A. Ritchie-Scott 
411 
The s + 1th term of the (a) line is (— lY — '7 , ^r-^- 
2* . s ' (ft — zs) ! 
s- 1th 
ih) „ i-lY 
2'-' (s - I) I {n - 2s) ! ■ 2 
W51). 
" ^ ^ ' 2'-^ ( ,s - 2 ) ! ( « - 2,s') ! ■ 2^ 2 ! / 
Hence the s + 1th term of the whole expansion is 
(-1) 
Hence 
s s . s — 1 s . * — 1 . s — 2 ,^ 
m rn ni 
+ 
?«! 2(?2 - 2)! 2-. 2 !(/? - 4) ! 
Putting p =1 
cc 00 — 
n\ ~^^~P'> 2{^ 2) ! 
2= . 2 ! (n - 4) ! 
T 
n! nl 2(n-2)\ 22.2!(n-4)! 
+ 
...(53). 
...(54) 
and 
= r„ + " • 7 1 + ^•n-l.^>^-2..-3 ^ 
which may be symbolically written 
.(56), 
if we apply suffixes to the T's on the right side instead of indices to an argument. 
In a similar manner it may be shown that 
_^ 71 .71 — 1 ^ W . ?l — 1 . /I — 2 . H — 3 . _w - 
= mn- 2 ® "-^ + 2~4 ® "-^ ^• 
which may be symbolically written 
w^^ = Tn{m(ic)] (58), 
and finally Win [T (x)] = T,, } a {x)] (59). 
From its mode of formation it is obvious that the functional equation of 
Tn {x) is 
Tn{x) = X.Tn-,{x)-{7l-l)T„_,{^) (60). 
If we make n = — 1 and expand we get 
T_, (*•) = i + 1 • 
tAy dU yAj 
.(61). 
But 
Hence 
e~-'^''dx = — e' 
+ ... 
e-i'^'dx 
E{x) 
.(02). 
