54 Rev. F. H. Jackson. [June 15, 



"A Generalisation of the Functions F (ri) and x n " By Eev. 

 F. H. JACKSON, RN. Communicated by Professor J. LARMOR, 

 Sec. RS. Received December 7, Read December 10, 1903. 

 Received in revised form June 15, 1904. 



CONTENTS. 



Introduction; 1. Difference-equation of the generalised gamma-function; 

 2. Weierstrassian forms of the function ; 3. Extension of the multiplica- 

 tion-theorem of Gauss and Legendre; 4. Definition of (#); 5. Extension 

 of Lommel's product of two Bessel functions ; 6. The generalisation of the 

 Beta function ; 7. Multiplication-theorem ; 8. Logarithmic derivatives and 

 other series ; 9. The form of a generalised G function. 



It is interesting to develope from simple principles a generalisation 

 of the functions x n and F (ri). Consider an infinite sequence 

 (1, p, f, f, . . . p n , . . .), then write 



[1] = 1, 



[w] = 1 +p+p* + . . . +pn-\ ( n p OS itive and integral), 

 [-TI] = -p-i-p-*- . . . , p -n ( w integral). 



In general for all values of x, we take [x] = (p* - l)/(p _ 1). The 

 object of this note is to carry on this extension, to determine the 

 generalised forms of the gamma function, and to investigate some of 

 its properties. 



1. Consider the expression 



We can form a function [n] \ in general, which is 



The infinite product is conversant if <n \ i \\T\. 



../*" 



But L M_. n _ T P* - 1 



