1200 



or if n -\- p is an uneven number; the integral vanishes also when 

 j) <^ n. Supposing therefore p -\- n even and n < p, we have 



00 00 



^xi> iT„ e-'^^dx = ( — 1)" I xp — {e-^') dx 

 J dx'< 



00 00 



r d rdn-^ \ 



= (—1)" .vp — e-^' dx 



^ 'J dx\dx^^-'^ J 



= (—1)"- 1 p I .r/'-i ^ e~^^dx 



dx»-^ 



or 



Hence 



iiiiii 



00 00 



I xP H,t e •'■(iï = ^ l^JT 



2 



1 -„-^^- 



'1)1 



2 



wliR'li aives 



(2c)y' = i^/, -i ^— iï^._o + ^/,_4 + ... 



11. Iji the second place ex|)andiiig 



the coefficients are given by 



1 1 C" 



00 



or, putting a = y -|- /?, by 



1 1 f* , 



^„ = — — :-- e-V- H„ {y + ^)dy. 



2" . n! VciJ 



— 00 



Now, expanding Hn (// -|- /?) bj Maclaikin's theorem we have 



H, {y^ ^) = H, iy) -f ,? H,: (y) + . . . ^ i?«^"^ (^) 



n.' 



