I9è 



equation, thus obtained, to the coefficient of 0' in the second menaber 

 of (4) we find : 



I 1 "1 1.3.5...(2A;-3) ) 



H^A^)=ir^\ 2-3",,/ ./,(..^)- ^ifn-xWY^^^ - . ^^ ^n-ik^-") (22) 



By means of this expression an integral ma}' be deduced. 



For if we multiply both members bj 



e-^V/',„ {x) dx 

 after replacing x" by x. and if we then integrate between and oo, 

 we find, using the following well-known formulae ^) : 



oo 



I e—^q^m {•^) (f n (•^') <^'''' = m =^ n 







30 



ƒ 



e-^ (f,n (a-) ^2. (l/.^•) ^.^ = ( -1)"+^ 2-^" . n! 



e-^qm"" {x) dx = 1 : 



1 1.3.5.. .(2»— 2m-3) 







or after some reduction 



<X1 



{e—^Up,n {x')Ho„{xyvd,c={- 1 )"+i2"+"'-i — !^ 1.3.5. . (2n — 2w— 3) (23) 

 J {n — m)! 







?n «\ « — 1 . 



In the same way we find 



e-^-r/„(.c-)iïo„(.t-).iY/.t- = (— 1)".22"-1 n! . . . (24) 







ƒ 







and 



a 



ƒ 



-^' ^„_i (a-^) ^2„ (^) '-cdx = (- 1)" - 1 . 22"-ii . y^/ . (25) 



If we write formula (22; in this form : 



i^2. {x) = {-\Y 22" . n! j <pn{x-) - i r/„_, {x^) - ^ ^ '^-^ qP. - 4^^) | 



I 2|/.Tjt=2 a;/ ) 



we know that ") 



9^„_;t(.^•')=,— yr-, h'-««" ^•-/.,(2.rp/«)6?«. 



1) Dr. Nijland's Dissprtalion page 11. 

 ~; These Proceedings XV. p. 1:247. 



