1196 



If A' = O we have 



2;/ H,, (0) L,._i (0) = C ill even) 

 • - 2/* Z„ (0) //,._, (0) = C 

 therefore in both cases 



2"+i w/ 



r. — 



thus finally 



2« (" — !)/ 

 Hn {x) />„_! (.7;) - /.„ {x) i/„_i (..) = ^••-' . . . (17) 



Now .1' having a large value, we may write aj)|)roximatelj 

 Hn {x) = (2a;V' Hn-\ {x) = (2.f)"-' 



and therefore 



n! é^^ 



6. Summation of some series containing the fnnotions //„(,r). 

 Let 



^' (_1U- -^^ ^ — -1^1 -^ p and ^ ( — 1)^- ^ ^ ^ — ^^ ^ ^ = O 

 (2^-)/ u (2A:4-1)/ 



and write H^k and //2^•-|-l fis definite integrals by means of (9), 

 then we have 



00 ,.2 00 ,,2 



1 ., ., f-- r-^ « 



ƒ= — g^"+=^" I g -^cos «i'G?u I g ^cos xudu ^ ( — 1)^ 



^ J J 



(z/?;)2^" 



00 ,,2 00 „2 



Q = — e^^+«' I e 4 5^^ auc^r | e ^ sin xudu 2 ( — 1)^" 

 ^ J J 



^j = — e-^-- r^" I e •» sin avar i e ** sin xuau ^^ ' "* ^ '" 

 ^ J J 







where 



(2^-+l)/ 







^ ( — 1 1'^ = cos uv ^ ( — 1 1'^ ^=. sm uv. 



^ ^ {2k)! (2A--f-l)/ 



Now 



\e * cos xu cos uvdu ^ ^ \e "* [cos {x -f '") u -\- cos (x — v) u] du 





 and 



