524 THE BELL SYSTEM TECHXICAL JOURNAL, MARCH 1957 



i.e., the m derivative of /„,{(.r) with respect to ^, with m any positive 

 integer less than the multiphcity of root ^^ . 



Before applying these results to c-codes in detail, let us derive a cer- 

 tain modification of (21). First, we see from (17) that if n is a positive 

 integer, then n is not one of the roots of <pe{n — 1, ^). If the roots ^3 

 of <pe{n — 1, ^) are distinct and ii A^ , 1 ^ i3 ^ e, are any e numbers 

 then a polj'nomial d{^) of formal degree e is uniqueh' determined by the 

 e + 1 conditions: 



e(h) = (b - n)<p/{n - 1, ^s)A^ , 1 ^ ^ ^ e,* 



(23) 

 d(n) = 1 



G(.r) = ± / ^\+ ■■■^;'^ - ^"^f d, (24) 



using, e.g., the Lagrange interpolation formula. It is obvious that G(x), 

 (21), may be expressed in terms of this polynomial as 



(^ - n)Mn - 1,0 



where T is any simple closed contour surrounding the roots: u, ^1 , ^2 , 

 • • • , ^e of the denominator of the integrand; (the numerator is an entire 

 function of ^ provided x' 9^ 1). 



Analysis a Httle more detailed shows that even if v^e(n — 1, ^) has 

 multiple roots the general solution of (11) can be represented in the 

 form (24), again with 0(^) any polynomial of formal degree e such that 

 0(n) = 1. The e constants of integration appear as the e + 1 parameters 

 of Qii) restricted by Q{ii) = l.f 



Expansion of the integrand in (24) according to (13) yields the form 



Vs 



1 { <Ps{n, ^diO J. , n 1 9 ro^^ 



7r~- / r^ s — r n^ "^ « = 0, 1, 2, • • • (2o) 



for the coefficients of G(x), (12). 

 If we denote bv 



LjGix) ^ Gj(x) = X vj,sx (26) 



s=0 



the result of apphdng the operator Lj to any solution (24) of (11), then 

 it is straightforward that 



r r..^ 1 f (1 + x)Hl - 2-)"-V,(n, m^) ., /..-^ 



27^^ Jr (| - 7i)<Pe{n - 1,^) 



* The prime denotes differentiation with respect to $. 



t If G{x) of (24) is to satisfj- (11) it is sufficient that d(^) he any function regular 

 within (and on) F and that 0{n) = 1, as may be easily verified. Since G{x) depends 

 on ei^) only by way of the values of d(^) at the zeros of the denominator in (24), 

 the condition that 0(?) be a polynomial of formal degree e serves mereh^ to deter- 

 mine 6(^) uniquely for a given solution G(x). 



