Exponent Recursion implementation

thetalist = {1.36386, 0.720838, 0.504584, 0.154796, 1.83031};

Method 1:

Clear[expR]

expR[0, theta_] = 1;

expR[m_Integer?Positive, theta_] := 
 expR[m, theta] = expR[m - 1, theta]*Exp[-I*theta]

list11 = expR[#, theta] & /@ Range[0, 20];

list11[[1 ;; 5]]

(* {1, E^(-I theta), E^(-2 I theta), E^(-3 I theta), E^(-4 I theta)} *)

More generally,

FindSequenceFunction[list11[[2 ;; 10]], m] // PowerExpand

(* E^(-I m theta) *)

list12 = expR[20, thetalist]

(* {-0.542712 - 0.839919 I, -0.275965 - 0.961168 I, -0.785742 + 
  0.618555 I, -0.998957 - 0.0456568 I, 0.459902 + 0.88797 I} *)

list12 == Exp[-I 20 thetalist]

(* True *)

Method 2 : Using RSolve

Clear[expR]

expR[m_, theta_] = 
 expR[m, theta] /. 
   RSolve[{expR[0, theta] == 1, 
      expR[m, theta] == expR[m - 1, theta]*Exp[-I*theta]}, 
     expR[m, theta], {m, theta}][[1]] // PowerExpand

(* E^(-I m theta) *)

which is identical to the result from FindSequenceFunction above

list21 = expR[#, theta] & /@ Range[0, 20];

list22 = expR[20, thetalist]

(* {-0.542712 - 0.839919 I, -0.275965 - 0.961168 I, -0.785742 + 
  0.618555 I, -0.998957 - 0.0456568 I, 0.459902 + 0.88797 I} *)

list22 == Exp[-I 20 thetalist]

(* True *)

Verifying that both methods are equivalent for nonnegative, integer m

list11 == list21

(* True *)

list12 == list22

(* True *)