NUMBER OF IMPEDANCES OF AN n TERMINAL NETWORK 311 



which shows their close relation with the impedance numbers. They 

 have the recurrence relation : 



€n+l = (e + 1)" 



and 



60 = ei = 1. 



Now, returning to the first of equations (6) and again differentiating: 



D exp tD = \[2{e' - l)e' + (e' - l)^^'] exp (g' - 1), 

 = (e' — l)e' exp (e* — 1) + e' exp tD, 

 = 2 exp tD + (e' - 1) exp (g' - 1) + exp t{D + 1), 

 = 2 exp tD + exp /(e + 1) - exp te + exp t{D + 1), 



from which, passing to the coefficient relation, comes the umbral 

 recurrence : 



J9„+i = 2D,, + (Z) + 1)» + e„+i - en. 



Similar recurrences for the T and U numbers are derived in the 

 same way; writing Ae„ = Cn+i — Cn, the results may be summarized 

 as follows: 



Dr^+X = 2Dn+ (D + 1)" + Ae„, 



Tn+i =4r„+ (r+l)" + 3Z)„, (8) 



Un+i = 6Un + (t/ + 1)" + 46r„ - 4r„+i 



+ 30Z)„ - 6Dn+i + 6Aen. 



The expressions in parentheses, it will be remembered, are short- 

 hand binomial expansions; thus: 



p+i)»=e(:)zp.. 



Relations with the Exponential Integers 



The generating identities in equations 6.1 furnish immediate rela- 

 tions with the exponential integers, e^. Writing exp (g' — 1) as exp le, 

 as above, and passing from generating relations to coefficient relations, 

 these results are as follows: 



D. = |[(6 + 2)" - 2(e + 1)" + 6„], 



r„ = |[(e + 4)" - 6(6 + 2)« + 8(6 + 1)» - 36,.], (9) 



t/n = i[(e + 6)" + 4(6 + 5)" - 15(6 + 4)" 



+ 35(6 + 2)« - 36(6 + 1)« + ll6„]. 



Expanding internal parentheses by the binomial theorem, the general 

 result is as follows: 



