Mr Rogers, On a Gaussian Series of Six Elements 257 



On a Gaussian Series of Si^ Elements. By L. J. Rogers. 

 (Communicated by Prof. G. H. Hardy.) 



[Read 26 January 1920.] 



§ 1. The symbol H (a, ^, \, jx, 7) will be used for the infinite 

 series 



^ + 1.7^-^0 + ^- + ^- + ^1>' 



where ^' = (« + ^0(/3 + rt)(\ + n)(/x + rt) 



Vn (1 + n) (7 + n) {k + u- 6) (K + n + S)' 



and 2K = a + ^+\ + fi+l-y. 



Since "^ = 1 + - (a + /8 + X + /i - 1 - 7 - 2«) + f-,) 



it follows that the series (1) is always convergent, provided that 

 the elements a, /3, . . . are all finite, and y, k ± 6 are not negative 

 integers. 



When ii =7 the series reduces to 1 + z — ' , ' ^-. + ..., 



1 . {k^ — a^) 



where 2k = a + ^ + \ + 1, 



which series will be written 



H{a,^.\) (2). 



It is not necessary to introduce 6 into the functional notation. 

 For the sake of conciseness it will be convenient to write 



a.n, ^n, '" fCn for « + H, ^ + n, ... K + U. 



§ 2. Corresponding to the well-known formula in hypergeometric 



series, it is easily seen that 



H(a, /3,\, tx,,y,)-H(a, ^,\, fi,y) 



a0X(y-/J,) rr/ o ^ X /1\ 



= r,iy+l)i>c^-e-^) ^^«- ^- ^- ^- y-^^ (!>• 



Moreover 



{(« - fiy- - e--] H («„ A, ^1, y^> 7i) - («^ - 0') H (a, /3, \ fx, y) 



^ (_ 7-«)(7-W7-^)/^ ^(,^^ ^^^ ^^^ ) (2). 



7(7+1) 



