Properties of the Integer-Order Generalized Bessel Integral Operator

This article examines the properties of solutions obtained for the Cauchy problem associated with ordinary differential equations that contain a higher-order singular Bessel differential operator together with a lower-order term. As part of the research, an integral expression for the unique solution to the Cauchy problem is derived, one that incorporates both the Gauss hypergeometric function and the second-kindHumbert hypergeometric function. It is rigorously proved that the obtained solution completely satisfies the prescribed initial conditions and is unique. Furthermore, based on this integral representation, an integer-order generalized Bessel integral operator is introduced. Several of its fundamental properties, including the existence of an inverse operator, integration-by-parts formulas, and the semigroup property of the operator, are established through a series of lemmas.

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