Al. a. TCIIOUPROFF 
147 
we find : 
^ /.->7 ; \ I ml. I \ } /si;. 
h, -1/1,-1 hk-lf-H-1 ^ 
^ j% ■ • ■ 0 j?o (2h,-l,)l{2L-lJl\..(2h,- k) I (2f- 2H-j) ! 
('•i) ()•..) ('•,,) ' 
r/S-^^h-hU [r^ - h, + -//,+ ... + - /, J t-'-^-^ 
/--ff-i 
s 
I ,ro {2K - I.) I {2JU - 4) ! . . . {^h a 9^^9^'- Ok-. ' (2/- - O) ! 
- Z'.]' 
where /S' denotes 
S denotes 
a 
and 
2f-2H-i -if-ui-j-fn tf-m-.j-a,'ih--.-Ok- 
t X ... X 
[-(■if-tH-j-a)] 
....(27), 
5' = <7i + ;72 + ..-+S'fc-i. 
If we note that V|^; ^ rit-i-'^'-^i'l [)\ - h^J-'J^^ = 0, when i\ > 2lt^ + (Ji, etc., we 
see without difficulty that, when 2/< R, the sum we are discussing is equal to 
zero. If 2f = R, then the only non-vanishing term in the sum is the one corre- 
sponding to l, = l^=... = lf.=j = 0 and r,- 2hi, g.,= r.,- 2li.„ (7i._,= /Vi- 2/a--i, 
2f— 2H — g = 7^ic— 21iic, and the sum reduces to 
-//,() 
.(28). 
If 2/=i^4-l, there are three types of non-vanishing terms: (1) terms for 
which 1^ = 1^= ... = lk=j = 0, and for which, of the quantities gi, g.,, ... g^+i, 
2/ - 2H — g, one, e.g. gi, is equal to r^ — 2h; + 1, and each of the rest is equal to 
r — 2}t\ (2) terms, for which Zj = ^ = ... = 4 = 0) i = I '^"^1 '^^^ the ({uantities 
gi = — 2hi; (3) terms, for which j = 0, one of the quantities I, e.g. ^^=1, and 
the other quantities I vanish, gi = ri — 2h! + l, and the remaining quantities 
g are each equal to /• — 2Ii. 
Noting that V.Y!-^''] [X - hj-^i 
becomes r ! h {r — 2k+ 1 ) when k = r — 2h -\- 1 and A' = /■, we can without difficulty 
reduce the sum we are considering, for the case 2f= 7^ 4- 1 to : 
H .C''' G^.'' ... C,.^'M/,j,o ^^2,0 ... Ah^,o ^R±i_ii y 
-II. (I 
,-(29). 
II, 0 
, Qih, ^2h^ f^2hk-i rrlhk-l A a A A U 
10—2 
