Expansions of analytic functions in products of Bessel functions
In a paper dedicated to F. W. Schäfke on his seventieth birthday and motivated by his work, the authors consider expansion of the form \[ \begin{aligned} f(z)= \sum^ \infty_{k=-\infty} & c^{(1)}_ k(f) J_{\nu+k/2}(\eta_{k,1,1}z)J_{\mu+k/2}(\eta_{k,2,1}z)+\\ \sum^ \infty_{k=-\infty} & c^{(2)}_ k(f)J_{\mu+k/2}(\eta_{k,1,2}z)J_{\nu+k/2}(\eta_{k,2,2} z).\tag{1}\end{aligned} \] Special cases are series studied by L. Carlitz and I. Ya. Berson. It is shown that such expansions are related to eigenfunction expansion associated with certain \(4\times 4\) linear first-order differential systems studied by the first three authors of the type \[ zy'(z)=\{\lambda A_ 1(z)+ A_ 0(z)\} y(z),\;y(ze^{2\pi i})= e^{2\pi i\nu} y(z),\tag{2} \] where \(z\in S^{\log}\), the Riemann surface of the logarithm over \(S=\mathbb{C}\backslash\{0\}\), the matrices \(A_ 1\), \(A_ 0\) are holomorphic on \(S\) and \(\nu\) is a complex parameter. The series (1) are the first components of eigenfunction expansions for pairs of functions \(\{f_ 1,f_ 2\}\). The question of how to choose \(f_ 2\), for a given \(f_ 1\), in order to obtain series of Carlitz and Berson type is discussed.
linear first- order differential systems
scientific article; zbMATH DE number 446739
scientific article