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

17-4, 17-5]

KAPTEYN SERIES

571

It is easy to deduce that, if the Maclaurin series for f(z) is
f(z)= E an,,n==O
then
(3) a0 = ao,
(4) 1sl <.i (n 2,n)2 (n.- -)! - an-1
17'5. Kapteyn series in which v is not zero.
The theory of Kapteyn series of the type
E amJvnm {(V + rn) Z},
'm=0
in which v is not zero or an integer, can be made to depend on the expansion of zv. The result of ~ 17'33 suggests that it may be possible to
prove that
(1) (1 )v 2 r ( + m) J, ( + 2m) z
m =0 (v + 2m)vSl. m!
throughout the domain K.
It is easy enough to establish this expansion* when t z < 0'6627434; but
no direct proof of the validity of the expansion throughout the remainder of
the domain K is known, and the expansion has to be inferred by the theory
of analytic continuation.
To obtain the expansion throughout the interior of the specified circle,
expand the series on the right in powers of z. The coefficient of ZV+2r is
r ( + m) (_)rm (v + 2m)V+2r
=0 (v + 2m) v+1 m!r' 22r-(r - n)! l(v + r++ M )
r1(v) (_)r-mm(v + 2m)2r-l r (v + m) r (v + 2r + 1)
2v+2 r( - 2r+ 1)0= m!(r-m) r() r(v + r + m +1)
When r 1, the last series is a polynomial in v of degree 3r - 1 which is
known to vanish identically whenever v is an integer. It therefore vanishes
identically for all values of v. The expansion (1) is therefore established
(inside the circle) by a comparison of the coefficient of zv on each side of the
equation.
From this result, we can prove that, under the conditions specified in ~ 17;4,
zv oo
(2) - = Z n, (t) Jv+nJ(v + n)z},
t- z.=0
where
I </ (v + - 2m)21 r (v + n -m)
v / ( 3)~ 2 no (,v_ + )v" n-2m. mn tn-2nm —1'
* This was done when Iz <0'659 by Nielsen, Ann. sci. de l'Ecole norm, sup. (3) xvIII. (1901),
pp. 42-46.