(10) 



108 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



Follo\Wng (2), this may also be written as 



00 



Mk = a - a) "Z cc^'mn.k , (9) 







wifh 



rrin.k = / wl-F^(w)] du, 

 Jo 



= k f u^~'Fn{u) du, k> 0. 

 Jo 



First, notice that 



mn.o = -f F'niu) du = FM = 1; 

 hence 



Mo = (1 - «) f: a" = 1, 







showing that F(u) is properly normalized. 



Next, by integrating both sides of (la) with respect to u from to « , 

 and using the second form of (10) (with k = 1) 



-(n + 1) = nmn-1,1 - (n + 1)(1 + a)mn,i + (n + l)amn+i,i (11) 



In the same way, after first multiplying (la) throughout by u^~^, it is 

 found that 



— k(n+ l)mn,k-i 



(12) 

 = nmn-i,k — (n-\- 1)(1 + a)mn,k + (n + l)amn+i,k 



Unfortunately, neither (11) nor any other instances of (12) have simple 

 solutions; nevertheless they may be used to determine Mk . 



Consider first the simplest case, Mi . If (11) is multiplied throughout 

 by a" and summed on n, the result may be written 



— Lio = aLn — (1 + a)Ln + Ln — Loi 

 = — Loi 

 where for convenience in writing and of later notation 

 Loi = S Q!"mn.i = (1 — oc)~^ Ml 

 Lii = ^{n -\- l)a"m„.i 



Lio = E (^ +i)«" = ^ Z «"'"' = (1 - «)" 



(13) 



