SEcTION III, 1916 [97] TRANS RSC 
The Algebraic Basis of certain Bessel Series, including more particularly 
the Neumann Series of the second kind. 
By JAMES HARKNESS, M.A., F.R.S.C. 
(Read May Meeting, 1916). 
§ 1. In this paper I propose to discuss certain series proceeding 
according to Bessel Functions and more particularly according to 
products of Bessel Functions—generally known as Neumann Series 
of the second kind—and to direct attention to the algebraic basis on 
which they rest. The definition adopted for Bessel’s Function will be 
pied: (2/08 C4 
D (Ls) (v+s+1)’ @ 
where » is an arbitrary parameter. References will be made prefer- 
ably to the exhaustive treatise of Dr. Niels Nielsen, Handbuch der 
Theorie der Cylinde:funktionen, as this work contains in a convenient 
form most of the general theorems so far discovered relating to our 
subject. Occasional reference will also be made to Gray and Mathews, 
Treatise on Bessel Functions. We shall cite these two books, for 
brevity, as N. or G. and M. Free use will be made of Schénholzer’s 
extremely useful theorem 
pepe pese 
JOC) Sexe) — face DRE EDR) T(p+s+1) ) G ne 
(N. p. 20.) 
J (x)= 
§ 2. The following two formulae 
~\Y TP x twee. do _T(p—vts)_ 
(3) A) = TG Ee G ) Fvts () (3) 
D > (+25) T(v+s) p-v 
L@=G) 2 oran (Feo & 
have a general external similarity (N. pp. 268, 275). By equating 
corresponding powers of x on each side of (3), (4) after the J’s have 
been replaced by series of powers of x, it appears that the two formulae 
depend on two sets of identities of similar appearance but of essentially 
distinct character. The earlier members of these two sets of identities 
are 

