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

 Therefore 



65 



Since the infinite product (1) is also convergent, if p<l, the expres- 

 sion for the function in this case is 



.(2). 



__ 

 = oo \n +T] [n + 2] . . . [n + K] L 



In the limit when p = 1, these expressions (1) and (2) reduce to 

 Gauss's expression for Euler's gamma function. 



2. The ffieierstrassian Form for the Function We obtain without 

 difficulty from (1) that 



-) J 





Since p>l, 



also the series 1 +_+_-+ ... is absolutely convergent, so that we 



L 2 J L 3 J 

 finally write 



in which 



In the case when p<l 



n p 2 p 3 1 



P and Q are extended forms of Euler's constant y. 

 3. Multiplication Theorem. Since T p l([nx + n]) may be written 

 ? . . . i Tg([nx]), in which, for brevity, q 





denotes ? and 



= = we have 



Gi 



F 2 



