On complex oscillation, function-theoretic quantization of non-homogeneous periodic ordinary differential equations and special functions (Q2895824)

Consider the differential equation NEWLINE\[NEWLINE f''+2Nf'+ (L^2M^2 e^{2Mz} +(N^2-\nu^2M^2)) f =\sum_{j=1}^n \sigma_j L^{\mu_j+1} M^2 e^{(M(\mu_j+1) -N)z} NEWLINE\]NEWLINE with \(L, M, N, \sigma_j, \mu_j, \nu \in \mathbb{C}\) satisfying \(LM\not=0\) and \(\sigma_j\not=0\) for some \(j\). This is equivalent to NEWLINE\[NEWLINE \zeta^2 y''(\zeta) + \zeta y'(\zeta) + (\zeta^2 -\nu^2)y(\zeta) = \sum_{j=1}^n \sigma_j \zeta^{\mu_j+1} NEWLINE\]NEWLINE via the substitution \(\zeta=Le^{Mz},\) \(y(\zeta)=e^{Nz}f(z),\) whose solution is expressible in terms of Bessel and Lommel functions. Under the condition that all the real parts \(\operatorname{Re} (\mu_j)\) are distinct, this paper presents a necessary and sufficient condition for the finiteness of \(\lambda(f)\) of a given solution \(f(z)\), where \(\lambda(f)\) denotes the exponent of convergence of zeros. This condition coincides with that for the subnormality of \(f(z)\), i.e., the property that \(\limsup_{r\to+\infty} r^{-1}\log\log M(r,f)=0\) with \(M(r,f)=\max_{|z|\leq r}|f(z)|\). The main part of the proof is devoted to the discussion on the asymptotic distribution of zeros of the solution \(f(z)\).