254 



THE REV. F. H. JACKSON 



Part I. 



Introduction. 



In this paper my object is, primarily, to investigate the properties of a certain 

 operative symbol A", which appears to be of great utility in discussing (/-functions. 

 The first part of the paper will consist of an investigation into the various forms of 



A>(*)}. 

 and the nature of the inverse operations symbolised by A "". With certain restrictions 

 as to continuity, etc., <p{x) will denote an arbitrary function of x. In the second part 

 of the paper such products as 



(1 + a 1 x + a 2 x 2 + . . . + a li x ll )(l + a 1 qx + a 2 q 2 x 2 + . . . + a n q n x n )(l + a 1 q 2 x + . . . + a n q"~"x n ) . . . . , 



< '^)( 1+ f)'( 1+ ?T 



<^>K*) 2 K?)° ' 



( 1 + 2(j xcoa6 + q 2 x 2 )( 1 + 2yV cos 9 + (/V) 2 (l + 2q*x cos 9 + q^x 2 f 



(1 +■ 2qy cos 6 + q 2 y 2 )(l + 2q 2 y cos 9 + q i y 2 ) 2 (l + 2q*y cos 9 + q 6 y 2 f ' 



= \ (x , y) + X^x , y) cos 9 + \ 2 (x ,y)cos29+ , 



will be discussed, and relations found between the coefficients X(x, y), and the q 

 generalisations of Bessel's Functions denoted by the symbols J [n] (x), *$ {n] {x), in previous 

 papers.* 



In the third section of the paper certain special A -equations will be discussed in 

 connection with their limiting forms as linear differential equations. 



In the appendix to Chapter ii. of Heine's Kugelfunctionen\ there is given a 

 discussion of the special (/-function <p(ci, b, c, q, x) analogous to the Hypergeometric 

 series F(a, b, c, x). Heine makes use of a difference symbol A, which he defines by 



the equation 



A</>(a ,b ,c,q,x) = <j>(a ,b,c,q, qx) - <£(a ,b,c,q,x). 



It is obvious that the repetition of the operation symbolised by A gives us 



A n &(x) = ^q n x)-n^q n - 1 x) + ?^^<f>(q n -' i x)+ ( - 1)"<%) . 



Now, in q series, such coefficients as the Binomial coefficients do not usually appear 

 The (/-equivalent of H is [1][2] . . . [r], in which [r] = (q r - 1 )/(</- l) ; therefore, in 

 discussing (/-functions, 1 think it desirable to use an operative symbol which gives rise 

 to a series in which the coefficients follow the (/-Binomial form, i.e. 



^ , ^x) = ^q n x)-[7i]^q n - 1 x) + q^[ Ti ^^(q n - 2 x)- ( - l)^'""" 1 '^) 



[2J! 



* Trans. Roy. Soc. Edin., vol. xli. pp. 105-118, 399-407, etc. Proc. L.M.S., series 2, vol. ii. pp. 192-221 ; vol. iii. 

 pp. 1-23. 



t Heine, Kuyelfunctionen, ed. 1878, pp. 99 et seq ; also Thomje, Crelle's Journal, vol. lxx. 



