268 ON CERTAIN FORMULAE FOR DIFFERENTIATION WITH APPLICATIONS [41 
from which the law is easily seen to be of the form 
(¿)" Pm = s > K ’.>O - U^ +1+e .^ +l+e 
(where the extreme values of 6 are 0 and (r — 1) respectively) and K r d is determined by 
K r+ i Ml = (r — 1 — \6) li r g +1 + (i — 3v + 2 + 20) K r ,e • 
This equation is satisfied by 
v(r-) E -6)T(2r-\-e)T(i-r + e+i) 
r ' d ~ r^ r(0 + 1) r(2r — 1 — 20) r(t — r +1) ; 
for in the first place this gives 
r(r-f-g)T(2r-2-g)r(t-r + g + 2) 
t r,o+1 r ^ r ^ + 2 ) r (2r — 3 — 26) r (i — r +1) 
_T (r-%- 0) V (2r - 1 - 0)T (i - r + 0 + 2) 
T(£) T(0 + 2)Y(2r-2-20)Y (i-r + l) ’ 
and hence the second side of the equation reduces itself to 
T(r - \ - 0) Y(2r -l-0)Y(i-r+6 + l) 
Y(I) r(0 + 2) r(2r — 1 — 20) Y(i-r+l) 
[2(r-l-6) (i- r + 0+l) + (0 + l)(i-3r + 2-20)}, 
where the quantity within brackets reduces itself to (t - r) (2r -1-6), so that the above 
value reduces itself to K r+i e+1 , which verifies the equation in question. Also by com 
paring the first few terms, it is immediately seen that the above is the correct value 
of K r g, so that 
(~) r (d\ r TT _ 0 Y(r-\-0)Y(2r-l-6)Y(i-r + 0+l) 
i \dx) °* 1 e Y($)Y(0 + l)Y(2r-l-0)Y(i-r + l) 
(\ - nY-'-o U^ +1+e ^ r+e+1 ...(1), 
0 extending as before from 0 to (r — 1). In particular if i be integer and r = i +1, 
(—) i+1 / d y +i 
[\/{x -I- X) — sj(x + fl)} 21 ■■ 
r(» + j) 
r(i) 
(A - /a) 2 
{(x + X)(x + /A)} i+i 
•(2), 
(since the factor F (i — r + 0 + 1) -r Y (i — r -f 1) vanishes except for 0 = 0 on account of 
r (i — r +1) = oo ). Thus also, if r be greater than (i + 1), = i + 1 + s suppose, then 
1 
{(x + X) (x + /¿)} i+i 
= Y(i + 8+l-0)Y(2i + 2s + l- 0)Y(0-s) 
9 Y(i+±)Y (0+1) Y(2i +2s+1-20) Y'(-s) [ ^ u +».-*+« 
(3),