PROPERTIES OF THE POLYNOMIALS DEFINED BY MORGAN-VOYCE
M.N.S. Swamy
Nova Scotia Technical College, Halifax, Canada
1.
Introduction
In dealing with e l e c t r i c a l ladder n e t w o r k s , A. M. Morgan-Voyce
defined a set of polynomials by:
(1)
b n (x) = x B n _ 1 (x) + b n _ x (x)
(n > 1)
(2)
B n (x) = (x + 1)
( n > 1)
Bn-1(x) + bn-1(x)
with,
b Q (x) = B Q (x) = 1
(3)
These polynomials
J
and B have a n u m b e r of v e r Jy f a s c i n
n
nating and i n t e r e s t i n g p r o p e r t i e s , and is the subject m a t t e r of this
article.
b
A few p r o p e r t i e s of t h e s e have been studied by Basin.
F r o m (1) and (2) we see that
(4)
b
= B
.
n~l
(5)
and,
x B = b ,, - b
n
n+1
n
Substituting (4)'in (1) we have that B s a t i s f i e s the difference equan
- B
n
tion,
B n (x) = (x + 2)
B n _ x (x) - B n _ 2 (x)
(n > 2)
with
(6)
B Q (x) = 1,
and B ^ x ) = x + 2
F r o m (1) and (2) it can e a s i l y be d e r i v e d that b (x) a l s o s a t i s f i e s the
s a m e difference equation, n a m e l y ,
b (x) = (x + 2)
n
b
, (x) - b ? (x)
n-1
n-2
with
(7)
b (x) = 1,
and b ^ x ) = x + 1
73
(n > 2)
P R O P E R T I E S OF THE POLYNOMIALS
74
The difference
Feb.
e q u a t i o n (6) m a y be e x p r e s s e d a s t h e c o n t i n u a n t ,
x+2
1
0
.
1
x+2
1
0
0
1
x+2
1
.
.
0
0
B (x)
n
(8)
0
1
0
and h e n c e we m a y study the p r o p e r t i e s
continuants.
1
of
x+2
B
n
(n > 1)
by u s i n g t h o s e of t h e
We s h a l l l i s t b e l o w o n l y s u c h of t h o s e p r o p e r t i e s of
w h i c h we w i l l u s e in studying
t h e sr
p o l yj n o m i a l s
b n (x)
(9)
B
and
b (x)
and in d e r i v i n g r e l a t i o n s b e t w e e n
B n (x):
B
, = B
m+n
m
n
- B
,
m-1
B
n-1
B, = B2 - B2 .
2n
n
n-1
(10)
(11)
J
(12)
(x
+ 2)
v
'
(13)
B
,
n-1
x(B
n
- B
)
n - 27 '
B9
, = B~
3 2 - B" 9
2n-l
n
n-2
n
B
n
B
2n-1
B
,xl
r-h+1
(14)
B
B
n-1
B
n-h+1
,, - B
n+1
n
h-2
B
n-r- 1
1
n-1
(15)
2.
E
dx
B (x) =
n
E
(B
r
B
n -i l - r )'
R e l a t i o n s b e t w e e n b (x) a n d B (x), a n d p r o p e r t i e s of b (x):
n
n
n
F r o m (5) a n d (7),
(16)
Also we have,
x
B
n
B (x)
(x + 1) b
n
- b •
n-1
1966
DEFINED BY MORGAN-VOYCE
(17)
B ,. - B . = b \ _ + b
n+1
n-1
n+1
n
75
F r o m (4) and (5),
(18)
b
- b
= x (B + B _)
n+1
n-1
n
n-1
By s u c c e s s i v e l y substituting 0, 1, 2, . . . for
n
in (5) and
adding we have,
n
19
X
< >
£Br=bn+l - 1
0
S i m i l a r l y from (4) we m a y deduce that,
(20)
V
b = B
L»J r
n
0
Now,
b
,_ = B , - B , 1 = (B B - B
,B , ) - ( B B . -B
, B J)
m+n
m+n m+n~l
m n
m - 1 n-1
m n-1
m - 1 n-Z
= B
(B
m
- B
n
J - (B , - B 9 )
n-1
n-1
n-2
B
,
m-1
Hence,
(21a)
b
•= B b - B
.b m+n
m n
m - 1 n-1
Interchanging m and n we have,
(21b)
b
, =b B - b
.B .
m+n
m n
m-1 n-i
Hence,
(22)
x
'
b B - B b = b
,B , - B
,b ,
m n
m n
m - I n-1
m - 1 n-1
We will see l a t e r that this is a p a r t i c u l a r c a s e of the m o r e g e n e r a l
r e l a t i o n s h i p (29).
Putting m = n in (21),
(23)
b^ = b B - b ..B ,
Zn
n n
n-1 n-1
Putting m = n+1 in (21),
76
P R O P E R T I E S OF THE POLYNOMIALS
(24a)
x
'
Feb.
b9 xl = b X1B - b B '
2n+l
n+1 n
n n-1
(24b)
= B
..b - B b
.
n+1 n
n n-1
F r o m (7) w e h a v e
(x + 2) b 9 ,, = b 9 , 9 + b 9
2n+l
2n+2
2n
b ^ B L1 - b B + b B
-b
_B
,
n+1 n+1
n n
n n
n-1 n-1
Hence,
(24c)
x
'
-
(x + 2) b 9 ,. = b , _ B ,. - b
B
,
' 2n+l
n+1 n+1
n-1 n-1
x
A l s o f r o m (12),
(x + 2) B 9 ,, = B 2 ,, - B 2 ,
'
2n+l
n+1
n-1
Hence,
(x + 2 ) ( B ? , 1 - b 9 ,, ) •= B ,, (B , . - b ,, ) - B
, (B
, - b • )
'
2n+l
2n+l
n+1 n+1
n+1
n-1
n-1
n-1
Hence,
(25)
'
(x + 2) B 0 = B . _ L 1 B - B
.B
2n
n+1 n
n-1 n-2
F r o m (23) a n d (24) w e d e d u c e t h a t ,
b9 - b9
. = b2 - b2
2n
2n-l
n
n-1
(26)
S u b t r a c t i n g (12) f r o m (25),
(x
+ 2 /)v( B 9 - B 9
J = B (B ,, - B ) - B 9 ( B
. - B
)
v
2n
2n-l
n n+1
n
n - 2 ' n-1
n - 29 '
Hence,
(27a)
(x + 2) b 0 = B b _,__ - B 9 b
.
2n
n n+1
n-2 n-1
(27b)
= b B
- b
_B
n n+1
n-2 n-1
We w i l l n o w d e r i v e a r e l a t i o n s h i p b e t w e e n t h e p o l y n o m i a l s
and
B (x), c o r r e s p o n d i n g to t h e r e l a t i o n (13) f o r
Consider the e x p r e s s i o n ,
B :
b (x)
1966
DEFINED BY MORGAN-VOYCE
77
b
=
B - B, 9 b
n-h+1 r
h-Z n - r - 1
(B , x l - B . )B - (B
,
n ~ r - 2 ' h-2
x
x
n-h+1
n-h' r
n-r-1
(B , , , B - B
B, 9 )
(B
B - B
B, 9 )
n-h+1 r
n - r - 1 h-2
n-h r
n - r - Z h-Z
from (13)
B B
n r-h+l ' Bn-lBr-h+l
= (B n - Bn - 1, )'B r-h+1
, , . = bn B r-h+1
, xl
Hence,
b B , ,. = b , , , B - B, 9 b
n r-h+1
n-h+1 r
h-2 n - r - 1
(28a)
Similarly,
(28b)
b B , x l = B , , . b - b, 9 B
,
n r-h+1
n-h+1 r
h-Z n - r - 1
Hence from (28a) and (28b) we get the r e l a t i o n ,
Changing
B r b n-h+1
,
n ,, - B, 0 b
h-2 n - r -n1 = b B , .- - b, 0 B
r n-h+1
h-Z n - r - 1
r to m, h - 2 to m - 4 , and n to m + n + l - r in the above
relation,
(29a)
B b - B
b
= b B - b
B
m n
m-r n-r
m n
m-r n-r
Using the r e l a t i o n (4) in (29a) we d e r i v e the c o r r e s p o n d i n g r e l a t i o n
for B (x) a s ,
n
(30a)
B
B
,
n-1
B
B
B
T = B B
,
1
n-r m-r-1
m-r n-r-1
n m-1
These r e l a t i o n s m a y be w r i t t e n n e a t l y in the form of d e t e r m i n a n t s :
m
B
B
B
B
n
m-r
b
m-r
b
m-1
Bm - r
B
(29b)
and
B
(30b)
B
n-r
i
n-r
m- 1 -r
B
n-r
n-1-r!
n- 1
Now putting h = 2, and n = r+1 in equation (28) we get,
(31)
B
b r B r - b r+1
, . B r - .i = 1
78
P R O P E R T I E S OF THE POLYNOMIALS
,Feb.
Putting m = n - 1 , and r = n - 1 in (29b) we get,
(32)
x
'
B b . - b B , =1
n n-1
n n-1
F r o m (31) and (32) we see that b (x) is p r i m e to b , (x), B (x)
n
^
n-1
n
and B , (x) for i n t e g r a l value's of x. Also, for i n t e g r a l values of x,
B x(x)' is ^p r i m e to B n - l, x(x),
.-.(x).
' bnx(x)' and b n+1
n
By s u c c e s s i v e l y substituting 1, 2, 3, . .„ for n in (10) and adding, we have
n
V B
L-J
?
= Bn2 - B02 = Bn2 - BCf
2r
1
Hence,
(33)
E B 2r
B2
n
=
0
Simi .larly, , using ( 1 1 ) , (23), (24) and (26) we d e r i v e :
n-1
(34)
£
B
2r+1
= Bn B n-1.
B
9
= b B
n n
0
n
E
(35)
2r
0
n-1
E
(36)
b
2r+l
= b B ,
n n-1
0
2n
2
("l)rbr = b
n
E
(37)
0
L e t us now find an e x p r e s s i o n for the d e r i v a t i v e of
n-1
b ' ( x ) = B 1 - B'
n
n
n-
n-2
B :
n- 1i - r
r
.i-L
B
0
A—t
B B
,
r
n-Z-r
n-2
n-2 B
*E
y
0
0
= Bn B
-1 0
b (x):
n
(B
r
n
.
, -B
o) = B , b n + Y^ B b
-r-1
n-r-2'
n-1 0 L*t r n - r - 1
0
1966
D E F I N E D BY M O R G A N - V O Y C E
79
Hence,
n-1
,w = E B
(38)
3.
b
E x p l i c i t p o l y n o m i a l e x p r e s s i o n s f o r B (x) a n d b (x):
We c a n e s t a b l i s h b y i n d u c t i o n t h a t ,
n
B (x) = V
(ck xk)
n
-*—^
n
k=0
,
where,
(39)
c
k
n
=
,n+k+l,
n - ki )'
(
Now
b x(x) = B v(x) - B
, (x) = V
n
n
n-1
L»J
(39)
n+k,
- z a>
x
,n+k+l.
* n+k .
n-k ' ' *n-k-l'
{
k
0
Therefore we have
bn (x) = V
L-J
(dknk
k,
x )
k=0
where,
d
(40)
k
/n+k
= ( , x)
n
n-k *
T h e e q u a t i o n s (39) a n d (40 a r e e x p l i c i t p o l y n o m i a l e x p r e s s i o n s
for
b
n
and
B , a n d s h o w t h a t t h e yJ a r e of d e g r e e
n
We s h a l l n o w d e r i v e a f o r m u l a
for
B (x)
dx
/
:
n.
P R O P E R T I E S OF THE POLYNOMIALS
80
Feb.
F r o m (39),
n
/ >\l3n(x)
k+ k+1
dx = ^2 <ckn x k+1
V ) +c
0
Now the coefficient of x
k+1
-
J_i
"
n+1
c
k+1
i
n-1
=
k+1
for the e x p r e s s i o n
r
,n+k+3.
(
i
)
,n+k+L
"
n-k
(
x
i
->)
.
-
(n
+
B ,, - B , i s ,
n+1
n-1
, ,x k
l
)
c
n-k-2
n
- (n + 1)(coefficient of x
k+1
in j B (x) dx . )
/•
Hence,
(41)
/ /«B„(x) dx =
n+
i ^1n"1
+c
It m a y a l s o be e s t a b l i s h e d that over the i n t e r v a l (-4, 0),
B (x)
n
and b (x) a r e orthogonal polynomials with r e s p e c t to the weight func-
tions
V 4 - <x + 2>2
and
V { x + 4)/-x
respectively
-
It m a y a l s o be seen from (6) that,
(42a)
B (x) = S (x+2)
n
n
and hence,
(42b)
where
4.
b (x) = S (x+2) - S . (x+2)
n
n
n-1
,
S (x) is the Chebyshev polynomial.
Conclusions:
The a r t i c l e deals with the p r o p e r t i e s of a set of polynomials
b (x)
and B (x) defined by (1), (2) and (3)„ Even though they a r e r e l a t e d to
the Chebyshev p o l y n o m i a l s , the author believes that B (x) and b (x)
a r e of use in the study of ladder n e t w o r k s and hence d e s e r v e a study of
this n a t u r e .
REFERENCES
1.
Ao M. M o r g a n - V o y c e , Ladder n e t w o r k a n a l y s i s using Fibonacci
n u m b e r s , IRE. T r a n s , on C i r c u i t Theory, Vol. C T - 6 , Sept. 59,
pp. 321-322 0
1966
2.
DEFINED BY MORGAN-VOYCE
S. L. Basin,
81
The a p p e a r a n c e of Fibonacci n u m b e r s and the Q
m a t r i x in e l e c t r i c a l n e t w o r k theory, Math. Magazine, Vol. 36,
M a r c h - A p r i l 63, pp. 8 4 - 9 7 .
3.
S. L. Basin, An application of continuants,
Math. Magazine,
Vol. 37, M a r c h - A p r i l 64, pp. 83-91..
4.
V. O. Mowery, Fibonacci n u m b e r s and Tchebycheff polynomials
in ladder n e t w o r k s , IRE. T r a n s , on C i r c u i t Theory, Vol. C T - 8 ,
June 61, pp. 167-168.
5.
T. R. Bashkow, A note on ladder n e t w o r k a n a l y s i s , IRE. T r a n s .
on C i r c u i t Theory, Vol. C T - 8 , June 61, pp. 168-169.
6.
T. R. O ' M e a r a ,
fer functions,
G e n e r a t i n g a r r a y s for ladder network t r a n s -
IEEE.
Trans,
on C i r c u i t Theory, Vol. C T - 1 0 ,
June 63, pp. 285-286.
7.
F . Bubnicki, Input i m p e d a n c e and t r a n s f e r function of a l a d d e r
network,
I E E E . T r a n s , on C i r c u i t Theory, Vol. C T - 1 0 , June
63, pp. 286-287.
8.
Solution to p r o b l e m 33, Canadian M a t h e m a t i c a l Bulletin, Vol.
4, No. 3, 1961, pp. 3 1 0 - 3 1 1 .
ACKNOWLEDGEMENTS
This w o r k was supported by the National R e s e a r c h Council of Canada,
under Grant No. A - 2 8 7 5 .
The author would like to thank the r e v i e w e r
and P r o f e s s o r V. E. Hoggatt for t h e i r useful suggestions and c o m ments.
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