ON 3-FUNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 267 



Then 



n,(¥A • • M=n(i + ^)n(i + *£) & „( 1+ ^)' 



m=0\ Q /?n=0\ <± / m=0\ ({ / 



i 1 + PnW ■ pA 2 * 2 , I 



1 (q-i) v-i)(?-i) r 



and each of these is absolutely convergent for all values of x in case | q | > 1 , as well as 

 for limited values of x in case | q | < 1 . 



§ 3. Product of n, Theta-Functions. 



In this section I propose to obtain a theorem, which may be stated as 



6(a! , x)Q(a 2 , x)Q(a 3 , x) Q(a n , x) = A + Aj cos 2x + A 2 cos 4a; + 



The functions (a 1 x)Q(a 2 , x) each reduce to Jacobi's function 0, in case 



the parameters a v a 2 , ct 3 , are each equal to unity. 



We have above obtained the expansion of the infinite product n <? ((x 1) a 2 , a 3 , . . . a n , x), 

 in the form of a power series ^c r a: r , viz. — 



q m q 2m q Zm q nm J ((/ - \) (cf ~ l)(q 



Change q into q' 2 , and let (l + p 1 x)( 1 + p 2 x) . . . (1 +p n x) = (1 + a 1 x + a 2 x 2 + . . . 

 + a n x n ). We write then 



?n=oo m=oo m=oo 



U q -2{a,a 2 a z a n x) = n (1 + Pl q* m x). II (\+p 2 q im x) n (l+p^x), 



m=0 m=0 m=0 



J\ q -'i(a x , a 2 , a 3 a B , a;<).II fl -2(a 1 , a 2 , a 3 a rt ,^" 7 )= II (1 +p 1 a»/ am (t + ^~ 1 ) 



+ l 9 1 2 a;Y m ) n(l+p n a^ 2m (^ + r 1 ) + Pn 2 a;Y m ). (32) 



m=0 



Now p v p 2 , p n are independent constants, for we may choose n arbitrary 



factors 



(1 +p 1 x)(l +p 2 x ) • • • (1+/V) to form a polynomial (l+a 1 x + a 2 x 2 + . . . +a n x n ). 



Replacing H q -z(xt) and H q -2(xt~ 1 ) by their series expressions ^?c r x r tf, and '^c r x r t" > \ we 

 obtain from the product of these series 



n 8 -2(a 1 a 2 a 3 . . . a n , a*)II ? -i!(a 1 a 2 . . . a„, a# _1 ) = J + 2J 1 (< + r 1 ) + 2 J 2 (i! 2 + f~ 2 ) + (33) 



J . J l5 J 2 , , are infinite series functions of x, reducing in special cases to 



Bessel's functions, 



J ° - ' + ggjy + {<t - dv - i/ ,jV + "* - ™ + <34> 



