Information for researches, when the differential equation

$f" + Af = 0$

has the fundamental system of solutions, where absent logarithmic solutions is known a long time. In the classical analysis for decision the equations received local expand in the integer degrees of an independent variable. In more difficult cases received local expand in fractional degrees of an independent variable, on so-called Newton – Poiseux series. A row of mathematicians for integration of linear differential equations applied a method of so-called generalized degree series, where meets irrational, in general real degree of an independent variable.

In this article we consider the question about construction of fundamental system of solutions of the above mentioned equation according to given sequences, where $A$ is meromorphic functions.

Earlier similar results considered by several authors. In particular they studied the zero sequences of the non-trivial solutions of the above mentioned equation, when $A$ is entire function or it is similar, when $A$ is analytic function in unit disc. Except this case, when above marked differential equation possesses one solution with given zero-sequence it is possible for consideration the case, when two linearly independent solutions of this equation possesses given zero-sequences, where $A$ is entire function. Also considered similar problems for differential equations of higher order. Let's note, that representation of function by Weierstrass canonical product is the basic element for researches in the theory of the entire functions.

Further in this article we consider the problem about construction of linearly independent solutions $\{f_1;f_2;f_3\}$ of equation

$f'''+A_2f''+A_1f'+A_0f=0,$

where $f_1$ is meromorphic function. Then we consider the question about construction of meromorphic functions $(f_2/f_1)'$ and $(f_3/f_1)'$ on the base of construction of fundamental system of solutions of linear differential equation of the second order according to given sequences.

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