1202 



o 



= Hn-x (0) — .-1 Hn-x (1) + 2 (n-1) fe—'H„^oda 



o 

 and 



/„ = iy„_i (0) - .-1 /ƒ,_, (1) (« > 0) 



Now //„- 1 (0) vanishes for odd values of n, therefore 



J,t= - ^ly^OA;-! (I) (A->0) 



/o^.+i = - e-^Hu (1) + i?^?it (0) {k ^ 0) 



Tlie following relations hold between three successive values of/: 

 Z2i+, - 2/2^. 4- 2 (2/— 1) /o/,_, (0) = {k > 0). 



[,k - 2 /.,;i._i + 2 (2^-2) /o^._o (0) = (- 1)^- 2 . ^^^— ^' (/: > 1). 



For 



/o;t+i - 2/.2A: + 2 (2/:-l) 7,i._, r= //o;fc (0) + 2 (2A— 1) ^o;t_., (0) — 



- .-' [//o;t(l) - 21I,k-x (1) + 2 (2^-1) //ot_o(l)| 



where the second member vanishes according to (5). 



In the sanie way the second relation may be proved. 



From this it is evident that all values of I depend upon the 

 values of J\ and f^, and these may be obtained directly for 



I^= f e ""(4«^ — 2) d(tz= - 2e ' 







7, = I e~''''2((daz=z l — e~^ 



■■-f 







If x = or .v=l the expansion does not hold. For these values 

 however we may easily \erify that the second member reduces to 

 the value ^. 



Taking .i' = 0, the second member reduces to 



^. 1 f ' -.-^ , ^^a"H„{0)H„{a) 

 Lim — = \ e da ^ ; 



H^i V:t.J 2" . n ! 







or, according to (22), to 



Lim I --^^ =e ^—'^- da' 



u 

 Assuming; 



