400 THE REV. F. H. JACKSON ON 
of q functions a great variety of notations has been used. I propose in this paper to 
bring before the Society a series of formule relating firstly to a function E,(x) analogous — 
to exp (x). These formule are supplementary to those given in Trans. Roy. Soc. — 
Edin., vol. xli. pp. 105-118, and will lead to one or two interesting properties of a 
function J?,,,(a), which may be termed a generalized Bessel-function of double order, — 
_ and to various novel expressions of elliptic functions in terms of the generalized Bessel- 
function. For example, Jaconi’s © function is expressed by the form 
U) Ara 
oS) — . an 
in which 
go= fA -¢*) 
pp alg 
q-1 
ee 
q-1 
po & 
It is noteworthy that n (the order of the J functions) is in (A) an arbitrary number. — 
It appears only in the expression on the right side of that equation. A definite integral 
expression for the functions J will also be given. 
2. 
Function E,(x). 
The series 
NPRRCR Aer Aig GEN are) 
ene eee ee 
and its equivalent product 
(1—a)(1—pa)\(l-p'e)...... 
are well known: we derive a function analogous to the exponential function. — 
(Ci: Trans. f.SE., xl. p. 116; and Proce, DMS series 2, vol. noup/ algae 
2 2 
[Be Sen ee WO 
p i ei a ale 
ce ee el ea BONA. Le he 
a iy an aor 
(7}=(p"-1)(p-1) 
The function E,(«) may be regarded, like the exponential function, either as the 
limit of a certain infinite series or a certain infinite product. The results numbered 
(2) . . . (26) are either easily obtained or are known in other forms. 
