (-21) = 0 . As a check, the r o o t s of the p r o b 3
3
3
2
lem equation a r e r ] = 7, r = 3, so f(xL = (x - 7 )(x + 3 ) = x - 3 l 6 x 3
21° = 0.
The s u m m a t i o n of t e r m s in the s u c c e s s i v e rows of Table 4 is s e e n
to yield the sequence of F i b o n a c c i n u m b e r s , defined by F , = F ? = 1,
F
= F , + F o8 This r e s u l t can be d e m o n s t r a t e d by7 substituting
&
n
n-1
n-2
p = q = 1 in the g e n e r a l e x p r e s s i o n (B. 1) and dividing by A = / 5 , for
the left hand m e m b e r b e c o m e s the Binet form, F = (a11 - p")/ ST. In
general,
178
QUADRATIC EQUATION
(B.2)
F
April
= 1 + (n-2) + (n-3)(n-4)/2! + (n-4)(n-5)(n-6)/3! + . . .
j=o
the sum of t e r m s in r i s i n g diagonals of P a s c a l ' s t r i a n g l e .
Also, the
q u a d r a t i c equation whose r o o t s a r e a and -(3 , nth p o w e r s of the
2
r o o t s of the q u a d r a t i c equation x - x - 1 = 0 with r o o t s
a = (1 + / 5 ) / 2
is x 2 - / 5 " F x - 1 = 0.
and (3 = (1 - / 3 ) / 2 ,
Expressing
r, - r ?
t e r m s a s we did for
in t e r m s of only p and A and a s s e m b l i n g
r, + r
in (A. 3), we a r r i v e at an a l t e r n a t e ex-
p r e s s i o n for the nth F i b o n a c c i n u m b e r ,
(B. 3) F
= 2 1 ~ n (n -I- 5 n ( n - l ) ( n - 2 ) / 3 ! + 5 2 n(n- l)(n-2)(n~3)(n-4)/5! +. . .
[(n-l)/2]
X
2 "
n
R 11
5 <(
1
i=0
n
2i+lJ
REFERENCES
le
S. L0 Basin,
"A Note on W a r i n g ' s F o r m u l a for Sums of Like
P o w e r s of R o o t s " ,
F i b o n a c c i Q u a r t e r l y 2:2, April, 1964, pp.
119-122.
2.
L. E. Dickson, F i r s t C o u r s e in the T h e o r y of Equations, John
Wiley, 1922.
3.
~
R. G. Buschman,
ials,
" F i b o n a c c i N u m b e r s , Chebyshev P o l y n o m -
Generalizations
and Difference E q u a t i o n s " ,
Fibonacci
Q u a r t e r l y 1:4, D e c , 1963, pp. 1-8.
4.
P a u l F . Byrd,
"Expansion of Analytic Functions in P o l y n o m -
i a l s A s s o c i a t e d with Fibonacci N u m b e r s " , F i b o n a c c i Q u a r t e r l y ,
1:1, F e b . 1963, pp. 16-29.
5.
P a u l F . Byrd,
"Expansion of Analytic Functions in T e r m s In-
volving Lucas N u m b e r s or S i m i l a r Number S e q u e n c e s " , F i b onacci Q u a r t e r l y 3:2, A p r i l , 1965, pp. 101-114.
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