[HARKNESS] THE ALGEBRAIC BASIS OF CERTAIN BESSEL SERIES 103 
(az 1): (re 
FSC tome 
(10) are the general terms in the expansions of 5 JC) ve), 
If we multiply both sides by , the various terms of 
(v+2) J,42 (x), etc.; hence we get Lommel’s theorem 
5 the al, (5) ti yst 2) Te ta ae (ved) Tee (oe) ee CD) 
EGNeip: 270): 
§ 5. In the kind of analysis that we have been considering it is 
useful to notice that the results arising from the formula in partial 
ac 
v+1. ————— Luce = = a let = ia 
fi) ee v+st+l1 
by making s=1, 2, 3, ..., have analogues in which two or more factors 
occur in the denominators. aa instance 
2.3 Sarl 
v+1.74+2.. Spee 0 y+ mo, me —= 
+ = ras oar: v+st+l1. —— 
and 
3 44e. 62 à 1 = 1 
p+1.y+2...v+54+3 v+l.r+2. 743 (Go ice, eme 
1 1 : 
T Chem RD eeepc 
Nielsen bases part of his discussion of Neumann series of the first 
kind on the identity 
D a5 ne seen ae 
ak 1)? Car Ro D Ever — mm —p) = 05m 
(N. p. 271). The algebraic equivalent is, of course, 
y+2r m p+2r—2 
y+2r.v+2r—1...v+2r—m = Voce —l...v+2r—m-1 
4 p+ 2r — im 
HO ee - © (12) 
His’method of proof depends on the use of A‘, the difference of the 
sth order. The following proof is based ultimately on the same order 
of ideas, but has a more elementary appearance. 
Replace » by y—2r—2m-—1 in the general formula patterned after 
the type of those given at the beginning of this paragraph. Employ 
m and m-+1 factors respectively in the denominators, replace s by m, 
m—1 respectively; and reverse the order of the series. We then get 
