524 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



which becomes, in our present example — 



r iib 11 U i 



9 (t + u)= ^{ 1 + IX3 + 1^6 + 17^9 + eta j 

 , f u , tt 4 u 2 n 10 | 



, f W 2 W 5 M 8 ?i u 1 



+ ^ \ O + TZb + TZTs + X3I + etc - j 



4. If A, B, C be the values which (pt, x <pt, 2 cpt assume when t = o, the above 

 expression becomes 



^= A { 1 + TX3 + etc -} + B {i + i34 + etc } + c {o+r! i 5 +etc } 



and thus it seems that all recurring functions of the third order are compounds 

 of multiples of the three functions — 



t 3 t e t 9 



t t* f t 10 



i + TZA + T^77 + T^To +etc > and 



1.2 + 1....5 + 1....8 + 1 11+ ' 



which may be called the fundamental functions of the third order. 



Similarly, the fundamental functions of any other order, say the fourth, are— 



t* t s t 12 



1 + T^i + rz8 + r^T2+ etc " 

 t t b t 9 t™ 



I + TZ5 + T779 + 1 13 + c- ' 



O + 1776 + 1....10 + 1....14 + ' 



t* f ii 11 t 15 



1.2.3 +1 7 + l....ll + l 15 + e a 



5. The most important of all recurring functions is that which is equal to its 

 own first derivative ; if (f>t represent such a function, we must have 



<pt = y <pt = 2 <pt = 3 <pt = etc., and 



r u u 2 ii ~\ 



<p(t + u) = <pt 1 1 + j + ^2 + ;f2~3 + etC ' I ' whence 



9 u = A ( 1 + - 1 + r2 + oT3 + etc -] ; 



* C-, t t 2 t 3 i 



* = A { 1 + , + + lX3 +etc -j ; 



and if A be unit, that is, if cpt represent the fundamental recurring function of the 

 first order, we have 



<p{t-\- u) = ft . fu ; 



