312 BELL SYSTEM TECHNICAL JOURNAL 



n ,• \ 



where the coefficients «{ for the three cases are as follows; 



An 



Dr. 



2»-i - 1 



(2^-1 - l)(2»'-2 - 1) 



i[6^ + 4-5^ - 15-4^ + 35-2' - 36] 



Note that in the first case (Z)„)q:i = 0, in the second (Tn)ai = a^ — 0, 

 in the third (f/n)«i = 0:2 = as = 0. Thus a given table of values of 

 e„ up to w = ^ determines Z)„ up to ^ + 2, Tn up to ^ + 3, and Un 

 up to ^ + 4. 



Somewhat simpler relations may be derived as follows. Repeated 

 differentiation of the generating identity of the e„ with respect to /, 

 and passage from the generating relations to coefficient relations leads 

 to the following: 



6n+l = (e + 1)", 



€n+2 = (6+ 1)"+ (e + 2)", 



e„+3 = (6 + 1)" + 3(e + 2)" + (€ + 3)", 



or, in general: 



m 

 1=1 



This formula may be inverted by the reciprocal relations for the 

 Stirling numbers of the first and second kinds ^ which run as follows: 

 If 



m 



then 



m 

 x=l 



where Sx, m is the Stirling number of the first kind defined by the 

 recurrence relation 



Sx, m+\ ^^ •^i— 1, TO Woj, rn 



and the boundary conditions Sm.m = '^, s^.m = x > m, So. m = 0, 

 m > 0. 



^ Nielsen; "Handbuch der Gamma Funktion," Leipzig, 1906, p. 69. 



