BINARY BLOCK CODING 523 



j valid for all n, ^ and non-negative integers s. We see, by the way, that 

 I (pe(n — 1, n) is simply the Hamming expression (1). 

 The function /„,{(a;) has the property that 



at least if, say, given x, \ y \ is small enough. From this and (8) for the 

 operator Lj it is apparent that 



L:fn,i{x) = <Pj{n, ^)fn,i(x) (19) 



Similarl}^, using (19) and (16), or directly from (10) for the operator i)/, 



Mf„,^(x) = [Lo + Lx + • • • + Le]fn,i(x) 



(20) 



= (pe(n - 1, ^)fnAx) 



If ^/3 is one of the roots of the polynomial <pe{n — 1, ^) then (20) be- 

 comes 



M(l + x)^^l - xy~^^ = 



If we assume for the moment for simplicity that <pe{n — 1, ^) has e 

 distinct roots ^/s , 1 ^ /3 ^ e, then (11) has as complementary function 



i; ^(1 + .r)f''(l - xy-^' 



where the A^ are e arbitrary constants. 



Fortunately, the function (1 -f- x)" = fn.n(x) is also a member of the 

 family (13); hence 



Mil + xy = <pe(n - 1, n)(l + xy 



and the function 



(1 + xy 



<Pe(n — l,n) 



is a particular integral of (11). [We see from (17) that (pe{n — 1, 7?) does 

 not vanish in cases of interest.] Finally, when the roots of (pe{n — 1, 

 are distinct, the general solution of (11) must be of the form 



G(x) = [^ + f . -f i: ^,(1 + xy^d - xy-^^ (21) 



(Pe{n — 1, n) ^=1 



If (pe{7i — 1, ^) has multiple roots then the general solution will con- 

 tain additional terms 



(const.) (1 + xy^a - xy-^' Tiog ^^^T 



(22) 



