On the Composition of a Prime Transcendental Function and a Prime Polynomial

where f is meromorphic and g is entire (g may be meromorphic when f is rational). A nonlinear meromorphic function F (z) is called prime (pseudo prime) if every factorization of form (1) implies that either f is bilinear or g is linear (either f is rational or g is a polynomial). Clearly, a prime function is an analogy of a prime number. Over the past thirty years, many classes of prime or pseudo-prime functions have been obtained (see [2]). As an analogue of the unique factorizability of natural numbers, one can also define that concept for entire functions. Suppose an entire function F has two factorizations f1 ◦ f2 ◦ · · · ◦ fm(z) and g1 ◦ g2 ◦ · · · ◦ gn(z) into nonlinear entire factors. If m = n and if there exist linear polynomials Lj (j = 1, 2, 3, . . . , n− 1) such that the relations