﻿Higher Derivative of a Function. 825 



Let us now assume 



Now if y==w K+1 , then from (1) follows 

 ___ = ^ x j^ Al + ^ ][ J w «- 1 A 2 +... + (^ J j^A K + A K+1 



But . ^ , lV/ /X\ v ^ A " y 



A,-s(-D"e) 



for values of X from 1 to a; inclusive. 

 There lore 



d^ 1 a=i 7 =o V ; \ X /\7/ ^" 



Denoting the double summation by S and letting \ = tc + 1 — a, 

 and in turn u-\-<y--=/3, we have 



= i 1 ,i ( - l) i - A *-« r~^ ? (3) 



since for j3=k + 1, d n u*^~P 



Now 2 2 M a ,^= 2 2M a , fl . 



a=l jS = a /3 — 1 a=l 



This can be shown in the following way; 



2 £ M fl>j8 = M 1 ,i + Mi f3 + Mi,8+ + M lf . 



+ M 2 ,3 + M 2 ,3+ +M 3 ,k 



+ M 3 , 3 +....+M 3 , K . 



+ 



+ *!«,«. 



