1194 

 tlierefore (his partioijlar inlcf^i-al is Hn ('') juid we have 



Hn {x) z=. ■ — — I e •* «" 005 j xu 1 du ... (9) 



o 



Choosing again 



( — 1)" . njt (—1)" n:!t 

 An =: sin — B , = COS — 



1/^2 y/^ 2 



Ihc second i)artic'ular integral may l)e writlen 



00 „2 



1 r- ^ . / nn\ 



L„ {x\ = ■ e^ \e •* ?<" sin ( xu \du . 



(10) 



ii. 'i'his second integral satislies also the relations (4) and (5). 

 For. ditferentiating, we iiave 



00 ,jï 



/;„ (,/•) = 2.1- L„ (.r) -i e^- I e ■» »"+• co« ( .tm — — J 



dii 



= 2xL„ {x) : e^-' Ce~^u"+^ si,i f xu — ^-^ilL.^i' J du 







or 



L\, (x) ^ 2xL„ (x) — L,+i (x) (11) 



Differentiating again, and remarking that />„ (.i*) satisfies the differ- 

 ential equation (3), we find 



{2n + 2) L„ - 2x L„^x + i.+o = 



or, changing n in n — 2 



Ln — 2x Ln-x + 2 (/l-l) Z„_2 = (12) 



which is in accordance with (5). 

 If now we substitute the value 



Ln-\-\ = 2x L„ — 2n Ln — \ 



from (12) in (11), we get 



Ln = 2n L„_i , . . (13) 



which is in accordance with (4). 



4. The function Ln {.v) may be expanded in series of ascending 

 powers of o). ^ 



If n is even, we have 



