/ b2, b3, ..., b_{n}\
K ( ) is also equal to the determinant
\a1, a2, a3, ..., a_{n}/
is also equal to the determinant
| a1 b2 0 0 . . . 0 |
| -1 a2 b3 0 . . . 0 |
| 0 -1 a3 b4 . . . 0 |
| 0 0 -1 a4 b5 . . -- |
| |
| u -1 a_{n-1} b_{n} |
| 0 0 -- -- 0 0 -1 a_{n} |,
from which point of view continuants have been treated by W.
Spottiswoode, J. J. Sylvester and T. Muir. Most of the theorems
concerning continued fractions can be thus proved simply from the
properties of determinants (see T. Muir's _Theory of Determinants_,
chap. iii.).
Perhaps the earliest appearance in analysis of a continuant in its
determinant form occurs in Lagrange's investigation of the vibrations of
a stretched string (see Lord Rayleigh, _Theory of Sound_, vol. i. chap.
iv.).
_The Conversion of Series and Products into Continued Fractions._
1. A continued fraction may always be found whose n^{th} convergent
shall be equal to the sum to n terms of a given series or the product to
n factors of a given continued product. In fact, a continued fraction
b1 b2 b_{n}
-- -- ----- can be constructed having for the
a1 + a2 + ... + a_{n} + ...
numerators of its successive convergents any assigned quantities p1, p2,
p3, ..., p_{n}, and for their denominators any assigned quantities q1,
q2, q3, ..., q_{n} ...
The partial fraction b_{n}/a_{n} corresponding to the n^{th} convergent
can be found from the relations
p_n = a_{n}p_{n-1} + b_{n}p_{n-2}, q_n = a_{n}q_{n-1} + b_{n}q_{n-2};
and the first two partial quotients are given by
b1 = p1, a1 = q1, b1a2 = p2, a1a2 + b2 = q2.
If we form then the continued fraction in which p1, p2, p3, ..., p_{n}
are u1, u1 + u2, u1 + u2 + u3, ..., u1 + u2 + ..., u_{n}, and q1, q2,
q3, ..., q_{n} are all unity, we find the series u1 + u2 + ..., u_{n}
equivalent to the continued fraction
u1 u2/u1 u3/u2 u_n/u_{n-1}
-- ------ ------ ----------
1 - u2 u3 u_{n}
1 + -- - 1 + -- - ... - 1 + -------
u1 u2 u_{n-1}
which we can transform into
u1 u2 u1u3 u2u4 u_{n-2}u_{n}
-- ------- ------- ------- ---------------,
1 - u1 + u2 - u2 + u3 - u3 + u4 - ... - u_{n-1} + u_{n}
a result given by Euler.
2. In this case the sum to n terms of the series is equal to the n^{th}
convergent of the fraction. There is, however, a different way in which
a Series may be represented by a continued fraction. We may require to
represent the infinite convergent power series a0 + a1x + a2x squared + ... by
an infinite continued fraction of the form
[beta]0 [beta]1 x [beta]2 x [beta]3 x
------- --------- --------- ---------
1 - 1 - 1 - 1 - ...
Here the fraction converges to the sum to infinity of the series. Its
n^{th} convergent is not equal to the sum to n terms of the series.
Expressions for [beta]0, [beta]1, [beta]2, ... by means of determinants
have been given by T. Muir (_Edinburgh Transactions_, vol. xxvii.).
A method was given by J. H. Lambert for expressing as a continued
fraction of the preceding type the quotient of two convergent power
series. It is practically identical with that of finding the greatest
common measure of two polynomials. As an instance leading to results of
some importance consider the series
x x squared
F(n,x) = 1 + --------------- + -------------------------------- + ...
([gamma] + n)1! ([gamma] + n)([gamma] + n + 1)2!