ON g- FUNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 269 



with for limited values of x a common region of convergence with E 7 (acc). We shall 

 subsequently require the properties 



■E q (ax)E q -i(-ax) = l (41) 



W Vfl«) - 1 + ffiW <" + 'X« + *% * + 



§ 5. "Double Product Theta-Functions." 

 Consider a function ^(x), of the form 



<t>(x).<t>(*).<t>(?-S ad inf. 



+W-%}+($ ad inf - 



In case the infinite product is convergent, we may write 



**>-$$$&<>»• ■ ■ ■ ■ <«> 



and by means of (15) 



*(*) - e + «*] ~ *«> «(«) + *W ; y ^ + (44) 



xf;(x) \qj Jx\ \qV 



Let us now consider the special function &(x), in which <p(x) = ~R q (ax),y\/(x) = 'E, q (bx), 

 so that 



•« -!©•«■ • • •_..■'• ■ <45) 



From (45) we have at once by means of (41) 



*/ \ / i , ( a - b) , (a-b)(a-qb) » „ , , (a-b) . . . (a-q'~ l b) ,. .. ) _, . .... 



*(</») = \ !+ V iyj -^+- r 2 \, V -' c2 + ■ • • • + V '- P^p = >-q'x + ■ ■ • [*(«), (46) 



so that, in case &(x) can be expressed as a series of positive powers of x, as seems 

 possible (a priori) from the nature of the infinite product, which is of the form 



\ + ax\(\+-xX(l + ^rx) 



'S tlS tl , .... (47) 



i + ^)(i + ^)(i + !*) 3 



we shall have the following relations to determine the coefficients in the expansion 



<&(x) — c + L\X + c 2 x 2 + , 



( 'o = l . 

 a-b 



Cl i 7TT = tvz ' 



, a-b Act- b)(a - qb) 9 ,, 



c ' 2 TiT^ 1 — [W~ = ' 



,. (a n _ n _ (a - &)-. . (a - ft) (« ~ ff& t 2 ,. , . (« - &)(« - qh) . . ■ (a- g-^) - , . 



TRANS. ROY. SOC. EDIN., VOL. XLVI. PART II. (NO. 11). 40 



