| K | ( | a1, | b2, b3, ..., bn a2, a3, ..., an | ) |
is also equal to the determinant
| a1-100 0 | b2a2-10 0 | 0b3a3-1 - | 00b4a4 - | ...b5 u0 | .... -10 | .... an-1-1 | 000- bnan |
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 nth 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 | bn | |||
| a1 | + | a2 | + ... + | an | + ... |
can be constructed having for the numerators of its successive convergents any assigned quantities p1, p2, p3, ..., pn, and for their denominators any assigned quantities q1, q2, q3, ..., qn ...
The partial fraction bn/an corresponding to the nth convergent can be found from the relations