A family of hyper-Bessel functions and convergent series in them

The Delerue hyper-Bessel functions that appeared as a multi-index generalizations of the Bessel function of the first type, are closely related to the hyper-Bessel differential operators of arbitrary order, introduced by Dimovski. In this work we consider an enumerable family of hyper-Bessel functions and study the convergence of series in such a kind of functions. The obtained results are analogues to the ones in the classical theory of the widely used power series, like Cauchy-Hadamard, Abel and Fatou theorem.

On a Three-Point Boundary Value Problem for Third Order Differential Equations with Singularities in Phase Variables

The singular boundary value problem (ϕ(𝑥″))′ = 𝑓(𝑡, 𝑥, 𝑥′, 𝑥″), 𝑥(0) = 𝑥(𝑇1) = 𝑥(𝑇) = 0 is considered. Here 0 < 𝑇1 < 𝑇, ϕ is an increasing homeomorphism from ℝ onto ℝ, positive 𝑓 satisfies the local Carathéodory conditions on [0, 𝑇] × (ℝ \ {0})3 and 𝑓 may be singular at the value 0 of all its phase variables. The conditions guaranteeing the solvability of the above problem are presented. The proofs are based on regularization and sequential techniques and in limit processes a combination of the Fatou theorem and the Lebesgue dominated convergence theorem is used.