In this article which is a detailed exposition of our notes (7) and (8), we consider the differential equation (7) in which At(p) are polynomials in a complex parameter, these being uniquely determined by the condition that the equation is satisfied by the family (depending on p) of the functions (1)
For non - singular values of p we build all linearly independent solutions (19) of the equation (7) and find for them integral representations (27) and (28), with the help of which we get addition theorems of type (40) and (41).
If m = 2 we get the known results of the Bessel functions theory.
Citation Example:Khriptun M. D. On an ordinary linear differential equation of higher order // Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 277-289.