( 399 ) 
XVII.—Theorems relating to a Generalization of Bessel’s Function, II. 
By the Rev. F. H. Jackson, R.N. Communicated by Dr W. Pepptn. 
(MS. received February 6, 1905. Read February 20, 1905. Issued separately April 18, 1905.) 
CONTENTS. 
PAGE PAGE 
§ 1. Introduction ; ; 2 2 , : 399 F ANd Gye : . : ; : 4 4 405 
§ 2. Function E,(x)  . é : 400 | § 5. eee series : 5 : : 4 ‘ 407 
§ 3. Expressions for JACOBI'S aiurictions : ; 403 | 
ie 
INTRODUCTION. 
The theory of the functions commonly known as q functions might perhaps be 
greatly developed, if investigators were to work on lines suggested by the functional 
notation of well-known analytic functions. For instance, the analysis connected with 
the circular functions sn x, cos x, ...., might be regarded as the theory of certain 
infinite products without using any special functional notation. It need not be 
explained however, how great was the gain to elementary algebra by the introduction 
of the exponential function (regarded as the limit of a certain infinite product, or as 
the limit of a certain infinite series) denoted e*, with certain characteristic properties, 
enabling the worker to make transformations easily and quickly. Of course, the vast 
store of interesting and in many cases useful results connected with the elementary 
_ functions of analysis might have been obtained without the introduction of any 
| notation capable of rapid and easy transformations, but I think it unlikely that they 
would have been obtained. 
In chapter xi. of Caytey’s Elliptic Functions the identity 
1 { y? ge" g gi” 
—________________, | ——= : aw a AE 
1—q?-1-q? die 1—q?" | aiere 1 = gent? + 1 —q?-1- 9 1 — g?"*2.] — gint4 
is used in order to express Jacosi's 9 function, in the well-known form 
1 — 2¢ cos 2a + 2q* cos 4a —2g9cos6u+ .... 
The likeness of the series (1) to BEssEL’s series is very obvious. It is a very special 
ease of the series which I have denoted J,,,; in previous papers, and in itself might have 
suggested a theory of q functions analogous to BrssEL’s functions. In the discussion 
TRANS. ROY. SOC. EDIN, VOL. XLI. PART II. (NO. 17). 59 
