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

5-41]

MISCELLANEOTUS THEOREMS

147

By considering the powers of z which occur in the product J^ (z) J, (z) it
is easy to infer that, if 2,, 2v and 2 (/ut + v) are not negative integers and if
At2 _ v2, then
(5) J )() J () () (. + v + 2n + 41)
(m5o Jm! (.J + v + m + )F. L + )r( + 1) I (v + m + 1)
In like manner, by solving (4) in series, we find that, when 2v is not a
negative integer, then
(6) J 2 (z)= ( (-)" (Z)2 + r (2v 2 + 1)
m=o01 1M (2v +in + 1) Jr(v n+ 1+)}2
and, when v is not a negative integer, then
(7) J,^(z)J (z-)= 2 (I > (() (I)2m (2m)!
7) O(Mn_ (v7 + n+) r(- v +n +1)
By reasoning which resembles that given in ~ 4'42, it may be shewn that
(6) holds when v is half of an odd negative integer, provided that the quotient
r (2v + 2mn + 1)/F (2Y + m + 1) is replaced by the product (2v + m + +)m
5'41. Products of series representing Bessel functions.
It is easy to obtain the results of ~ 5'4 by direct multiplication of series.
This method has the advantage that special investigations, for the cases in
which /Lt2 = v2 and those in which f + v is a negative integer, are superfluous.
The coefficient of (-)m (pz)l+v+2~" in the product of the two absolutely
convergent series
() (_ ) (),' X ()n (iZ)v+,n, =om! F r (t + mi + 1) = n! P (v + n +l)
qua 1
is 1
~=o i t P (v + ~ + 1).+(~, - %! P (/ + mn - ii + 1)
=,,~, p (t +,, + l) F (. +,, + i)nz.(
_ ()* ( -m ( - - -2, 2)n
m! r (f/ + n + 1) r (, +, + 1)
(fk + V + t + 1 ),,
~! r (At + m 1) r (v + m + 1)'
when Vandermonde's theorem is used to sum the finite series.
Hence, for all values of p/ and v,
(1) J J, m(z() J (Z)= ( )++2r( + v + 1)
m, =0oin! F (/t+ in + 1) (v + rn + 1)'
and this formula comprises the formulae (5), (6) and (7) of 5'4.
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