1904.] A Generalisation of the Functions T (n) and x n . '69 



corresponds to 



x / * V- r(r- 



[2]lr([f])J 



It is easily established that 



-IW... (13). 



... (14). 



This reduces to 



J M J_, l+ i + J_ n J M _i = sin TITT. 



TTiC 



As remarked in article 



is a pseudo-periodic 



function of n analogous to sin mr. The period is given by 



6. TAe Function E p ([x] [>]). Let F(|>- l]a^'") denote the conver- 

 gent infinite series 



i [ % -!] 



I -19 L 



p [i] 



If p = 1, this series reduces to (1 - x) n ~ l * 

 Consider 



r 1 



F ([n - 



Jo 



(15). 



(16). 



Integrating the series term by term, we obtain, after obvious 

 reductions 



m] 



1] [w + 1] [1] [2] [m + 1] [m + 2] 



...}. 



The series within the large brackets is a particular case of Heine's 

 series 



. . . = n 



whence 



} 



1 T p ([m])T p ([n]) _ B , f , f ,, 

 - 



