Here we compute the needed derivatives to derive the results in Appendix 6

Notation:  t=theta, T=Capital Theta, P=Capital Phi

F[
          B_] := ((1 - t B  -  T (B^12) + t T (B^13))/(1 - P (B^12)));

F[0]

D[F[B], {B, 1}] /. {B -> 0}

(1/(12!)) D[F[B], {B, 12}] /. {B -> 0} // Simplify

(1/(13!)) D[F[B], {B, 13}] /. {B -> 0} // Simplify

For gamma_  1 we need E (w_t a_ {t - 2})

(1/(2!)) D[F[B], {B, 2}] /. {B -> 0} // Simplify

(1/(14!)) D[F[B], {B, 14}] /. {B -> 0} // Simplify

(* 
      lets do all of the expressions we might need at one time *)

f[k_] := (1/(k!)) D[F[B], {B, k}] /. {B -> 0} // 
        Simplify

tmp = Map[f, Range[0, 15]]

tmp[[12]]

tmp[[10 + 1]]