[alpha].[beta] [alpha]([alpha] + 1)[beta]([beta] + 1)
--------------x + -------------------------------------- x squared + ...
1.[gamma] 1.2.[gamma].([gamma] + 1)
gave the expression for F([alpha], [beta] + 1, [gamma] + 1, x) /
F([alpha], [beta], [gamma], x) as a continued fraction, from which if we
put [beta] = 0 and write [gamma] - 1 for [gamma], we get the
transformation
[alpha] [alpha]([alpha] + 1)
1 + -------x + --------------------x squared +
[gamma] [gamma]([gamma] + 1)
[alpha]([alpha] + 1)([alpha] + 2)
---------------------------------x cubed + ... =
[gamma]([gamma] + 1)([gamma] + 2)
1 [beta]1 x [beta]2 x
-- --------- --------- where
1 - 1 - 1 - ...
[alpha] ([alpha] + 1)[gamma]
[beta]1 = -------, [beta]3 = --------------------------, ...,
[gamma] ([gamma] + 1)([gamma] + 2)
([alpha] + n - 1)([gamma] + n - 2)
[beta]_{2n-1} = ------------------------------------,
([gamma] + 2n - 3)([gamma] + 2n - 2)
[gamma] - [alpha] 2([gamma] + 1 - [alpha])
[beta]2 = --------------------, [beta]4 = --------------------------,
[gamma]([gamma] + 1) ([gamma] + 2)([gamma] + 3)
n([gamma] + n - 1 - [alpha])
..., [beta]_{2n} = ------------------------------------.
([gamma] + 2n - 2)([gamma] + 2n - 1)
From this we may express several of the elementary series as continued
fractions; thus taking [alpha] = 1, [gamma] = 2, and putting x for -x,
x 1 squaredx 1 squaredx 2 squaredx 2 squaredx 3 squaredx 3 squaredx
we have log(1 + x) = -- --- --- --- --- --- ---
1 + 2 + 3 + 4 + 5 + 6 + 7 + ...
Taking [gamma] = 1, writing x/[alpha] for x and increasing [alpha]
indefinitely, we have
1 x x x x x
e^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
b5 +
b4 + ----
a5
b3 + ---------
a4
b2 + --------------
a3
a1 + -------------------
a2