Permutable functions concerning differential equations II

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Let f(z) and g(z) be two permutable transcendental entire functions with where p(z), p i and q i are polynomials, i = 1, …, n. It is interesting to find the explicit form for g, and it is also interesting to ask if J( f) = J(g). In this article, we shall prove that J( f) = J(g) and there exists a polynomial q(z) and two constants c and d such that q(g) = cq( f) + d under some conditions. This extends some previous results and it relates to an open question due to I.N. Baker.