A new method for solving the linear recurrence relation with nonhomogeneous constant coefficients in some cases

Recursive relation mainly describes the unique law satisfied by a sequence, so it plays an important role in almost all branches of mathematics. It is also one of the main algorithms commonly used in computer programming. This paper first introduces the concept of recursive relation and two common basic forms, then starts with the solution of linear recursive relation with non-homogeneous constant coefficients, gives a new solution idea, and gives a general proof. Finally, through an example, the general method and the new method given in this paper are compared and verified.


Introduction
Recurrence relation, which mainly characterizes the unique laws satisfied by a sequence, plays an important role in almost all branches of mathematics. This is especially true for combinatorial mathematics, because each combinatorial counting problem has its combinatorial characteristics and laws. In many cases, recursive relationship is one of the most appropriate tools to describe this combinatorial law. Because of its importance, it has been deeply studied and applied in many aspects of engineering and life [1][2][3][4].
How to establish the recurrence relation, what are the properties of the known recurrence relation and how to solve the recurrence relation are several basic problems in the recurrence relation. Consider a sequence  : 0 n a n , where n is the number of schemes to be calculated in a specific mathematical model. Actually, it's a function of n . For such an unknown sequence, in order to calculate the number of schemes, a natural idea is to directly find its general term expression and then find all its terms. However, when the general term n a cannot be obtained directly, if we can find a relationship between n a and its several adjacent terms, and this relationship is always true when n is greater than or equal to a fixed positive integer, we have the possibility of finding the sequence n a by using this relationship.
Depending on whether there is a nonhomogeneous term, recursive relations are generally divided into two types: r-order homogeneous constant coefficient linear recursive relations and r-order nonhomogeneous constant coefficient linear recursive relations. The solution of r-order homogeneous constant coefficient linear recursive relation is relatively simple. As long as the characteristic polynomial is listed to solve the characteristic root, and then the formula is applied to solve the undetermined coefficient, the solution of the recursive relation can be obtained. This paper mainly studies a special case of non-homogeneous recurrence relation, gives a new solution method from another point of view, and gives a general proof. Finally, an example is given to verify the correctness of the method.

Common solving methods and new ideas of recursive relations
The general form of r-order non-homogeneous linear recursive relation with constant coefficients is: For the r-order linear recursive relation with non-homogeneous constant coefficients, the general solution is divided into three steps. The first step is to find the special solution of the non-homogeneous linear recursive relation; The second part is to solve the general solution of its corresponding derived recurrence relationship; Finally, the third step is to add the two solutions, and the result of the addition is the general solution of the original r-order non-homogeneous linear recurrence relationship with constant coefficients. The general solution of deducing the recurrence relation is actually to solve an rorder homogeneous linear recurrence relation. There are specific solutions in the corresponding textbooks [5,6]. Next, we mainly study the solution method of the special solution of the nonhomogeneous recurrence relation. However, in the common solution, this method can only find the special solution for some special ( ) g n , which refers to the part of the linear recursive relationship with non-homogeneous constant coefficients excluding the derived recursive relationship. In order to facilitate the solution, formula (1) Next, a brief proof of the feasibility of this method is given.

Case 1: ( ) g n is polynomial
Suppose if g n a n a n a n a In this case, it is required to prove the existence of i a (  a n a n a n a c a n j a n j a n j a Comparing the coefficients of m n , we can get The value of 1 1 , , , that is, we can prove that these m+1 coefficients do exist, so this new method is feasible when g(n) is a polynomial.