A treatise on the theory of Bessel functions, by G. N. Watson.

102

THEORY OF BESSEL FUNCTIONS

[CHAP. IV

When p has any of the values 1,, 2,..., the solutions which contain z-5
as a factor have to be replaced by series involving logarithms (~ 3'51, 3'52),
and there is only one solution which involves only powers of z. By the
previous reasoning, equation (1) still holds.
When p has any of the values 0, 1, 2,... a comparison of the lowest powers
of z involved in the solutions shews that (1) still holds; but it is not obvious
that there are no relations of the form
-P oF (i -p; c2z) = z-eZF, (-p; - 2p; - 2cz) + -/czP+l0F (p +; c z2)
=-Z-Pe-ClF, (-p; - 2p; 2cz) + e2zP+loFl (p + 3; IC2Z2)
where k,, ck are constants which are not zero.
We shall consequently have to give an independent investigation of (1)
and (2) which depends on direct multiplication of series.
NOTE. In addition to Kummer's researches, the reader should consult the investigations of the series by Cayley, Phil. acg. (4) xxxvi. (1868), pp. 348-351 [Collected Papers,
vii. (1894), pp. 9-12] and Glaisher, Phil. lMag. (4) XLIII. (1872), pp. 433-438; Phil.
Trans. of the Royal Soc. CLXXI. (1881), pp.. 759-828.
4'42. Relations between the solutions in series.
The equation
eczF, (p + 1; 2p + 2; - 2cz) = e-czFl( (p + 1; 2p + 2; 2cz),
which forms part of equation (1) of ~ 4'41, is a particular case of the more
general formula due to Kummer*
(1) 1F((a; p; =)= e;lF(p-a; p; -'),
which holds for all values of a and p subject to certain conventions (which will
be stated presently) which have to be made when a and p are negative integers.
We first suppose that p is not a negative integer and then the coefficient of
[n in the expansion of the product of the series for e; and ~Fl (p - a; p; -f) is
a () \m (p.(p-)(1 — )_ n
-_( )o. (1 - a -~ n)(
n!(p)n
if we first use Vandermonde's theorem t and then reverse the order of the factors
in the numerator; and the last expression is the coefficient of "n in 1F, (a; p; C).
The result required is therefore established when a and p have general complex
values+.
* Journal flr Math. xv. (1836), pp. 138-141; see also'Bach, Ann. Sci. de Vl'cole norm. sup. (2)
II. (1874), p. 55.
f See, e.g. Chrystal, Algebra, ii. (1900), p. 9.
+ Another proof depending on the theory of contour integration has been given by Barnes,
Trans. Camb. Phil. Soc. xx. (1908), pp. 254-257.