100 BELL SYSTEM TECHNICAL JOURNAL 



While the integrals in equations (1) and (2) are well known equiva- 

 lent forms for the complete and incomplete Beta functions, the equa- 

 tions themselves are not so familiar although one or the other will 

 be found in Laplace, Poisson, Boole (Differential Equations) and at 

 least two other places. 



Approximate Formula 



The approximate formula derived from equation (1) and submitted 

 herewith for consideration is 



(3) 





where Si is the ith approximation to the infinite series 



^R.T^-'{\ + (5 - \)T,-- + (5 - 1)(5 - 3)Tc' 



(4) S = 



s=l 



1 + Z^2.[l-3-5 ••• (25 - 1)]2-'' 



Ti = rV2, 



(5) r- = {n - 1) log -^ +{c-\) log^ + {n - c) log ^ , 



Jl L Ct ft' (J/ 



and a = up; T to be taken negative when a < (c — l)w/(w — 1). 

 The first, second and third approximations to the infinite series S are 



^ - J? ^ _ i^i + RiT _ i?i + RoT + i?3(l + T'-) 



^1 - Ai. ^2 - J _^ ^^^2 ' ^' - 1 + R,/2 



where 



Ri = 4[(« - c) - (t: - l)]/3-V2(w. - l)(w - c){c - 1), 



R^- = (l/6)[l/(;/. - + l/(c - 1) - 13/(« - 1)], 

 R,= - (4/15)i?i[i?o + 6/in - 1)]. 



It will be noted that i?2, \Ri\ and ji^s] are symmetric functions of 

 (n — c) and (c — 1). 



For the limiting case (Poisson's Exponential Binomial Limit) 

 where w = 'x , p = but np = o, we have 



r^ = 1 + (r - 1) log (c - I) fa + (a - r), 



i?i = 4/3V2(c - 1), 

 i?o = l/6(c - 1), 

 7?3 = _ (4/15)i?ii?2. 



