+- 



to Hypergeometric Series. 357 



The form of the identity thus arrived at will be best perceived 

 by considering a particular case. Thus, comparing the coeffi- 

 cients of x^, wo have 

 a.u + l.u + 2 . /8 . ^ + l./3 + 2 



1.3 . 3 . ry+i.y + |.,y + | 



a.a + 1 . /3 . /8+I7— «.7 — /3 



+ 



2 



« ■ /3 



7+^y+l 1 • 



y — a. 7 — a.-{-l .<y- 



7 + i 



-;8.7-/3+l 



y+^ • r+; 



+ 



7 — a.y — a + 1.7 — a + 2.y — /3.7— /S + 1 .7-/3 + 2 



1.2 . 3 . 7 + 1 . 7 + 1- 

 2a. 2a + 1.2a + 2. 2/3. 2/3 + 1. 2;8 + 2 



7 + t 



+ ■ 



1 . 2 



2a.2a + l 



3 . 27 

 .2/3.2/3 + 1 



+ 



1.2. 27.27 + 1 

 2a. 2/3 

 1.27 



+ 



27 + 1.27 + 2 

 7— ct— j5 



1 

 7 — a — /S.7 — a— /3+1 



172 

 y — a— /3. y — a — /3 + 1.7 — «-)i3+2 

 1.2.3 



7 . 7 + 1 .7 + 2 

 y+j. 7 + 1-7 + 1' 



It may b^ observed that the function on the right-hand side 

 is, as regards u, a rational and integral function of the degree 

 3, and as such may be expanded in the form 

 A« . « + 1 . a + 2 

 + Ba . a+ 1 . 7 — a 

 + C«. y — a. 7 — « + l 

 + D7 — a. 7 — a + 1 .y — « + 2, 

 and that the last coefficient D can be obtained at once by writing 

 a = ; this in fact gives 



y— /3.y— /3+I.7— ;8 + 2 7 . 7 + 1. y + 2 



2 



7 + ^-7 + ^7 + v} 



7— yS+1 .7-/34-2 



07.7 + 1.7 + 2 = 

 and thence 



1.2.3.7+^.7+1.74 I' 

 which agrees with the left-hand side of the equation : and the 

 value of the first coefficient A may be obtained in like manner 

 with a little more difficulty; but 1 have not succeeded in obtain- 

 ing a direct proof of the equation. Tlie form of the equation 

 shows that the left-hand side should vanish for y=— 2, which 

 may be at once verified. 

 Grassmere, August 25, 1858. 



