VARIABLES IN CERTAIN LINEAR DIFFERENTIAL OPERATORS. 
35 
lu like manner m{0, 1 ; m, — m — r]^ whose symbolical form is 
I — m — r cl (m) ’b st (™) m + 1 
n T“C — Xn — X. 7] — 
' ^ m ' -/H + 1 ' 
. - X 
(m) 
m + r 
. (57) 
a form proceeding by positive integral powers of t), and therefore of transforms 
into 
- m — 1 l — m — r ~^(rn) — r cl'Yj , i -i \ ^ ’’ . 
mi T, +™X.J? +(™+1) V+i’' ,75 + 
^ ' m + 1 ' 
+ + r - 1) X^™^ 7)"' + (m + ^0 % . (58) 
' ' ' ' 'nx-VT-\> ^ ' rti + r ^ ' 
of which the terms in zero and negative powers of ^ must, as before, disappear, leaving 
as the result of transformation 
m (0, 1 ; m, —ra—T\-,— —?h( 1 —m —r) ]'l, 0; 1—m —r, m—1 
Sy 
+ mx;f {0, 1; 1-r, _ i] + («,+1)X;;;;JO, 1; 2-r, -1] + • • • 
(m) 
+ (™ + >'-l)X“,_,{0,l;0,-l}+(m + r)ry{0,l;l,-l} (59) 
(in) 
By addition of p, times (56) to p times (59) the transformation of the more general 
(p, f ; m, — m — 7’}^ is at once deduced. 
15. The forms taken by (56) and (59) for the case r ~ 1, i.e., 7n n ~ — 1, since 
— TO — 2 
'‘^y^ y^, 
m {1, 0; TO, — m - 1] = — [0, 1 ; - to, m — 1] + y"™ [0, 1; 0, — 1} 
_ — o 
- ^y^ y^ ~ 
and 
[0, 1 ; TO, — TO — = TO [1, 0; — TO, TO — 1}^ + 7/“™ {0, 1; 0, —1}^ 
1; 1, -1]^ . (61) 
One or two particular cases of these formulae deserve mention. The value zero 
of m makes (60) an identity. In (61) the substitution of the same value produces 
[0, 1 ; 0, L};;; — (0, 1 ; 0, l]y 7/^ {O, 1 ; 1, 1 
F 2 
X (to) V,. — 
z= X = IJ 
TO 1 *^1 
and 
may be written 
X \Ul) 11 
= mx 
in +1 1 
