﻿226 On an Equation in Differences of the Second Order. 



Now I have been able to establish the following theorem of 

 derivation as a particular case of a more general one of which 

 the clue is in my hands : — 



[1, X, p, v, , . .] = [X, p,v;.. .] + 2(X 2 — X) [1, X-2, p, v, . . .] 



+ 22X//,[1.X-1, / 16-1,V,...]. 



Suppose now that X, /uu,v, . . . all become unity, and that we call 



[1, 1, 1, ... to n terms] =£l n , 

 then the theorem above stated gives the relation 



o n =a w _ 1 +(^-i)(^~2)ri„_ 2 . 



But by virtue of the form of the equations for finding foe, I 

 know independently that fl n is the product of n terms of the pro- 

 gression 



1,1,2,2,3,3,4,... 



Hence we have one particular solution of the above equation in 

 differences. To find the second, if we makeXli and X2 2 , 1 and 2 

 respectively instead of 1, 1, it will be found that the nth. term 

 becomes the product of n terms of the analogous progression 

 1,2,2,4,4,6, 6.... Thus, then, we are in possession of the com- 

 plete integral of the equation 



u x + 1 — u x ~T \% —~SC) U A >—D 



viz. 



u 2x =C . I 2 . 3 2 . 5 2 . . . (2c£-l) 2 + K2 2 . 4 2 . . .(2a— 2) 2 2<z>, 



w 2x+1 = C.l 2 .3 2 .5 ...(2a-l) 2 (2^+l)+K.2 2 .4 2 ...(2^) 2 . 



Writing u x — \ .2.3... (x — \)v xy the above equation takes the 

 remarkably simple form 



X 



of any two of the quantities i, j, Jc, I happens to become equal to the sum 

 of the other two, the order sinks and is either 8 or 7 ; I am not quite cer- 

 tain which at present, although it is more probably the former. 



* Whether taken under this or the original form, the equation will be 

 found to lie outside the cases of integrable linear difference of equations of 

 the second order with linear or quadratic coefficients given by the late Mr. 

 Boole in his valuable treatise on finite differences. In the second form the 

 solution ought by Laplace's method to be represen table by a definite inte- 

 gral. Expressed under the more ordinary form the integral is as follows : 



_ c 3.5.7...(2g-l) , 2.4.6... (2x) 

 2x 2 . 4 . 6 . . . (2a? -2) " 1 . 3 . 5 . . . (2x- 1)' 



% '- 1 " b 2.4.6...(2.~2) + 1.3.5...(2*-3)' 



