and integrate between — oo and -f- ^^ ^hen we find by application 

 of tlie well-known integrals : 



+ 00 



I Hjn (a-) Hn (a*) e-^' d.v = md(=n 



— 00 



I Hm^ (x) e-^"" dx = 2'" . wi/ \/n'. 



00 



r°° (2h-2«)/(2;)/ ^ ^^^ 



jy,. (..■) //.»-.,(.«) e-'^d. = (-1)» |t;^5^— j7^ I/" • (7) 



00 



Prof. Kapteyn deduces the following representation by means of 

 an integral for Hermite's polynomia ^) : 



CO „2 



Hn {.X) = \e ■* M" COS I xu — — du, 



]/jtj V ^ J 







If we substitute this expression in (6): 



oa ,,2 



cpJx^)= I e 4 ^ (_ 1 \n—k l_i u^n-2k cos (xu— (n—k) n) du 







or, if we work out the cosine 



00 „2 



gx3 r - - « (2/5;)/ 

 g)„(^') = \ e '^ cos xudu ^ tt^tzz rr, "^^""^^ 



(8) 







Now is 



8 



(2y^^)/=: ie-yy'^f'dy 







consequently 



00 



^_A:^i^ u^(n-k)—:s;— le-l/y^l^dy=. 



(^/)^(n-/t)/ {k/y{n-k)/J ' 



Ü 

 



For Abel's polynomia we have : 



(9) 



y ^2^- 





1) I.e. p. 1194 (9). 



