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XL — On g-Functions and a certain Difference Operator. By the Rev. F. H. 

 Jackson, M.A. Communicated by Professor Chrystal. 



(MS. received January 16, 1908. Read February 17, 1908. Issued separately April 21, 1908.) 



CONTENTS. 



Part I. 



Introduction .......... 



§ 1. Definitions of A*> A*- 



§ 2. Equation A x y = y- Transformation of the Power Series 2«nX" 

 § 3. Elementary properties of the operative symbol A. Comparison with the differential 

 Analogue of ~Duv = vDu + uDv, etc. ...... 



§ 4. Interpretation of A~", q- Finite Integration ..... 



§ 5. Examples of Inverse operations ...... 



§ 6. Solutions of A Equations. Function of multiplicative period q, F(qx) = F(x) 



§ 7. Forms of k n u x , & n u x v z ' . 



§ 8. Maclaurin's Series and Conjugate Form ..... 





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il operator 



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Part II. 



§ 1. Infinite product of which the general factor is (l+a 1 xq r + a 2 x 2 q 2r + . . . a n x"q rn ) 



§ 2. Expression of this product as a power series ........ 



§ 3. Expression of the product of n functions Q(a 1 x) Q(a 2 x) .... 0(«„a;) as Aj + A 2 cos 2x + A 3 cos 4x +• 



§4. Special Series for J acobi's function ©"(a;). ........ 



, (l+ax)(l + aqx) 2 (l+aq 2 xf .... „ 



§ 5. Expression of {l+bx){l+biX y (1 + bq 2 zf _ . .. as 2 V • 



. (1 + 2 ^ cos e + a 2 a- 2 g 2 )(l + 2(ta;g 2 cos 6 +a 2 x 2 q i )\l + '2,axq 3 cos e + a 2 x 2 q % f . , 



3 6. Expression of (Y^ bxq cos g + tftf^yi + 2bxq 2 cos e + ^^ + 2bx f cos B + b 2 xhff . 



§ 7. Coefficients /j, n , v n allied to X n . 



§ 8. Relations of these with q generalisations of Bessel's coefficients 



as 



2a„ 



cos ne 



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271 

 271 



Part III. 



§ 1 . The general hypergeometric series 5-form .... 



§ 2. Special g-difference equations allied to Bessel's differential equation 



§ 3. Solutions corresponding to Hankel's Y„ 



§ 4. Solutions of the equation satisfied by J,,; . J„ . 



§ 5. Relations between solutions ...... 



§ 6. Recurrence and special A relations satisfied by the solutions . 



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



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