342 BELL SYSTEM TECHNICAL JOURNAL 



We therefore start with the operational e(|uation 



/; - 1 \ " 





^P + 



Now the solution of this operational equation is very easy and can 

 be expressed in a number of ways. We require it expressed in the 

 form most easily computed. The following appears best adapted 

 for our purposes. Consider the auxiliary operational equation :— 



+ ...+(- 1)"2")^„. 



The explicit solution is gotten by replacing 1/p" by t"/nl and p" by 

 d"/dt", whence 



_ n2t n{n-l) {2tY „ (2/)» 



But writing 



P J H{p) 



fp-l\" 1 



(P-2Y _ 1 



it follows that 



'"=(^+1) 



.p+lJ H{p+1) 



^P+1 P 1 



'p 'p+lH{p+l) 



V ^ p) p+iHip^iy 



Referring now to theorem VH, we see that 



<^«(0 = (l+ / dty„(t) ' e-'. 



Since we haye already solyed for <t„(/), this determines 4>n(t) and 

 hence Vn(t). The functions z'o,^i,t'2 • • . are therefore determined. 



