ON ^-FUNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 265 



Part II. 



§ 1. Certain Infinite Products. 



In this section of the paper I propose to investigate the expansion, in series form, 

 of such products as 



(l + UyC + a 2 x 2 +■'/.. + a„j: n V 1 + a x — + a 2 — +,...+ a,,-^ V 1 + a x —^ ■ 



+ aX 



toW 1 + W 



q J V 2 

 (1 + 2qx cos + (fx 2 )(l + 2q 2 x cos 6 + g 4 x 2 ) 2 ( 1 + 2q*x cos 6 + q 6 x 2 ) 3 



( 1A )( 1+ ^( 1+ -)' 



( 1 + 2qy cos 6 + ij V)( 1 + V.? cos 6> + ? y ) 2 ( 1 + 2q 3 y cos + q 6 y 2 ) 3 ' 



= \ (x , y) + X^x , y) cos 2$ + \ 2 (x , y) cos 36 + 



The coefficients X , X x , . . . . X n (x , y) will be seen to have an intimate connection 

 with the q generalisation of Bessel's Functions denoted by the symbols J w , ^ [n] , . . . 

 in previous papers. 



It is easy to see that the A -equation 



A.u = {c + c } x + c 2 x 2 + .... +c n _ l x n - 1 }u . . . (27) 



is satisfied by the convergent infinite product 



U = n s(«l > a 2 ' a 3 » • • • ■ «» » X ) » 



= (l W+ « 2 * 2 + . . +«„*»)(! +^ + ^2+ .... + |;,»)(l + 3., + |2 x2+ ."..+*) , 



in which 



"l — C Q > 



1 



_(?-!) 



"2 ^2~ C l' 



«„ = <i^>e 



n 1 ■ 



The general factor of the function is 



(1 4- a x xq~ r + a 2 x 2 q~' lr + . . . + a n x n q~ nr ) . 



§ 2. Expansion as an Infinite Series. 

 An examination of the infinite product shows us that in case the product can be 

 represented by a series of powers of x , such series will be of the form '^ i c l x r , 



= l + -?±2-x + c 2 x 2 + c l x s + 



