40 BELL SYSTEM TECHNICAL JOURNAL 



gives us, as one computing formula, 



WiXu X,) = ^="'\y|^. , (2) 



n + 1 



which is fairly manageable so long as the range Xi to X2 is not too 

 great and we have tables for the logarithms of the factorials involved. 

 When X2 — X\ is large compared with c, one may use the equivalent 

 formula 



'^ /Xi\/7V+1-Xi\ _ /Xo+l\/ N-X, 



w{x,, X,) = '-^^ — ^^ 7iv+r^ ' ^ 



This transformation may, as Molina has shown, be effected as 

 follows : ^ 



A'=Xi \ ^ / \ ^ ^ ' X=Xi \ <^ / \ ^ ^ / X=Xi+l \ ^ / \ " ' / 



Now 



:i::o(r-f)=xri)(S(-^")r"-r' 



^ <^Y^2+l\p'=^+' /N-X\/X-X.-\ 



Likewise 



^ ^*/X2+l\/iV-X2 



^■■r(^)(rf)=s(t)(V+\-t 



x=A-i \ ^ / \ f^ ^ I r=o\'/\'^i'- '^ 

 If in (2a) we let X\ = c and Xi = X, we obtain 



g/^" + l\/iV+ 1 - X + 1\ 



« + 1 

 from which we may compute 'W{X\, X2) from the equation 



T^(X,, X2) = W{c, X2) - W{c, X, - 1). 



^ Sec also Netlo, "Lelirbiul) dcr Konihinatorik," ]). 12, \'a\. 6; p. 15, 1".(|. 11. 



