BIXAKY BLOCK CODIXG 521 



(w, s integers, ^ s ^ n) it follows that 



f. fs\{n - s\ ^.+i_2, _ 1 f {x + yYil + xyy-' . 

 k \q)\j -q) ~ 2^- h W' ^ 



where contour C is, say, a small circle around the origin, taken posi- 

 tively. Thus 



_ 1 f a+xyr ( x + y \ (8) 



= LjGix) 



where the operator Lj is thus defined. Change of integration variable 

 gives 



T .rM - (1 - •^')""'' f G{z) dz 



(9) 



2Tri JcA^ - xzy-''+^{z - xy+^ 

 (1 - xY''' d' G(z) 



^0 \j - vl 



dz' (1 — xzy-'+^ 



(1 - xYd^Gix) 



pi dxP 



(with Cx a small circle enclosing x but not x~^, x" 7^ 1). Thus Lj may be 

 regarded as a Hnear differential operator of order y, (Lo = 1). 

 Using this result, (5) may be given the form 



(1 + xT = [Lo + Li + • • • + UG{x) 



= 1 [ ql^l (1 + xyTGi^^A dy (10) 

 2x1 Jc 1 — y \1 + xy/ 



= MG{x) 



this last expression as a definition of operator M. Written as a differential 

 equation, (10) is 



^ (1 - .V -g /n - A ^, d^^ 



^ p! ^o\ r / dxP 



It is straightforward that the only singularities of this equation are 

 regular singularities^ at .r = ±1, =c. 



III. THE DIFFERENTIAL EQUATION 



In this section we discuss (11) without reference to the fact that G{x) 

 is supposed to be a generating function. That is to say, with n and e 



