Associated operator Legendre function and the incomplete Cauchy problem

Using the concept of a fractional integral of a function with respect another function, a new operator function, called the associated operator Legendre function, is introduced, which makes it possible to find a solution to the incomplete Cauchy problem for the abstract singular Legendre equation. It is found that when the requirements for the operator coefficient of the equation under consideration are weakened, fractional powers of this operator coefficient should be used to construct solutions.