1192 



— 2.f + 2n //„ = O 



(3) 



dlJn 



dx 



- 271 Il„^x — o 



(4) 



//„ - 2x //„ _, + 2 (« - 1) //„_, = ... (5) 



/CC 



1I„, {x) Un (.v) *'-'"' 'lx = O m =1= n . . . (6) 



/CO 

 Iln' (.f) ^-^' t^^ == 2» . n / l/jr 



(7) 



The object of this paper is lo examine these polynoniia and tlie 

 series connected witii these, wliich also satisfy the difïerential equa- 

 tion (3). 



2. To integrate the differential equation (3) by means of definite 

 integrals, put 



til en we have 



d'z dz 



h 2.1' h 2 (n 4- 1) -- = . 



dx" dx 



To solve this, we assume 



u 



-I 



e--*' T dt 



where T is a function of t, and P and Q are constants. The result 

 of this substitution is 



Q. 



r dT 1 



dt— . 



2(^7'.-^-')J!+j -^' 



-2t~- + {e + 2n) T 

 at 



Now this equation will be satisfied, if we make 



7' =: P> e^ 

 and 



P=: Qz= ± ioo . 



Hence the general integral is 



J ^ , t- 



-xt-\ 



z =:c, i e ^ V' dt -\- c. 



f 



I 



too (* 



—xt-\ 



e 4 <" dt 



