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62
ON THE LENGTH OF THE EUCLIDEAN ALGORITHM
(m = 0, 1, 2, ••• , n), where F
acci number (2).
= F
0
-A
1
-1
=0
Feb. 1973
and F . for each j > 0 is the j
" 31
Fibon-
The results in Theorem 2 were suggested to the authors by considering a number of
special c a s e s on an IBM 360/65 computer.
REFERENCES
1. R. L. Duncan,
"Note on the Euclidean A l g o r i t h m / ' The Fibonacci Quarterly, Vol. 4
(1966), pp. 367-368.
2. O. P e r r o n , Die Lehre von den Kettenbruchen, Vol. 1, Teubner, Stuttgart, 1954.
3. J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, 1939.
LETTERS TO THE EDITOR
Dear Editor:
In the paper (*) by W. A. Al-Salam and A. V e r m a , "Fibonacci Numbers and Eulerian
Polynomials," Fibonacci Quarterly, February 1971, pp. 18-22, an e r r o r occurs in(9), which
is readily corrected.
I will generalize their (4) by defining a general polynomial operator
M by
(I)
Mf(x) = Af(x + ct) + Bf(x + c 2 ) ,
where f(x) is a polynomial and A, B, cl9
we note that M = A e °
lD
+ Be°
2D
ct £ c 2 ,
and c 2 a r e given numbers.
With D = d/dx,
so that
QO
OO
n
Mf(x) = A J^ nT ^ W
+ B
2
n=0
n
nT D*f(x) '
n=0
or
OO
(II)
Af(x +
Cl)
W
+ Bf(x + c 2 ) = J^ -£
n=0
Dnf(x) ,
where Wn = Ac*1 + BcL2 is the solution of Wn+2
,- - ^QWn and c \4 T^ cL2 are the roots
i r = PWn+1
2
of x = Px - Q. In (*), Eq. (4) is a special case of (I) with A = fi and B = 1 -1±, There
are two c a s e s of (II) to consider:
Case 1. A + B / 0 , If A = B, we obtain from (II)
V_
v
)
+
f(x
+
c
)
=
2
-5?
Cl
2
n=0
Dllf x)
(III)
f(x +
(
>
where V0u = 2,» Vii = P ,
[Continued on page 71. ]
and V +2
,n = PVn+1
,- - QVn . If Cj-1 and c*2 are roots of x 2 = x + 1,
n