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

17-33]

KAPTEYN SERIES

567

In like manner, we deduce from (1) that
o00 1; J2m+l {(2m + l)} =
m=O - Z
and hence that
(=.=2 9 J.m+l {(2m + l) z
(3) (2m+1)2
M-o (2m + 1)'
The expansions of z" when n is 1 or 2 are therefore constructed.
Now assume that, for some particular value of n, z" is expansible in the
form
z2 =n2 2 bm,n Jn(mz),
n =1
and consider the function q (z) defined by the equation
00o 2 _ 2
4 (z) = (n ~+ 2)21 M n- bm J. (mz).
m==l m
By the process of differentiation already used, we have
2 d ) d ( () ( + 2)2(1 - 2) (M2 - n2) b,n Jm (ntz)
dz dz:=1
=(t + 2)2 {+2 2 - (n + 2)2 (1 - 2)
dz2 dz) n2
= (n + 2)2 zn+2
On integration we deduce that
b (z)= Zf+2 + A'log z + B'.
It is obvious that A' = B' =0 from a consideration of the behaviour of b (z)
near the origin.
Hence the expansion of zn+2 is
z'+2 (n + 2)2 E bm,n+2 Jm (mz),
m=l
/t12 - n2
where bmn, n+2 = m2 bm, nIt follows at once by induction that
221-1 F (m + bn
bm, 2m = n - r ( m - + 1)m 2
a2 2 ~ 22n r (m + n) J2m (2mg)
and so z. = n2 I )2-).T-i)
m=1 (2nM)212-l 2 r (m-n + 1)
That is to say
z= 2 r (m + n) J2r, (2mz)
~= ~ m22+i r (m - n+1)'
and this is equation (3) of ~ 17'32. The expansion of Z2n-1 is obtained in the
same' way from the expansion of z; the analysis in this case is left to the
reader.