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# 数学代写|组合学代写Combinatorics代考|MAT492 Orthogonal polynomials

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## 数学代写|组合学代写Combinatorics代考|Orthogonal polynomials

In this section, we consider an irreducible tridiagonal matrix in general:
$$B=\left[\begin{array}{ccccc} a_0 & b_0 & & & \ c_1 & a_1 & b_1 & & 0 \ & \ddots & \ddots & \ddots & \ & & c_{d-1} & a_{d-1} & b_{d-1} \ 0 & & & c_d & a_d \end{array}\right], \quad b_{i-1} c_i \neq 0(1 \leq i \leq d) .$$
We set the complex field $\mathbb{C}$ as the base field. The intersection matrix $B_1$ of a Ppolynomial scheme and the dual intersection matrix $B_1^$ become irreducible tridiagonal matrices. If we consider P-polynomial schemes and Q-polynomial schemes, since $B_1$ and $B_1^$ are real matrices and their eigenvalues are also real, it is enough to consider the real field as a base field. If we consider L-pairs later, we need to consider complex matrices.

For $B$, we define the polynomial $v_i(x)(0 \leq i \leq d+1)$ of degree $i$ by the following three-term recurrence:
$$x v_i(x)=b_{i-1} v_{i-1}(x)+a_i v_i(x)+c_{i+1} v_{i+1}(x) \quad(0 \leq i \leq d),$$
where $v_{-1}(x)=0, v_0(x)=1, b_{-1}$ is an indeterminate, and $c_{d+1}=1$. We also define a sequence $k_0, k_1, \ldots, k_{d+1}$ by $k_0=1$, and
$$k_i=\frac{b_{i-1}}{c_i} k_{i-1}=\frac{b_0 b_1 \cdots b_{i-1}}{c_1 c_2 \cdots c_i} \quad(1 \leq i \leq d+1),$$
where $b_d=c_{d+1}=1$. Then we have
$$x \frac{v_i(x)}{k_i}=c_i \frac{v_{i-1}(x)}{k_{i-1}}+a_i \frac{v_i(x)}{k_i}+b_i \frac{v_{i+1}(x)}{k_{i+1}} \quad(0 \leq i \leq d),$$
where $c_0, k_{-1}$ are indeterminate. We call $\left{v_i(x)\right}_{i=0}^{d+1}$ and $\left{k_i\right}_{i=0}^{d+1}$ a system of polynomials and a sequence of degrees determined by $B$, respectively. Conversely, if polynomials $v_i(x)(0 \leq i \leq d+1)$ of degree $i$ satisfy the three-term recurrence $(6.74)$ for $i=0,1, \ldots, d$, the corresponding tridiagonal matrix $B$ is uniquely determined.

## 数学代写|组合学代写Combinatorics代考|Tridiagonal pairs (TD-pairs)

In the previous section, we stated some properties on irreducible representations of the Terwilliger algebra of a P- and Q-polynomial scheme. In this section, we axiomatize the essential part of them and introduce the concept of tridiagonal pairs (TD-pairs). Some TD-pairs do not arise from irreducible representations of Terwilliger algebras. However, this wider framework of TD-pairs is suitable for the study of irreducible representations of Terwilliger algebras.

The classification of TD-pairs implies the determination of all the irreducible representations of Terwilliger algebras. The classification of TD-pairs is almost completed [260, 253]. In this book, we deal with so-called L-pairs, a special class of TD-pairs. Using the classification of L-pairs we can determine the principal representations of Terwilliger algebras. This implies, if we ignore combinatorial structures, we can determine the Bose-Mesner algebras of P- and Q-polynomial schemes at the algebraic level.

In Section 6.2.1, we discuss weight space decompositions of TD-pairs, and in Section 6.2.2, we discuss TD-relations following [256]. We also introduce Askey-Wilson parameters (AW-parameters). The classification of L-pairs will be discussed in Section $6.3$.

## 数学代写|组合学代写COMBINATORICS代考|ORTHOGONAL POLYNOMIALS

$$B=\left[\begin{array}{llllllllllllll} a_0 & b_0 & c_1 & a_1 & b_1 & 0 & \ddots & \ddots & \ddots & c_{d-1} & a_{d-1} & b_{d-1} 0 & c_d & a_d \end{array}\right], \quad b_{i-1} c_i \neq 0(1 \leq i \leq d) .$$

$$x v_i(x)=b_{i-1} v_{i-1}(x)+a_i v_i(x)+c_{i+1} v_{i+1}(x) \quad(0 \leq i \leq d),$$

$$k_i=\frac{b_{i-1}}{c_i} k_{i-1}=\frac{b_0 b_1 \cdots b_{i-1}}{c_1 c_2 \cdots c_i} \quad(1 \leq i \leq d+1),$$

$$x \frac{v_i(x)}{k_i}=c_i \frac{v_{i-1}(x)}{k_{i-1}}+a_i \frac{v_i(x)}{k_i}+b_i \frac{v_{i+1}(x)}{k_{i+1}} \quad(0 \leq i \leq d),$$

## 数学代写|组合学代写COMBINATORICS代考|TRIDIAGONAL PAIRS $T D$ – pairs

TD 对的分类意味着确定 Terwilliger 代数的所有不可约表示。TD-pairs的分类基本完成
$$260,253$$
. 在本书中，我们处理所谓的 $L$ 对，一种特殊的 TD对。使用 $L$ 对的分类，我们可以确定 Terwilliger 代数的主要表示。这意味着，如果我们忽略组合 结构，我们可以在代数级别确定 $P$ 和 $Q$ 多项式方案的 Bose-Mesner 代数。

$$256$$
. 我们还介绍了 Askey-Wilson 参数 $A W$ – parameters. L对的分类将在第 $6.3$.

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