1904.] A Generalisation of the Functions T(n) and x n . 



67 



The expression (1 +p) (1 +p 2 ) ... (1 +p n ) is written (2) n , this nota 

 tion being both natural and convenient in investigating the properties 

 of the generalised Bessel function* 



Similarly 



When x and n are both positive integers, 



The function we are seeking is clearly for all values of x and 'ft 

 (subject to limitations of convergence) 



which reduces if p = 1, to the function x 11 . 



5. Extension of LommeVs product. 



(m + n + 2r\ 



m++2r 



To illustrate the use of the function (2) n , we take 



r = QO 



JM(Z) = S(-i) r r T 



(m+r) 



in which m and % are not restricted to integral values, so that \n + r\ \ 

 denotes T p ([n + r+ 1]). The function |[ m ] may be derived from J[ m ] by 



inverting the base p, when J^ becomes p m *$[ m ] (-) We proceed to 

 show that 



{2m + 2?i + 2r} ! {2m + 2r} ! 

 ! denoting (2).r jl ([s + l]) J or 



{2r} \ 



(7), 



* 'Boy. Soc. Edin. Trans.,' vol. 41, part 1, Nos. (1), (6). ' Lond. Math. Soc. 

 Proc.,' Series 2, vols. 1 and 2. 



