Transcendental entire solution of some \(q\)-difference equation. (Q1396061)

The authors consider the following \(q\)-difference equation \[ \sum^p_{k=0} b_k(z) f(q^k z)= \beta(z), \] where \(b_j(z)\), \(\beta(z)\in C[z]\) with \(b_j(z)= \sum^{\beta_j}_{k=0} b^{(j)}_k z^k\) \((b^{(j)}_{\beta_j}\neq 0)\), \(0\leq j\leq p\) and \(q= e^{2\pi i\lambda}\), \(\lambda\in (0,1)\setminus Q\). They study some properties of the solutions of this equation. In particular, the following result is proved. ``Suppose the above \(q\)-difference equation admits a transcendental entire solution \(f(z)\) and \(\phi(z)\) has only one root of the modulus one. Then, in any sector, \(f(z)\) takes any finite value infinitely often.'' The function \(\phi(z)\) in the above result is defined as \[ \phi(z)= \sum^\tau_{t=1} b_t z^{j_t- j_1}= 0, \] where \(b_t= b^{(j_t)}_{B^*}\), \(B^*= \max_{0\leq j\leq p}\,B_j\) (\(B_j= \deg[b_j(z)]\)) and \(j_1<\cdots< j_\tau\) be such that \(B^*= B_{j_t}\) \((1\leq t\leq\tau)\) with \(B_j< B^*\) \((j\neq j_t)\).