308 BELL SYSTEM TECHNICAL JOURNAL 



interval, that is, of the operator defined by 



A/(2) =fiz+\) -}{z). 

 For convenience of reference, a short table of the numbers follows: 



1 



1 



2 



3 



4 



5 



1 



3 1 



7 6 1 



15 25 10 



The table may be verified and extended readily by the recurrence 

 relation. 



With this table (extended to w = 10) and corresponding tables of 

 dx, tx and Ux running to x = 10, the values given in Table I may be 

 calculated by equations (2) and in this sense this paper is completed 

 at this point. The sections below contain an algebraic and arith- 

 metical examination of the numbers. 



Generating Identities 

 The generating identity for the function 



n 

 «=0 



exp [a(g' - 1)] = E -, L a-Sx, n. 



„=0 ^ • 1=0 



This leads, by differentiating 5 times with respect to a and setting a 

 equal to unity, to the generating identity: 



°° fn " 



(e' - 1)^ exp (e' - 1) = E -, L {x).Sx, „. 



n=0 '* • «=0 



This relation may be rendered more summarily by introducing the 

 notation of the symbolic or umbral calculus « of Blissard ; the expression 

 on the right is written exp tb where 6 is an umbral symbol standing 

 for the sequence (6o, 5i, • • -Sn- ■ •) in this case infinite, through the 

 relation 5" = 5„ and: 



*E. T. Bell, "Exponential Polynomials," Annals of Math. 35, 2 (April, 1934) p. 

 265- or J. Riordan, "Moment Recurrence Relations . . . ," Annals of Math. Statistics 

 8, 2', pp. 103-111 (June, 1937), eq. 3.4. 



6 Cf. Bell, I.e. p. 260 where further references are given. 



