Representation of sums of one class of polyparametric functional series by the method of finite hybrid integral transform of Legendre of the 2nd kind -- Kontorovich-Lebedev of the 2nd kind (Q2703321)

Using the Cauchy method and the method of finite hybrid integral transforms of Legendre of the 2nd kind -- Kantorovich-Lebedev of the 2nd kind, the authors construct in the region \(\{r:\;r\in(R_{0},R_{1})\cup(R_{1},R_{2})\); \(R_{0}>0\), \(R_{2}<\infty\}\) the bounded solution of separate system of Legendre and Bessel differential equations \((\Lambda_{\mu}-q_{1}^{2})u_{1}(r)= -f_{1}(r)\), \(r\in(R_{0},R_{1})\), \((B_{\alpha}-q_{2}^{2})u_{2}(r)= -f_{2}(r)\), \(r\in(R_{1},R_{2})\), with boundary value conditions \((\alpha_{11}^{0}{d\over dr}+\beta_{11}^{0})u_{1}(r)|_{r=R_{0}}=g_{0}\), \((\alpha_{22}^{2}{d\over dr}+\beta_{22}^{2}) u_{2}(r)|_{r=R_{2}}=g_{2}\) and conjugation conditions NEWLINE\[CARRIAGE_RETURNNEWLINE\left.\left[(\alpha_{j1}^{1}{d\over dr}+\beta_{j1}^{1})u_{1}(r) -(\alpha_{j2}^{1}{d\over dr}+\beta_{j2}^{1})u_{2}(r)\right]\right |_{r=R_{1}}=0,\quad j=1,2.CARRIAGE_RETURNNEWLINE\]NEWLINE From the uniqueness of the solution the representation of sums of one class of polyparametric functional series is obtained.