Al. a. Tchouproff 
153 
When r is even, 
T's,.^ = 1 . 3 . 5 ... (2r- 1)^/, 
- (Z7 ) . ^ ^r-h-i)\-jjj,l...jfl[h, !] [A. :]-" ... [A, 
-1.3.5...(2?--l)2/-f-<*+'']2''+>,'-''-'S 
(17), 
where the summation for i extends to all integ'er values from 1 to the smaller of 
the numbers 2h and r — h, and the last summation to all positive integei' values 
^^juji, ••■jf> satisfying the condition : 
ji +i2 + ••• +:if = i 
and to all positive integer values of h^, lu, ... hj, satisfying the conditions: 
3 < //o< ... < hf, 
h,j, + h,j. + ...+ hfjf= 2h + ii. 
When is odd, 
V:.r = 1.3.5...(2r + l)-yt../-V3 
r—h—i 
= 1.3.5... (2r + 1) S 2''+' /ti,'-''"' S — 
where the summation for i extends to all integer values from 1 to the smaller of 
the numbers 2/; + 1 and r-h, and the last summation extends to all jiositive 
integer values of j,, js, ...jf, satisfying the condition : 
:h +h + ■■■ +jf=i 
and to all integer values of h.,, ... hf, satisfying the conditions: 
3 <//.,< ... < }if, 
Jhj, + h,j, + ... + hfjf = 2h + 2 / + 1. 
The coefficients T may also be calculated by means of recurrence formulae. 
Putting 7?), = 0 in (8) and replacing the ?u's by /x's, we find : 
If ''"1 
and hence (or directly from (9)) 
r-1 
.(19). 
h = i-l 
