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Solutions of singular differential equations with the Bessel operator of negative order are studied. In this regard, of great interest are the solutions of the singular differential Bessel equation , which are presented in the paper as linearly independent functions and , . Some properties of the functions expressed in terms of the properties of the Bessel–Levitan -function are considered. The direct and inverse -Bessel transforms are introduced, and the -pseudoshift operator that commutes with the Bessel operator is defined. The fundamental solution of the ordinary singular differential operator is found. A representation of the fundamental solution of the Kipriyanov -operator with a singularity at the point and on the cone in the Euclidean -dimensional half-space is given. [ABSTRACT FROM AUTHOR]

For the Euler–Poisson–Darboux type equations with the Bessel operators with negative index we consider the Cauchy problem with the boundary conditions. We obtain a representation of the solution in the form of the Poisson formula. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the Euler–Poisson–Darboux equation with boundary conditions. The Bessel operators in the equation can have different parameters. We establish a representation of the solution in the form of the Poisson formula with a special shift generated by the product of cylindrical functions of the first kind and different orders. [ABSTRACT FROM AUTHOR]