# 数学代写|离散数学作业代写discrete mathematics代考|MATH300

## 数学代写|离散数学作业代写discrete mathematics代考|Solving Linear Non-homogenous Recurrence Relations

The solution of a linear non-homogenous recurrence relation with constant coefficients is the sum of the two parts, the homogenous solution, which satisfies the recurrence relation when the right-hand side of the equation is set to 0 , and the particular solution, which satisfies the difference equation with $f(n)$ on the right-hand side.

There is no general procedure for determining the particular solution of a difference equation. However, in simple cases, this solution can be obtained by the method of inspection. To determine the particular solution, we use the following rules:
Rule 1:
When $f(n)$ is of the form of a polynomial of degree $m$ in $n$,
$$k_0+k_1 n+k_2 n^2+k_3 n^3+\cdots+k_{m-1} n^{m-1}+k_m n^m,$$ the corresponding particular solution will be of the form
$$Q_0+Q_1 n+Q_2 n^2+Q_3 n^3+\cdots+Q_{m-1} n^{m-1}+Q_m n^m .$$
Rule 2:
When $f(n)$ is of the form
$$\left(k_0+k_1 n+k_2 n^2+\cdots+k_{m-1} n^{m-1}+k_m n^m\right) a^n,$$
the corresponding particular solution is of the form
$$\left(Q_0+Q_1 n+Q_2 n^2+\cdots+Q_{m-1} n^{m-1}+Q_m n^m\right) a^n$$
if $a$ is not a characteristic root of the recurrence relation.

## 数学代写|离散数学作业代写discrete mathematics代考|Generating Functions

In this section, we will show how recurrence relations can be solved using the powerful generating function method. Generating function is an important tool in discrete mathematics, and its use is by no means confined to the solution of recurrence relations.

If $a_0, a_1, a_2, \ldots, a_n$ is a finite sequence of numbers, the generating function for the $a_n$ ‘s is the polynomial
$$G(z)=\sum_{k=0}^n a_k z^k=a_0+a_1 z+a_2 z^2+\cdots+a_n z^n$$
where $z$ is an indeterminate (that is, an abstract) symbol. If $a_0, a_1, a_2, \ldots, a_n, \ldots$ is an infinite sequence of numbers, its generating function is defined to be
$$G(z)=\sum_{k=0}^{\infty} a_k z^k=a_0+a_1 z+a_2 z^2+\ldots$$
The symbol $z$ is just the name given to a variable and has no special significance. For any sequence $\left{a_n\right}$, we write $G(z)$ to denote the generating function of $\left{a_n\right}$. Clearly, given a sequence, we can easily obtain its generating function and its converse. For example, the generating function of $a_n=\alpha_n$, $n \geq 0$ is
$$\alpha^0+\alpha z+\alpha^2 z^2+\alpha^3 z^3+\ldots$$
We note that the infinite series (2.26) can be written in closed form as $\frac{1}{1-\alpha z}$ which is a rather compact way to represent the sequence $\left{a_n\right}$ or $\left(a, \alpha, \alpha^2, \ldots\right)$.

# 离散数学代写

## 数学代写|离散数学作业代写discrete mathematics代考|Solving Linear Non-homogenous Recurrence Relations

$$k_0+k_1 n+k_2 n^2+k_3 n^3+\cdots+k_{m-1} n^{m-1}+k_m n^m,$$

$$Q_0+Q_1 n+Q_2 n^2+Q_3 n^3+\cdots+Q_{m-1} n^{m-1}+Q_m n^m .$$

$$\left(k_0+k_1 n+k_2 n^2+\cdots+k_{m-1} n^{m-1}+k_m n^m\right) a^n,$$

$$\left(Q_0+Q_1 n+Q_2 n^2+\cdots+Q_{m-1} n^{m-1}+Q_m n^m\right) a^n$$

## 数学代写|离散数学作业代写discrete mathematics代考|Generating Functions

$$G(z)=\sum_{k=0}^n a_k z^k=a_0+a_1 z+a_2 z^2+\cdots+a_n z^n$$

$$G(z)=\sum_{k=0}^{\infty} a_k z^k=a_0+a_1 z+a_2 z^2+\ldots$$

$$\alpha^0+\alpha z+\alpha^2 z^2+\alpha^3 z^3+\ldots$$
$\left(a, \alpha, \alpha^2, \ldots\right)$

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