6g Rev. F. H. Jackson. [June 15, 



Multiplying together the two infinite series J and |, the coefficient 

 O f ,,,.+n+2r i n the resulting series is 



' 



We can sum this series of r+ I terms simply, as follows : 



In the case (p = 1), we see that the expression is Vandermoride's, 



and in the general case it is included under the following extension of 



Vandennonde's theorem 



Substituting m + r f or x, n + r for y, and changing the base p into p 2 , we 

 find 



= (2m + 2r} r + jp*< + '> _.__ {2m + 2r} r _ s {2n + 2r} g (10), 



in which, since r - s is integral, 



{2m + 2r] r _ 8 denotes [2m + 2r] [2m + 2r - 2] . . . to r - s factors. 



Dividing both sides of (10) by {2m + 2?-} I {2n + 2r} ! (m and n 

 unrestricted) we obtain 



{2m + 2/1 + 4r} r 



\{2n + 2r}\{2r}\ 



{2m + 2s} ! {2n + 2r - 2s} ! {2r - 2s} ! {2s} ! '" 



which series we have seen to be coefficient of x m+n+2r , so that (7) is 

 established. In the notation of the generalized gamma function 



I-/' ' , ' 



r = ^ ([m + n + r + !])!> ([m + r + 1]) !>([ + r + l])Fy ([r + 1]) 



By means of this product various series of squares of Bessel 

 ctioiis, and series of products of pairs of Bessel functions, may be 

 generalized ; for example, 



^ r IJi) 2 + 3 {Ja} 2 + 5 {J ,} 2 + _ _ (Lommel.) 



