784 On Products of Legendre Functions. 



all be expressed by the single formula 



r /7-"-ff\ r ( y + « + 8 + 2 \ 



Ppp", v 2 ; I 2 ; 



which is valid for all integer values of a, /3, y whose sum is 

 odd. This result is symmetrical in ex. and /3, as of course it 

 should be. 



Since 7 + a + /3 is odd, the arguments of the Y functions 

 in the numerator,, capable of: becoming negative, are 



-Hy-«-/3, 7 + *-/3 + l, 7 + /3-a-fl} 

 or 



i( 7 + a+ /3) _(* + £), i( 7 + a+ £ + i)_& 



i(y + « + £+l)-«. 



The first cannot be an integer, 7 + a-|-/3 being odd. The 

 others can, but if they are integers of negative sign, the 

 function 



in the denominator is of the same type, so that the limiting 

 value is always finite. 



There are interesting cases in which the integral is zero 

 when « + /8 f 7 is odd. This occurs, for example, if 



7~«— ft + 1 

 2 



is a negative integer (or zero); or if a + /3 — y is an odd 

 positive integer. This in fact includes all cases, when 

 a + /3 + y is odd. Thus the integral 



f'p^Q^-O 



when (1) u + j3±y is even and* (2) « + /3 + 7 is odd, and 

 a + ft — 7 is positive. It is always zero if u + /3 is greater 

 than y. 



