310 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Moreover, if <£ ( T) is any polynomial in T, we have 



4> (r ) = s, * (c ) + (A - o —q£— + 21 ^T" 



+ "- + 0<-i)! s^- 1 J ,c ' 



In particular, if £ k =. ± 1 , 



F k (T~*) = F k (T) >, 

 and if 4 ^ ± 1> and 4 = fit" 1 has the same multiplicity as &., then 



F k > (T-i) = F k (T) . 



"We have now the following theorems : 



I. If £7 is real and T= e uV ~ l is also real, then T 2 = 1 * 



II. If T is real and no negative number is a root of the characteristic 

 equation of T, there is a real polynomial f(T) satisfying the condition 

 T=e /{T) f 



III. If T is real and each negative root other than —1 of the char- 

 acteristic equation of T is paired with its inverse, there is a polynomial 

 / (T) satisfying the equation T= e fiT) and such that f(T) — /(7 7-1 ) is 

 real.f 



* For if T = e^-i is real, then 



U* IP 



3! + 5! 



r/ 3 U 5 I r— i 



is real. Therefore, if U is real, T - T~ l = ; that is T 2 = 1. 



t If T is real, each imaginary root of the characteristic equation of T is paired 

 with its conjugate imaginary. Therefore, if this equation has no negative root, 

 f(T) may he taken real by a proper choice of m 1; m 2 , etc. 



If the distinct negative roots of the characteristic equation are 



and f 2> /'+l — ~~ *> and moreover (for i — 1, 2, . . . /) £• ar "3 £ _1 have the same 

 multiplicity, then, for a proper choice of m x , m 2 , etc., the imaginary part of the 

 polynomial/^) is 



n <f=\ Y (Fi{T) +F yl + i (T)) + U \T=\F 2v , + 1 {T) ; 



and, since now 



Fi (T~ 1 ) = F v , +i ( T), F v , +i ( r- 1 ) = F { ( T), 

 (i = 1, 2, . . . /) 



therefore f(T) -f{T~ l ) will be real. 



