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BELL SYSTEM TECHNICAL JOURNAL 



In the first type of recurrence only successive values of the numbers 

 themselves appear. The derivation is illustrated for the Dn, the 

 simplest case. Differentiating the first of equations (6) leads to the 

 relation : 



D exp tD = \{e^' - e') exp {e' - 1), 

 or 



(e' - \)D exp tD = e'{e' + \)W - ^f exp (e' - 1) 

 = (e2' + e') exp tD. 



Equating coefficients of /"/n! in this relation gives the umbral re- 

 currence : 



D{D + 1)" - Z?„+i = {D + 2)« + (^ + 1)", 



which in ordinary form is: 



{n - 2)Dn = L 



(';)(2'+»-(,;0]-"- 



The process is common to the three classes of numbers and pro- 

 duces similar results which may be put in general form as follows: 



a„An = E 



n\ , n 



A n—it 



(7) 



where A „, a„, 6, and c, are defined for the three cases by the following 

 table: 



Somewhat more convenient recurrences may be obtained by allowing 

 the presence of numbers other than those for which the recurrence is 

 sought. For this purpose it is expedient to introduce the exponential 

 numbers €„ of E. T. Bell. 



These are defined by the generating identity: 



exp /e = exp (e' — 1) 



or by the equivalent formula: 



(/" 

 e„ = lim^exp (e' - 1) = Z ^x, n, 



