1208 

 Putting .V =: O in tliis relation, we obtain 



T„(0) = {n odd) 



If now we compare the two forms 



we may determine the relation which exists between C and k. For 

 putting X = 0, we have / = and 



C 



= /;L„_j_i(0) (n even^ 



C 1 



-— (?z odd) 



7'„_i(0) kLn{0) 

 thus 



Therefore if C=cp the continued fraction (24) represents — — 



„,,... Ai+i 

 and if (;=0 the value of this traction is —— . 



From (e) we may deduce a new form for L,{x). For introducing 

 C instead of k, we have 

 2Ce^' {HnTn—Hn+iT,,^i) - ]/^{C-J){Hn^iLn—H„L,^i) - 



- Vn e^\LnTn-U^\Tn-,) = 0. 



Now the relations 



Hn+i = 2wH„-2nHr,-i 

 L„_)_i =r 2xLn —2nLn—i 



lead easily to 



LnT„ — LnJ^\Tn-\ = 2" .u! L^. 



With these values, and (17) therefore, we find 



2C . 2" . /I / — (C— 7) 2"+i7i / — |/?r . 2" . « / jL^ =r 

 or 



a; 



2 2 r 



(26) 



