ON ^-FUNCTIONS AND A CERTAIN DIFFERENCE OPERATOR. 



271 



in which 



/jl q (x) = c 2 + Cj 2 x 2 + c?x A + + c r 2 x' r + . . . 



pjp) = c n c x n + c n+i c i x n+2 + + Cu+rCrX n+2r + 



c r being determined by the relation 



ft, r (I-? 2 )'- 1 (l-r/)(l -^)'-' i (1 



;i-a a )(i-2*)(i-s e ) 



c,_, + 



+ (a- b)(q*a-b) ... (g 2 '- 2 a-b ) 



(l-2 s )(l-2 4 ) • ■ (1-2) 



2F~ '0' 



§ 7. Coefficients /«„, ^ related to X n . 

 Putting 6 = 0, we obtain 



(1 4- 2axcf cos 28 + a^V) 1 ^ 4- 2axq i cos 28 + aWq s ) 2 

 v (x) + 2v 1 (x) cos 28 + 2v 2 (x) cos 40 + 



in which 



v„(x) = c n c x n + c n+1 c 1 x n+2 + 



l\-q lr ) ac r _, , , a 2 c r 



«* (1-2 2 "(1 -g 2 )(l -g 4 ) 



Similarly, we may show that 



+ q r 



a'c n 



(l-g 2 )(l-g 4 ) (l-2 2 



1 



( 1 + 2axg 2 cos 26 + u*sfyf(l + 2axq i cos 2d + a 2 x 2 q s ?^- + ^axq 6 cos 26 + aV^ 18 ) 3 

 = A o (a;) + 2A 1 (a;)cos204-2A. 2 (a;)cos404- , 



in which 



Xjx) = C n C<p n + c n _ l c l x n+ ' 2 + 



a 2r r ((fi-iyitfl- l)(^-l)7 („2-\)(ai-i) .. 



(g 2 -l)(g 4 -l) • • • (2 2r "l) 



(55) 



(56) 



(56a) 



(57) 



T ( 2 2_i) (gt-l)(2*-l) 



It is not difficult to see that 



"ofc) = \(*)/*o(«) + 2X l(-«>l(-« ; ) + ^fcW*) + » 



CO 



"rfc) = 2, 0*r+l(»)M«) + /V(*)^r+l(*)). 



We shall now obtain certain relations connecting these coefficients with the q 

 generalisation of Bessel's Function. 



§ 8. Connection of Coefficients with ^-Bessel Coefficients. 



To show the connection between the coefficients A., n, u, and the q generalisations 

 of Bessel's coefficients we proceed as follows : — 



*«(*)- 



\ l _ ax(q-l) ' 

 2 - 



E>x) Ei («*)e,(^) 



'.(|W? 



j + ax(q - I) 



i+: 



*(g - i) T 

 <f J 



"l + Kg -1)1 _ ! + Mg-i) ! 2 ^ + Mg - 1) 1 



g JL g 2 J l g 3 J 



(58) 



(5») 



