( 354 ) 





x]ints^2)b 



r\A2i 



With the aid of these values we iiud 



A, B, + C,D,E, = 0, 



A 



" (n—s)! , "= if>(14-s)62s 



^ "^ "^ . = 0^ ^ ^/ ^ ^ s = o s!(n^s+l)!' 



(-l)n 



(n+1)/ 



hence 



&2«+2 



C, D, E, + C„ D, E, + C„ i>„ E, = 

 2 /r/ h f(n 4- 2, ^n — 2 ^-^— — 



" (n — 6')/ 



S.' 



+ (-1)" />2.+2 1 ^'' [2^5'^' -^(1 4- .^) - iK'^ +« + 2)]. . .(1) 



Let ns now determine Un in another wa^' to give this result 

 another form. To this end we dilFerentiate the equation 



00 t2 



Un= \ e ■'' .r" dx , 



we then get 



1 .dir,, r"-'-'- 



=r I e 



26 dh J 



^ .f«-i dx. 



(a) 



1 drUn 1 dU^^ 



26 dh"" "' 26^ "rfï" 





b^ 



X „,,,_9 



.t'«-2 ^,,;. 



Out of the identity 



b"' 



-X 



b"- 



b^ 



.^.« d{e '') — —€ ^\r" dx + 6^ e ^" .^•"-2 c?.v , 



we moreover deduce by integrating between the limits and 00 



cc 6- 00 6^ CO b'^ 



— nie "' "" x'^- Ulx = — i e "' \v'^ dx- ^- V i e ''x'^-^dx, 



00 



hence we find for Un the differential equation 

 d''U„ 'In^ldUn 



dh' 



h dh 



4 Un = 



(2) 



