The continuant K ( b2, b3, . . ., bn
a1, a2, a3, . . ., an) is also equal to the determinant
|
a1 −1 0 0
0 |
b2 a2 −1 0
0 |
0 b3 a3 −1
— |
0 0 b4 a4
— |
. . . b5
u 0 |
. . . .
−1 0 |
. . . .
an−1 −1 |
0 0 0 —
bn an |
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.).
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
and the first two partial quotients are given by
If we form then the continued fraction in which p1, p2, p3, . . ., pn are u1, u1 + u2, u1 + u2 + u3, . . ., u1 + u2 + . . ., un, and q1, q2, q3, . . ., qn are all unity, we find the series u1 + u2 + . . ., un equivalent to the continued fraction
| u1 | u2 ⁄ u1 | u3 ⁄ u2 | un ⁄ un−1 | ||||||
| 1 | − | 1 + | u2 | – | 1 + | u3 | – . . . − | 1 + | un |
| u1 | u2 | un−1 | |||||||
which we can transform into
| u1 | u2 | u1u3 | u2u4 | un−2un | , | ||||
| 1 | − | u1 + u2 | − | u2 + u3 | − | u3 + u4 | − . . . − | un−1 + un |
a result given by Euler.
2. In this case the sum to n terms of the series is equal to the nth 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 + a2x² + . . . by an infinite continued fraction of the form
| β0 | β1x | β2x | β3x | ||||
| 1 | − | 1 | − | 1 | − | 1 | − . . . |
Here the fraction converges to the sum to infinity of the series. Its nth convergent is not equal to the sum to n terms of the series. Expressions for β0, β1, β2, . . . 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
| F(n,x) = 1 + | x | + | x² | + . . . |
| (γ + n)1! | (γ + n)(γ + n + 1)2! |
We have
| F(n + 1,x) − F(n,x) = − | x | F(n + 2,x), |
| (γ + n)(γ + n + 1)2! |
whence we obtain
| F(1,x) | = | 1 | x ⁄ γ(γ + 1) | x ⁄ (γ + 1)(γ + 2) | |||
| F(0,x) | 1 | + | 1 | + | 1 | + . . ., |
which may also be written
| γ | x | x | |||
| γ | + | γ + 1 | + | γ + 2 | + . . . |
By putting ± x² ⁄ 4 for x in F(0,x) and F(1,x), and putting at the same time γ = 1 ⁄ 2, we obtain
| tan x = | x | x² | x² | x² | tanh x = | x | x² | x² | x² | ||||||||
| 1 | − | 3 | − | 5 | − | 7 | − . . . | 1 | + | 3 | + | 5 | + | 7 | + . . . |
These results were given by Lambert, and used by him to prove that π and π2 incommensurable, and also any commensurable power of e.
Gauss in his famous memoir on the hypergeometric series
| F(α, β, γ, x) = 1 + | α · β | x + | α(α + 1)β(β + 1) | x² + . . . |
| 1 · γ | 1 · 2 · γ · (γ + 1) |
gave the expression for F(α, β + 1, γ + 1, x) ÷ F(α, β, γ, x) as a continued fraction, from which if we put β = 0 and write γ − 1 for γ, we get the transformation
| 1 + | α | x + | α(α + 1) | x2 + | α(α + 1)(α + 2) | x3 + . . . = | 1 | β1x | β2x | |||
| γ | γ(γ + 1) | γ(γ + 1)(γ + 2) | 1 | − | 1 | − | 1 | − . . . |
where
| β1 = | α | , β3 = | (α + 1)γ | , . . ., β2n−1 = | (α + n − 1)(γ + n − 2) | , |
| γ | (γ + 1)(γ + 2) | (γ + 2n – 3)(γ + 2n − 2) |
| β2 = | γ − α | , β4 = | 2(γ + 1 − α) | , . . ., β2n = | n(γ + n − 1 − α) | . |
| γ(γ + 1) | (γ + 2)(γ + 3) | (γ + 2n – 2)(γ + 2n − 1) |
From this we may express several of the elementary series as continued fractions; thus taking α = 1, γ = 2, and putting x for −x, we have
| log(1 + x) = | x | 12x | 12x | 22x | 22x | 32x | 32x | |||||||
| 1 | + | 2 | + | 3 | + | 4 | + | 5 | + | 6 | + | 7 | + . . . |
Taking γ = 1, writing x ⁄ α for x and increasing α indefinitely, we have
| ex = | 1 | x | x | x | x | x | ||||||
| 1 | − | 1 | + | 2 | − | 3 | + | 2 | − | 5 | + . . . |
For some recent developments in this direction the reader may consult a paper by L. J. Rogers in the Proceedings of the London Mathematical Society (series 2, vol. 4).
Ascending Continued Fractions.
There is another type of continued fraction called the ascending continued fraction, the type so far discussed being called the descending continued fraction. It is of no interest or importance, though both Lambert and Lagrange devoted some attention to it. The notation for this type of fraction is
| b4 + | b5 + | |||
| b3 + | a5 | |||
| b2 + | a4 | |||
| a1 + | a3 | |||
| a2 | ||||
It is obviously equal to the series
| a1 + | b2 | + | b3 | + | b4 | + | b5 | + . . . |
| a2 | a2a3 | a2a3a4 | a2a3a4a5 |
Historical Note.
The invention of continued fractions is ascribed generally to Pietro Antonia Cataldi, an Italian mathematician who died in 1626. He used them to represent square roots, but only for particular numerical examples, and appears to have had no theory on the subject. A previous writer, Rafaello Bombelli, had used them in his treatise on Algebra (about 1579), and it is quite possible that Cataldi may have got his ideas from him. His chief advance on Bombelli was in his notation. They next appear to have been used by Daniel Schwenter (1585–1636) in a Geometrica Practica published in 1618. He uses them for approximations. The theory, however, starts with the publication in 1655 by Lord Brouncker of the continued fraction
| 1 | 12 | 32 | 52 | ||||
| 1 | + | 2 | + | 2 | + | 2 | + . . . |
as an equivalent of π ⁄ 4. This he is supposed to have deduced, no one knows how, from Wallis’ formula for 4 ⁄ π viz.
| 3 . 3 . 5 . 5 . 7 . 7 . . . |
| 2 . 4 . 4 . 6 . 6 . 8 . . . |
John Wallis, discussing this fraction in his Arithmetica Infinitorum (1656), gives many of the elementary properties of the convergents to the general continued fraction, including the rule for their formation. Huygens (Descriptio automati planetarii, 1703) uses the simple continued fraction for the purpose of approximation when designing the toothed wheels of his Planetarium. Nicol Saunderson (1682–1739), Euler and Lambert helped in developing the theory, and much was done by Lagrange in his additions to the French edition of Euler’s Algebra (1795). Moritz A. Stern wrote at length on the subject in Crelle’s Journal (x., 1833; xi., 1834; xviii., 1838). The theory of the convergence of continued fractions is due to Oscar Schlömilch, P. F. Arndt, P. L. Seidel and Stern. O. Stolz, A. Pringsheim and E. B. van Vleck have written on the convergence of infinite continued fractions with complex elements.
References.—For the further history of continued fractions we may refer the reader to two papers by Gunther and A. N. Favaro, Bulletins di bibliographia e di storia delle scienze mathematische e fisicke, t. vii., and to M. Cantor, Geschichte der Mathematik, 2nd Bd. For text-books treating the subject in great detail there are those of G. Chrystal in English; Serret’s Cours d`algèbre supérieure in French; and in German those of Stern, Schlömilch, Hatterdorff and Stolz. For the application of continued fractions to the theory of
