certain Factorial Expression. 421 



functions H and 11 1 their values, the formula is 



^+3.(r-A) + 3.4(r-^)(r-f)"^3.4.5(r-i)(r-f)(r-|) 



which is a formula obviously belonging to the theory of the hy- 

 pergeometric series 



F(., A ,, .) = 1 +«4.H- 5if±l^^.H fa., 



but the formula applicable to the case in hand has probably not 

 been given. It may be proved as follows, premising that I dis- 

 regard all difficulties arising from infinite values of the functions 

 in the definite integrals, convergency, &c. We have 



n(a-l)n(-«-7-l) A I cc.^ «.« + l./3.^-l o N 



- n(-y-i) 1^ + 17^'^+ 1.3.7.7-1 ^"') 



Now we have 



11a? 



(/3+i)(/3 + 2) 02-^(^ + 1)^; 

 and hence multiplying by dx and integrating from x = 0, and 

 again multiplying by dx and integrating from ^=0 to x=-\y 

 we find 



{'de . d''-\\-e)-''-''^^'''-\de . e^'-'ii-ey"-'-' 



Jo Jo 



+ {^+2)^'dd.e'-\i-e)-"-'-' 



(/3+i)(°^+2)n(a-i)n(-a-7-i) ,^ 

 - n(-7-i) ^""^ 



if for shortness 



3.7 3.4.7.7 — 1 

 Substituting for the definite integrals their values, 



n(«-3)n(-«-7+/3+i) n(«-3)n(-«-y-i) 



n(-y + ^-l) n(-7-3) 



n(«-2)n(-a-y-l) 

 + {P + '^) n(-y-2) 



( y8+i)(^+2)n(a-i)n(-«-7-i) ,g 

 = n(-7-i) ^''' 



