144 Expectatioii of Moments of Frequencij Distributions 
When /<. > 0 und 2/; — // < k wc have : 
= 0 
v'':+" a. 
(/. ) ^Ic. k 
^' (/,) "yl-, k-n~ . , , II, i ^k-tii + h 
1=0 {or -ill- k) 
(/,) k,k — ii II, H 
a)'^k,k-ii ii,in-k 
1.3.5 ...(2/1-1) 
.(10). 
V,,, Of, , =0 
(A) k,k-il J 
(8) Of the properties of the /3 cueffieients it is essential to note the foUowing 
/St. ; = fik^,, i + 1 ) /^.t-i, J- 1 [ (11), 
/3,,,_, = 1.2.3...(/.--l) 
•(12), 
where the coefficients Bj ; are independent of k and are determined by the relations 
Bj,, =1.3.5...(2j-l) j 
Bj, I = (2/ - t - 1 ) [i>'M,i-i + i^j-M] (13). 
i^,,,_, = 1.2.3...j ) 
Hence we have : 
5,„=1.3.5...(2y-l) 
5,, = 1.3. 5 ...i2j-i)if[.y-i] 
B,., = 1 . 3 . 5 . . . (2y - 3) (1 - 1 + [./ - l]f-^'i 
i?,,= 1.3.5...(2y-3){^[i-2]M + f [j-2][-J + ^J[i-2]i-: 
5, , = 1 . 3 . 5 . . . ( 2; - 5) . U - n-''' + - ^]'-*^ + tUi - -n-'' I 
+ i}[i-2]i-]} I 
iJ. , = 1 . 3 . 5 . . . ( 2y - 5) {,1^. [ ; - 3][-] + ^ [j - 3][-J + f 1 [j - 3][-J | 
From (12) we find, wheli h > 0, 
,.(14). 
r'A/-/' /D _ V 7? (7''-' 
(A) 
/^..i =i^,o = l-3-5...(2y-l) 
(k) 
fk + h 
ik) 
^k,i=^ 
.(15). 
