56 



Mr Berry, Note on a property 



Applying the annihilator and equating coefficients of the 

 separate powers of x and y to zero we get the system of 

 equations : 



AG.+ C^O 



- mC + A0 X + 2C 2 = 



- (to - 1) G 1 + AC 2 + 30 3 = 



-(m-r)C r + A<7,. +1 + (r + 2) C r+2 = 



•(I). 



— G m _ 1 + A(7 m — 0^ 



Eliminating G 1} C 2 , ... G m we see that G satisfies the partial 

 differential equation 



4> m (A)C„ = • ( 2 X 



where <£ m (A) is the symbolic determinant 



A, 1, 0, 0, 0, 



-to, A, 2, 0, 0, 



0, -(m-1), A, 3, 0, 



0, -2, A, 

 0, 0, -1, 





 to 



A 



In this determinant the elements in the leading diagonal are 

 all A, those in the parallel line immediately above and on the 

 right are 1, 2, ... m, those in the parallel line immediately below 

 and on the left are — to, — (to — 1), ... —2,-1, and all the other 

 elements are zero. 



To reduce this determinant let us first add the 3rd column to 

 the 1st, the 4th to the 2nd, the 5th to the 3rd and so on, thus 

 obtaining a modified determinant which it is easy to write but 

 troublesome to print. In this new determinant let us subtract 

 the 1st and 2nd rows respectively from the 3rd and 4th ; the 3rd 

 and 4th thus modified respectively from the 5th and 6th, and so 

 on. The last two rows now become 



o, 



o, . 



.. 0, 



A, 



TO, 



o, 



o, . 



.. 0, 



— TO, 



A, 



so that A 2 + to 2 is a factor, and the remaining factor, i.e. the 

 minor obtained by suppressing the TOth and to 4- 1th rows and 

 columns, is 



<£m- 2 (A). 



