522 THE BELL SYSTEM TECHNK'AL JOURNAL, MARCH 1957 



fixed but arbitrary non-negative integers, we denote by 



G(x) = f: UsX (12) 



s=0 



any solution of (11) regular in the unit circle. 



It proves convenient to introduce certain functions /„, {(a;) defined by 



/„.,(.r) = (1 + x^d - xy-' 



A (13) 



s=0 



where the coefficients <Ps(n, ^) are given by 



...(«,«= t (-!)'(" 7 %i,) (14) 



Here, ^ is to be regarded as a free complex variable. By (1 + .r)^(l — x)"~^ 

 we mean exp (^ log (1 + x) + (n — ^) log (1 — x)), each logarithm 

 vanishing at x = 0. As a function of x this function is single valued in, 

 say, the rc-plane cut on (— «=, —1] and [1, '-^-), and the series (13) con- 

 verges to it in: I X- I < 1. ^ 

 Binomial coefficients are defined by 1 



/r\ _ r(f + 1) 



,sj s!r(r + 1 - s) 



f (r - 1) • • • (f - s + 1) 



s > 



s! 



when f is not an integer, and <Ps(n, ^) is seen to be a polynomial in ^ of 

 degree s: 



<Ps{n,^) =?jf'-f ... + (-l)^(^') (15) 



The recurrence relation 



^o(n, + fiin, ^) + • • • + fsin, ^ = <Ps{n - 1, ^) (16) 



obtained by expanding the various factors [ ] in the identity 



[(1 + .r)^(i - xy-'m - xy'] = [(1 + x)'(i - xy-'-^] 



is an important one. We note also for reference that 



<Po{n, = 1 

 <Ps{n, n - ^) = (-l)V«(w, ^) 



^.(«, ») = (;;) (17) 



^.{n- 1,») = l+(i)+ ••• +''" 



s 



