Darboux transformation of symmetric Jacobi matrices and Toda lattices

Let J be a symmetric Jacobi matrix associated with some Toda lattice. We find conditions for Jacobi matrix J to admit factorization J = LU (or J = 𝔘𝔏) with L (or 𝔏) and U (or 𝔘) being lower and upper triangular two-diagonal matrices, respectively. In this case, the Darboux transformation of J is the symmetric Jacobi matrix J^(p) = UL (or J^(d) = 𝔏𝔘), which is associated with another Toda lattice. In addition, we found explicit transformation formulas for orthogonal polynomials, m-functions and Toda lattices associated with the Jacobi matrices and their Darboux transformations.

1 Introduction

Let a sequence of real numbers $\text{s}={\left\{s_n\right\}}_{n=0}^{\infty }$ be associated with a measure μ on (−∞, +∞), i.e.

$\begin{array}{l}{s}_n={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}{\lambda }^{n}d\mu (\lambda ),n\in {\mathbb{Z}}_{+}.\end{array}$

However, in the general case, $\text{s}={\left\{s_n\right\}}_{n=0}^{\infty }$ is associated with a linear functional 𝔖 by

$\begin{array}{ll}{s}_n=𝔖({\lambda }^n),n\in {\mathbb{Z}}_{+}.& (1.1)\end{array}$

We consider the sequence $\text{s}={\left\{s_n\right\}}_{n=0}^{\infty }$ such that

$\begin{array}{l}{D}_n\ne 0,\text{ for all }n\in \mathbb{N},\end{array}$

where $D_n=\mathrm{\text{det}}{(s_{i+j})}_{i,j=0}^{n-1}$. Note, if D_n > 0 for all n ∈ ℕ, then there exists measure μ associated with $\text{s}={\left\{s_n\right\}}_{n=0}^{\infty }$, otherwise, the sequence $\text{s}={\left\{s_n\right\}}_{n=0}^{\infty }$ is associated with only linear functional 𝔖.

On the other hand (see [1, 2]), the real sequence $\text{s}={\left\{s_n\right\}}_{n=0}^{\infty }$ is associated with the symmetric Jacobi matrix J and the sequence of orthogonal polynomials of the first kind ${\left\{P_n(\lambda )\right\}}_{n=0}^{\infty }$, which can defined by

$\begin{array}{l}{P}_0(\lambda )\equiv 1\text{and }{P}_n(\lambda )=\frac{1}{\sqrt{D_{n-1}{D}_n}}\left|\begin{array}{cccc}{s}_0& s_1& \dots & s_n\\ s_1& s_2& \dots & s_{n+1}\\ \dots & \dots & \dots & \dots \\ s_{n-1}& s_n& \dots & s_{2n-1}\\ 1& \lambda & \dots & {\lambda }^n\end{array}\right|.\end{array}$

[3, 4] Moreover, the sequence ${\left\{P_n(\lambda )\right\}}_{n=0}^{\infty }$ satisfies a three-term recurrence relation

$\begin{array}{ll}\lambda P_n(\lambda )=a_{n+1}{P}_{n+1}(\lambda )+b_n{P}_n(\lambda )+a_n{P}_{n-1}(\lambda )& (1.2)\end{array}$

with the initial conditions

$\begin{array}{ll}{P}_{-1}(\lambda )\equiv 0\text{and }{P}_0(\lambda )\equiv 1.& (1.3)\end{array}$

In the short form we can rewrite Equation (1.2) as

$\begin{array}{l}JP(\lambda )=\lambda P(\lambda ),\end{array}$

where $P(\lambda )={(P_0(\lambda ),\dots ,P_n(\lambda ),\dots )}^T$ and the symmetric Jacobi matrix J is defined by

$\begin{array}{ll}J=\left(\begin{array}{cccc}{b}_0& a_1& & \\ a_1& b_1& a_2& \\ & a_2& b_2& \ddots \\ & & \ddots & \ddots \end{array}\right).& (1.4)\end{array}$

On the other hand, the symmetric Jacobi matrix J is associated with the moment sequence $\text{s}={\left\{s_n\right\}}_{n=0}^{\infty }$, the following relation holds (see [2, 5])

$\begin{array}{ll}{s}_n=(e_0,J^n{e}_0)\text{ for all }n\in {\mathbb{Z}}_{+},& (1.5)\end{array}$

where $e_0={(1,0,\dots )}^T$ and m– function of Jacobi matrix is found by

$\begin{array}{ll}m(z)={\displaystyle {\int }_{ℝ}}\frac{d\mu (\lambda )}{\lambda -z}.& (1.6)\end{array}$

There exist two type transformations of orthogonal polynomials, which are the Christoffel and Geronimus transformations. One are studied in the paper Zhedanov [6]. The Christoffel transformation is defined by

$\begin{array}{ll}\tilde{P}(\lambda )=\frac{P_{n+1}(\lambda )-A_n{P}_n(\lambda )}{\lambda -\alpha },n\in {\mathbb{Z}}_{+},& (1.7)\end{array}$

where $A_n=\frac{P_{n+1}(\alpha )}{P_n(\alpha )}$ and α is arbitrary parameter. Moreover, Equation (1.7) can be rewritten as follows:

Theorem 1.1. ([7, Theorem 1.5]) Let ${\left\{P_n(\lambda )\right\}}_{n=0}^{\infty }$ be the sequence of the orthogonal polynomials associated with Equation (1.2). Then the Christoffel–Darboux formula takes the following form

$\begin{array}{ll}{\displaystyle \sum _{i=0}^n}{P}_i(x)P_i(t)=a_{n+1}\frac{P_{n+1}(x)P_n(t)-P_n(x)P_{n+1}(t)}{x-t}.& (1.8)\end{array}$

The second transformation is a Geronimus transformation of the orthogonal polynomials [6], one is defined by

$\begin{array}{l}\tilde{P}(\lambda )=P_n(\lambda )-B_n{P}_{n-1}(\lambda ),B_n\in ℝ\text{and }n\in \mathbb{N}.\end{array}$

Toda lattice. The Toda lattice is a system of differential equations

$\begin{array}{ll}{x}_n^{″}(t)=e^{x_{n-1}-x_n}-e^{x_n-x_{n+1}},n\in \mathbb{N},& (1.9)\end{array}$

which was introduced in Toda [8].

We study the semi-infinite system with x₋₁ = −∞. [9, 10] Flaschka variables are defined by

$\begin{array}{ll}{a}_k=\frac{1}{2}{e}^{\frac{x_{k-1}-x_k}{2}}\text{and }{b}_k=-\frac{1}{2}{x}_k^{\prime }.& (1.10)\end{array}$

Therefore, we obtain the following system in terms of Flaschka variables

$\begin{array}{ll}{a}_k^{\prime }=a_k(b_k-b_{k-1})\text{and }{b}_k^{\prime }=2(a_{k+1}^2-a_k^2),a_0=0.& (1.11)\end{array}$

Hence, the semi-infinite Toda lattice is associated with the symmetric Jacobi matrix J and Lax pair (J, A), such that

$\begin{array}{l}\left[J,A\right]=JA-AJ,\end{array}$

where the matrix A = J₊ − J₋, where J₊ and J₋ are upper and lower triangular part of J, respectively and

$\begin{array}{l}A=\left(\begin{array}{cccc}0& a_1& & \\ -a_1& 0& a_2\\ & -a_2& 0& \ddots \\ & & \ddots & \ddots \end{array}\right).\end{array}$

As is known (see [8, 11]), the system (1.11) is equivalent to the following

$\begin{array}{l}{J}^{\prime }=-\left[J,A\right].\end{array}$

Darboux transformation of the monic classical and generalized Jacobi matrices were studied in Bueno and Marcellán [12], Derevyagin and Derkach [13], and Kovalyov [14, 15]. Darboux transformation involves finding a factorization of a matrix from a certain class such that the new matrix is from the same class. There are two types of Darboux transformation: transformation with and without parameter. Jacobi matrix is associated with many objects. There are moment sequence, measure, linear functional orthogonal polynomials and Toda lattice. Hence, in the current paper, we study not only Darboux transformation of the symmetric Jacobi matrices, but we also study the transformation of the associated objects. Hence, we investigate the Darboux transformation of the symmetric Jacobi matrices J and find relations between associated Toda lattice, orthogonal polynomials, moment sequences and m–functions. We obtain that the Darboux transformation without parameter of the symmetric Jacobi matrices has more additional existence conditions in contrast to case of the monic Jacobi matrices. On the other hand, the Darboux transformation with parameter of the symmetric Jacobi matrices is generated more easily. The results obtained can be applied for further research related to symmetric Jacobi matrices, Toda lattices and inverse problems. Of course, it can also be applied to the Toda lattice hierarchy.

Now, briefly describe the content of the paper. Section 2 contains Darboux transformation without parameter of the symmetric Jacobi matrix J. We find LU–factorization of J and the transformed matrix J^(p). Relation between Toda lattices, moment sequences and m–functions associated with the Jacobi matrices was obtained. In this case, the orthogonal polynomials are transformed by the Christoffel formula (1.7). In Section 3, we study the Darboux transformation with parameter of the symmetric Jacobi matrix J. We find 𝔘𝔏–factorization of J and transformed matrix J^(d). Moreover, the relations between orthogonal polynomials, m–functions, moment sequence and Toda lattices are found according to explicit formulas.

2 Darboux transformation without parameter of symmetric Jacobi matrix

Now we study a Darboux transformation without parameter of symmetric Jacobi matrix J. The goal is to find the transformations of polynomials of the first kind, m-functions, measure, moment sequence and Toda lattice, which are associated with the transformed Jacobi matrix.

2.1 LU–factorization

Lemma 2.1. Let J be a symmetric Jacobi matrix. Then J admits LU–factorization

$\begin{array}{ll}J=LU,& (2.1)\end{array}$

where L and U are lower and upper triangular matrices, respectively, which are defined by

$\begin{array}{ll}L=\left(\begin{array}{cccc}1& & & \\ l_1& & 1\\ & l_2& 1\\ & & \ddots & \ddots \end{array}\right)\text{and }U=\left(\begin{array}{cccc}{u}_1& v_1& & \\ & u_2& v_2& \\ & & u_3& \ddots \\ & & & \ddots \end{array}\right),& (2.2)\end{array}$

if and only if the following system is solvable

$\begin{array}{ll}\begin{array}{l}{b}_0=u_1,v_1=a_1,v_j=a_j,l_j{u}_j=a_j,\\ l_j{v}_j+u_{j+1}=b_j,u_j\ne 0\text{and }{l}_j\ne 0,j\in \mathbb{N}.\end{array}& (2.3)\end{array}$

Proof. Let us calculate the product LU

$\begin{array}{l}LU=\left(\begin{array}{cccc}{u}_1& v_1& & \\ l_1{u}_1& l_1{v}_1+u_2& v_2\\ & & l_2{u}_2& l_2{v}_2+u_3& \ddots \\ & & \ddots & \ddots \end{array}\right).\end{array}$

Comparing the product LU with the Jacobi matrix J

$\begin{array}{l}\left(\begin{array}{cccc}{b}_0& a_1& & \\ a_1& b_1& a_2& \\ & a_2& b_2& \ddots \\ & & \ddots & \ddots \end{array}\right)=\left(\begin{array}{cccc}{u}_1& v_1& & \\ l_1{u}_1& l_1{v}_1+u_2& v_2& \\ & l_2{u}_2& l_2{v}_2+u_3& \ddots \\ & & \ddots & \ddots \end{array}\right),\end{array}$

we obtain the system (2.3).

If the system (2.3) is solvable, then J admits the factorization J = LU of the form (2.1–2.3), where L and U are found uniquely. Conversely, if J admit LU—factorization then the system (2.3) is solvable. This completes the proof.

Lemma 2.2. Let J be the symmetric Jacobi matrix and let J = LU be its LU– factorization of the form (2.1–2.3). Let P_j be the polynomials of the first kind associated with the matrix J. Then

$\begin{array}{ll}\frac{P_n(0)}{P_{n-1}(0)}=-\frac{1}{l_n},n\in \mathbb{N}.& (2.4)\end{array}$

Proof. Let J admit the LU–factorization of the form (2.1–2.3). Setting λ = 0 in Equation (1.2), we obtain

$a_{n+1}{P}_{n+1}(0)+b_n{P}_n(0)+a_n{P}_{n-1}(0)=0.$

By induction, we prove Equation (2.4).

1. Let n = 0, then

$a_1{P}_1(0)+b_0{P}_0(0)+a_0{P}_{-1}(0)=0$

and due to the initial condition (1.3) and (2.3), we get

$a_1{P}_1(0)+b_0{P}_0(0)=0\Rightarrow \frac{P_1(0)}{P_0(0)}=-\frac{b_0}{a_1}=-\frac{u_1}{l_1{u}_1}=-\frac{1}{l_1}.$

2. Let n = 1, then

$a_2{P}_2(0)+b_1{P}_1(0)+a_1{P}_0(0)=0$

and by Equation (2.3), we have

$\begin{array}{l}\frac{P_2(0)}{P_1(0)}+\frac{b_1}{a_2}+\frac{a_1}{a_2}\frac{P_0(0)}{P_1(0)}=0\Rightarrow \frac{P_2(0)}{P_1(0)}\\ =-\frac{b_1}{a_2}+\frac{a_1{l}_1}{a_2}=\frac{-l_1^2{u}_1-u_2+l_1^2{u}_1}{l_2{u}_2}=-\frac{1}{l_2}.\end{array}$

3. Let Equation (2.4) hold for n = k − 1.

4. Let us prove Equation (2.4) for n = k, we obtain

$a_{k+1}{P}_{k+1}(0)+b_n{P}_k(0)+a_k{P}_{k-1}(0)=0.$

$\frac{P_{k+1}(0)}{P_k(0)}+\frac{b_k}{a_{k+1}}+\frac{a_k}{a_{k+1}}·\frac{P_{k-1}(0)}{P_k(0)}=0.$

Consequently

$\begin{array}{c}\frac{P_{k+1}(0)}{P_k(0)}=-\frac{b_k}{a_{k+1}}-\frac{a_k}{a_{k+1}}·\frac{P_{k-1}(0)}{P_k(0)}\end{array}$

= {by Section (2.3)}

$\begin{array}{c}=\frac{-b_k+a_k{l}_k}{a_{k+1}}=\frac{-l_k^2{u}_k-u_{k+1}+l_k^2{u}_k}{l_{k+1}{u}_{k+1}}=-\frac{1}{l_{k+1}}.\end{array}$

So, Equation (2.4) is proven. This completes the proof.

Corollary 2.3. Let J be the symmetric Jacobi matrix and let J = LU be its LU–factorization of the form (2.1–2.3). Let P_j be the polynomials of the first kind associated with the matrix J. Then

$\begin{array}{ll}{P}_n(0)={(-1)}^n{\displaystyle \prod _{i=1}^n}\frac{1}{l_i}.& (2.5)\end{array}$

Proof. Let J admit the LU–factorization of the form (2.1–2.3) and let P_j be the polynomials of the first kind associated with J. By Lemma 2.2, Equation (2.4) holds and we obtain

$P_n(0)=\frac{P_n(0)}{P_{n-1}(0)}·\frac{P_{n-1}(0)}{P_{n-2}(0)}·\dots ·\frac{P_1(0)}{P_0(0)}={(-1)}^n{\displaystyle \prod _{i=1}^n}\frac{1}{l_i}.$

So, Equation (2.5) is proven. This completes the proof.

Corollary 2.4. Let J be the symmetric Jacobi matrix and let J = LU be its LU–factorization of the form (2.1–2.3). Let P_j be the polynomials of the first kind associated with the matrix J. Then

$\begin{array}{ll}{P}_n(0)={(-1)}^k\frac{1}{l_n}·\frac{1}{l_{n-1}}·\dots ·\frac{1}{l_{n-(k-1)}}{P}_{n-k}(0).& (2.6)\end{array}$

Proof. Let J admit the LU–factorization of the form (2.1–2.3). By Lemma 2.2, we obtain

$\begin{array}{c}{P}_n(0)=\frac{P_n(0)}{P_{n-1}(0)}·\frac{P_{n-1}(0)}{P_{n-2}(0)}·\dots ·\frac{P_{n-k-1}(0)}{P_{n-k}(0)}·P_{n-k}(0)\\ ={(-1)}^k\frac{1}{l_n}·\frac{1}{l_{n-1}}·\dots ·\frac{1}{l_{n-(k-1)}}{P}_{n-k}(0).\end{array}$

Hence, Equation (2.6) is proven. This completes the proof.

Theorem 2.5. Let J be the symmetric Jacobi matrix and let P_j be the polynomials of the first kind associated with J. Then J admits LU—factorization of the form (2.1–2.3) if and only if

$\begin{array}{ll}{P}_j(0)\ne 0\text{ for all }j\in {\mathbb{Z}}_{+}.& (2.7)\end{array}$

Furthermore,

$\begin{array}{ll}{b}_0=u_1,v_j=a_j,l_j=-\frac{P_{j-1}(0)}{P_j(0)}\text{and }{u}_j=-\frac{a_j{P}_j(0)}{P_{j-1}(0)}.& (2.8)\end{array}$

Proof. Let P_j(0) ≠ 0 for all j ∈ ℤ₊. By Lemma 2.2 the system (2.8) is equivalent to the system (2.3). Consequently, by Lemma 2.1 the Jacobi matrix J admits LU—factorization of the form (2.1–2.3). Conversely, if the Jacobi matrix J admits LU—factorization of the form (2.1–2.3), then by Lemma 2.1 and Lemma 2.2 the polynomials of the first kind P_j satisfy (2.7). This completes the proof.

2.2 Transformed Jacobi matrix J^(p) = UL

Definition 2.6. Let the symmetric Jacobi matrix J admit LU—factorization of the form (2.1–2.3). Then a transformation

$\begin{array}{l}J=LU\to UL=J^{(p)}\end{array}$

is called a Darboux transformation without parameter of the symmetric Jacobi matrix J.

Theorem 2.7. Let J be the symmetric Jacobi matrix (1.4) and let J = LU be its LU–factorization of the form (2.1–2.3). Then the Darboux transformation without parameter of the matrix J is the symmetric Jacobi matrix

$\begin{array}{ll}{J}^{(p)}=UL=\left(\begin{array}{cccc}{b}_1& a_1& & \\ a_1& b_2& a_2& \\ & a_2& b_3& \ddots \\ & & \ddots & \ddots \end{array}\right)& (2.9)\end{array}$

if and only if

$\begin{array}{ll}{u}_j=b_0\text{and }\frac{a_j^2+b_0^2}{b_0}=b_j\text{ for all }j\in \mathbb{N}.& (2.10)\end{array}$

Proof. Calculating UL, we obtain

$\begin{array}{c}{J}^{(p)}=UL=\left(\begin{array}{cccc}{u}_1& v_1& & \\ & u_2& v_2& \\ & & u_3& \ddots \\ & & & \ddots \end{array}\right)\left(\begin{array}{cccc}1& & & \\ l_1& 1& & \\ & l_2& 1& \\ & & \ddots & \ddots \end{array}\right)=\\ =\left(\begin{array}{cccc}{u}_1+v_1{l}_1& v_1& & \\ l_1{u}_2& u_2+v_2{l}_2& v_2& \\ & l_2{u}_3& u_3+v_3{l}_3& \ddots \\ & & \ddots & \ddots \end{array}\right)\end{array}$

= {by Equation (2.3)}

$\begin{array}{c}=\left(\begin{array}{cccc}{u}_1+v_1{l}_1& a_1& & \\ l_1{u}_2& u_2+v_2{l}_2& a_2& \\ & l_2{u}_3& u_3+v_3{l}_3& \ddots \\ & & \ddots & \ddots \end{array}\right).\end{array}$

Consecuently, J^(p) is the symmetric Jacobi matrix if and only if

$\begin{array}{ll}{l}_j{u}_{j+1}=a_j\text{ for all }j\in \mathbb{N}.& (2.11)\end{array}$

Comparing Equation (2.3) with Equation (2.11), we get

$l_j{u}_j=a_j=l_j{u}_{j+1}\Rightarrow u_j=u_{j+1}\Rightarrow u_j=b_0\text{ for all }j\in \mathbb{N}.$

By Equation (2.3), u_j + v_jl_j = b_j for all j ∈ ℕ, we obtain Equations (2.9, 2.10) and J^(p) is the symmetric Jacobi matrix. This completes the proof.

Theorem 2.8. Let the symmetric Jacobi matrix J satisfy (2.7) and let J = LU be its LU–factorization of the form (2.1–2.3). Let J^(p) = UL be the Darboux transformation without parameter of J. Then the polynomials of the first kind $P_n^{(p)}$ associated with J^(p) can be found by Christoffel–Darboux formula

$\begin{array}{ll}{P}_n^{(p)}(\lambda )=\frac{1}{P_n(0)}\frac{P_{n+1}(\lambda )P_n(0)-P_n(\lambda )P_{n+1}(0)}{\lambda },& (2.12)\end{array}$

where P_j are the polynomials of the first kind associated with the symmetric Jacobi matrix J.

Proof. Let the Jacobi matrix J satisfy (2.7) and admit LU–factorization of the form (2.1–2.3). Calculating the inverse matrix of L, we obtain

$\begin{array}{l}{L}^{-1}=\left(\begin{array}{c}1\\ -l_1& 1\\ l_1{l}_2& -l_2& 1\\ -l_1{l}_2{l}_3& l_2{l}_3& -l_3& 1\\ \ddots & \ddots & \ddots & \ddots & \ddots \\ {(-1)}^n{\displaystyle \prod _{i=1}^n}{l}_i& {(-1)}^{n-1}{\displaystyle \prod _{i=2}^n}{l}_i& \dots & l_{n-1}{l}_n& -l_n& 1\\ \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \end{array}\right).\end{array}$

On the other hand,

$\begin{array}{l}{J}^{\left(p\right)}P\left(\lambda \right)=UL{P}^{\left(p\right)}\left(\lambda \right)=\lambda P^{\left(p\right)}\left(\lambda \right)\Rightarrow \\ \Rightarrow LUL{P}^{\left(p\right)}\left(\lambda \right)=J\left(L{P}^{\left(p\right)}\left(\lambda \right)\right)=\lambda \left(L{P}^{\left(p\right)}\left(\lambda \right)\right)=\lambda P\left(\lambda \right).\end{array}$

Consequently, we obtain the relation between the polynomials of the first kind

$\begin{array}{c}{P}^{(p)}(\lambda )=L^{-1}P(\lambda )=\left(\begin{array}{c}1\\ -l_1& 1\\ l_1{l}_2& -l_2& 1\\ -l_1{l}_2{l}_3& l_2{l}_3& -l_3& 1\\ \ddots & \ddots & \ddots & \ddots & \ddots \end{array}\right)\left(\begin{array}{c}{P}_0(\lambda )\\ P_1(\lambda )\\ P_2(\lambda )\\ P_3(\lambda )\\ ⋮\end{array}\right)\\ =\left(\begin{array}{c}{P}_0(\lambda )\\ P_1(\lambda )-l_1{P}_0(\lambda )\\ P_2(\lambda )-l_2{P}_1(\lambda )+l_1{l}_2{P}_0(\lambda )\\ P_3(\lambda )-l_3{P}_2(\lambda )+l_2{l}_3{P}_1(\lambda )-l_1{l}_2{l}_3{P}_0(\lambda )\\ ⋮\end{array}\right)\\ =\left(\begin{array}{c}{P}_0^{(p)}(\lambda )\\ P_1^{(p)}(\lambda )\\ P_2^{(p)}(\lambda )\\ P_3^{(p)}(\lambda )\\ ⋮\end{array}\right).\hfill \end{array}$

By Corollary 2.4, we obtain

$\begin{array}{ll}{P}_n^{(p)}(\lambda )=P_n(\lambda )+{\displaystyle \sum _{i=0}^{n-1}}{(-1)}^{n-i}{P}_i(\lambda ){\displaystyle \prod _{j=i+1}^n}{l}_j=P_n(\lambda )+{\displaystyle \sum _{i=0}^{n-1}}\frac{P_i(0)}{P_n(0)}{P}_i(\lambda ).& (2.13)\end{array}$

However, we can rewrite Equation (2.13) and by Christoffel–Darboux formula (1.8), we obtain

$\begin{array}{c}{P}_n^{(p)}(\lambda )=P_n(\lambda )+{\displaystyle \sum _{i=0}^{n-1}}\frac{P_i(0)}{P_n(0)}{P}_i(\lambda )=\frac{1}{a_{n+1}{P}_n(0)}{\displaystyle \sum _{i=0}^n}{P}_i(0)P_i(\lambda )=\\ =\frac{1}{P_n(0)}\frac{P_{n+1}(\lambda )P_n(0)-P_n(\lambda )P_{n+1}(0)}{\lambda }.\end{array}$

Hence, Equation (2.12) holds. This completes the proof.

In the following statements we find the connection between orthogonal polynomials, moment sequences, measures, linear functionals, m–functions and Toda lattices according to the transformation Darboux transformation without parameter of the symmetric Jacobi matrix.

Proposition 2.9. Let the symmetric Jacobi matrix J admit LU– factorization of the form (2.1–2.3) and let the symmetric Jacobi matrix J^(p) = UL be the Darboux transformation without parameter of J. Let $\text{s}={\left\{s_n\right\}}_{n=0}^{\infty }$ and ${\text{s}}^{(p)}={\left\{s_n^{(p)}\right\}}_{n=0}^{\infty }$ be the moment sequences associated with the matrices J and J^(p), respectively. Then the moment sequence ${\text{s}}^{(p)}={\left\{s_n^{(p)}\right\}}_{n=0}^{\infty }$ can be found by the following formula

$\begin{array}{ll}{s}_{n-1}^{(p)}=\frac{s_n}{b_0}\text{ for all }n\in \mathbb{N}.& (2.14)\end{array}$

Proof. Let the symmetric Jacobi matrix J admit LU–factorization of the form (2.1–2.3) and let the symmetric Jacobi matrix J^(p) = UL be its Darboux transformation without parameter. By Equation (1.5), we obtain

$\begin{array}{l}{s}_n=(e_0,J^n{e}_0)=(e_0,{(LU)}^n{e}_0)=(e_0,L{(UL)}^{n-1}U{e}_0)=\\ =(L^T{e}_0,{(J^{(p)})}^{n-1}{b}_0{e}_0)=b_0(e_0,{(J^{(p)})}^{n-1}{e}_0)=b_0{s}_{n-1}^{(p)}.\end{array}$

Consequently, the moments $s_{n-1}^{(p)}$ can be found by Equation (2.14). This completes the proof.

Corollary 2.10. Let the symmetric Jacobi matrix J admit LU–factorization of the form (2.1)–(2.3) and let the symmetric Jacobi matrix J^(p) = UL be the Darboux transformation without parameter of J. Let 𝔖 and 𝔖^(p) be the linear functionals associated with the matrices J and J^(p), respectively. Then

$\begin{array}{ll}{𝔖}^{(p)}=\frac{\lambda }{b_0}𝔖.& (2.15)\end{array}$

Proof. Let 𝔖 and 𝔖^(p) be the linear functionals associated with the symmetric Jacobi matrices J = LU and J^(p) = UL, respectively, where L and U are defined by Equations (2.1–2.3). By Equation (1.1), we obtain

${𝔖}^{(p)}({\lambda }^{n-1})=s_{n-1}^{(p)}=\frac{s_n}{b_0}=\frac{1}{b_0}𝔖({\lambda }^n)\text{ for all }n\in \mathbb{N}.$

Consequently, Equation (1.19) holds. This completes the proof.

Corollary 2.11. Let the symmetric Jacobi matrix J admit LU–factorization of the form (2.1–2.3) and let the symmetric Jacobi matrix J^(p) = UL be the Darboux transformation without parameter of J. Let dμ and dμ^(p) be the measures associated with the matrices J and J^(p), respectively. Then

$\begin{array}{ll}d{\mu }^{(p)}(\lambda )=\frac{\lambda }{b_0}d\mu (\lambda ).& (2.16)\end{array}$

Proof. Let μ and μ^(p) be the measures associated with the symmetric Jacobi matrices J = LU and J^(p) = UL, respectively, where L and U are defined by Equation (2.1–2.3). Then

${\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}{\lambda }^{n-1}d{\mu }^{(p)}(\lambda )=s_{n-1}^{(p)}=\frac{s_n}{b_0}=\frac{1}{b_0}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}{\lambda }^{n}d\mu (\lambda )\text{ for all }n\in \mathbb{N}.$

Consequently, we find transformation of the measure and Equation (2.16) holds. This completes the proof.

Proposition 2.12. Let the symmetric Jacobi matrix J admit LU– factorization of the form (2.1–2.3) and let the symmetric Jacobi matrix J^(p) = UL be the Darboux transformation without parameter of J. Let m and m^(p) be the m–functions associated with the matrices J and J^(p), respectively. Then

$\begin{array}{ll}{m}^{(p)}(z)=\frac{s_0+zm(z)}{b_0}.& (2.17)\end{array}$

Proof. By Equation (1.6)

$\begin{array}{l}{m}^{(p)}(z)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}\frac{d{\mu }^{(p)}(\lambda )}{\lambda -z}=\\ =\frac{1}{b_0}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}\frac{\lambda d\mu (\lambda )}{\lambda -z}=\frac{1}{b_0}{\displaystyle \underset{ℝ}{\int }}\frac{\lambda -z}{\lambda -z}d\mu (\lambda )+\frac{1}{b_0}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}\frac{zd\mu (\lambda )}{\lambda -z}\\ =\frac{1}{b_0}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}1d\mu (\lambda )+\frac{z}{b_0}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}\frac{d\mu (\lambda )}{\lambda -z}=\frac{s_0+zm(z)}{b_0}.\end{array}$

Hence, m–function is transformed by Equation (2.17). This completes the proof.

Toda latice. The last statement is the following theorem of this section. One is described the Toda lattice associated with the symmetric Jacobi matrices J^(p).

Theorem 2.13. Let the symmetric Jacobi matrix J admit LU–factorization of the form (2.1–2.3) and J be associated with the Toda lattice (1.9–1.11). Let the symmetric Jacobi matrix J^(p) = UL be the Darboux transformation without parameter of J. Then J^(p) is associated with the following Toda lattice

$\begin{array}{ll}{x}_k^{″}(t)=e^{x_{k-1}-x_k}-e^{x_k-x_{k+1}},& (2.18)\end{array}$

$\begin{array}{ll}{a}_k=\frac{1}{2}{e}^{\frac{x_{k-1}-x_k}{2}}\text{and }{b}_{k+1}=-\frac{1}{2}{x}_k^{\prime }.& (2.19)\end{array}$

$\begin{array}{ll}{a}_k^{\prime }=a_k(b_{k+1}-b_k)\text{and }{b}_{k+1}^{\prime }=2(a_{k+1}^2-a_k^2),a_0=0.& (2.20)\end{array}$

Furthermore, the matrix A does not change.

Proof. Let the symmetric Jacobi matrix be associated be associated with the Toda (1.9–1.11) and let J = LU, where L and U are defined by Equations (2.2, 2.3, 2.10). Consequently, the symmetric Jacobi matrix J^(p) = UL is the Darboux transformation without parameter of J. By Equation (2.9), we obtain $J_{+}=J_{+}^{(p)}$, $J_{-}=J_{-}^{(p)}$ and the matrix A does not change in the Lax pair, i.e.

$A=J_{+}-J_{-}=J_{+}^{(p)}-J_{-}^{(p)}.$

Moreover, similar to Equation (1.9–1.11), the symmetric Jacobi matrix J^(p) = UL is associated with the Toda lattice (2.18–2.20). This completes the proof.

3 Darboux transformation with parameter of the Jacobi matrix

The next step is the Darboux transformation with parameter of the symmetric Jacobi matrix J. We study the transformations of the polynomials of the first kind, m–functions, measure, moment sequence and Toda lattice, which are associated with the transformed Jacobi matrix.

3.1 𝔘𝔏–factorization

Theorem 3.1. Let J be the symmetric Jacobi matrix and let S₀ be a some real parameter. Then J admits the following 𝔘𝔏–factorization

$\begin{array}{ll}J=𝔘𝔏,& (3.1)\end{array}$

where 𝔏 and 𝔘 are lower and upper triangular matrices, respectively, which are defined by

$\begin{array}{l}𝔏=\left(\begin{array}{cccc}1& & & \\ \frac{S_0+b_0}{a_1}& 1& & \\ & \frac{S_1+b_1}{a_2}& 1& \\ & & \ddots & \ddots \end{array}\right)\\ \text{and }𝔘=\left(\begin{array}{cccc}-S_0& a_1& & \\ & -S_1& a_2\\ & & -S_2& \ddots \\ & & & \ddots \end{array}\right),& (3.2)\end{array}$

if and only if the following system is solvable

$\begin{array}{l}{S}_i(S_{i-1}+b_{i-1})=-a_i^2,S_{i-1}+b_{i-1}\ne 0\text{and }{S}_{i-1}\ne 0,\\ \text{for all }i\in \mathbb{N}.& (3.3)\end{array}$

Proof. Let J be the Jacobi matrix. Let 𝔏 and 𝔘 are defined by Equation (3.2), where the parameter S₀ ∈ ℝ \ {0, −b₀}. Calculating the product 𝔘𝔏, we obtain

$\begin{array}{c}𝔘𝔏=\left(\begin{array}{cccc}-S_0& a_1& & \\ & -S_1& a_2& \\ & & -S_2& \ddots \\ & & & \ddots \end{array}\right)\left(\begin{array}{cccc}1& & & \\ \frac{S_0+b_0}{a_1}& 1& & \\ & \frac{S_1+b_1}{a_2}& 1& \\ & & \ddots & \ddots \end{array}\right)\\ =\left(\begin{array}{cccc}{b}_0& a_1& & \\ -\frac{S_1\left(S_0+b_0\right)}{a_1}& b_1& a_2& \\ & -\frac{S_2\left(S_1+b_1\right)}{a_2}& b_2& \ddots \\ & & \ddots & \ddots \end{array}\right)\end{array}$

Comparing the product 𝔘𝔏 with the Jacobi matrix J, we obtain the system (3.3). This completes the proof.

3.2 Transformed Jacobi matrix J^(d) = 𝔘𝔏

Definition 3.2. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3). Then a transformation

$\begin{array}{l}J=𝔘𝔏\to 𝔏𝔘=J^{(d)}\end{array}$

is called a Darboux transformation with parameter of the Jacobi matrix J.

Theorem 3.3. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) with parameter S₀ ∈ ℝ\{0, −b₀}. Then the Darboux transformation with parameter of the Jacobi matrix J is the symmetric Jacobi matrix

$\begin{array}{ll}{J}^{(d)}=\left(\begin{array}{cccc}-S_0& a_1& & \\ a_1& b_0& a_2& \\ & a_2& b_1& \ddots \\ & & \ddots & \ddots \end{array}\right)& (3.4)\end{array}$

if and only if

$\begin{array}{ll}{S}_0=S_i\text{ for all }i\in \mathbb{N}.& (3.5)\end{array}$

Proof. Let J admit 𝔘𝔏—factorization of the form (3.1–3.3). Calculating the product 𝔏𝔘, we obtain

$J^{(d)}=𝔏𝔘=\left(\begin{array}{cccc}-S_0& a_1& & \\ -\frac{S_0(S_0+b_0)}{a_1}& S_0+b_0-S_1& a_2& \\ & -\frac{S_1(S_1+b_1)}{a_2}& S_1+b_1-S_2& \ddots \\ & & \ddots & \ddots \end{array}\right).$

Hence, J^(d) is the symmetric Jacobi matrix if and only if

$-S_{i-1}(S_{i-1}+b_{i-1})=a_i^2\text{ for all }i\in \mathbb{N}.$

On the other hand, by Equation (3.3), we know

$-S_i(S_{i-1}+b_{i-1})=a_i^2\text{ for all }i\in \mathbb{N}.$

Consequently, we obtain Equation (3.5). This completes the proof.

Theorem 3.4. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let J^(d) = 𝔏𝔘 be its Darboux transformation with parameter. Then the polynomials of the first kind transform by the Geronimus formula

$\begin{array}{ll}{P}_0^{(d)}(\lambda )\equiv P_0(\lambda )\text{and }{P}_i^{(d)}(\lambda )=P_i(\lambda )+\frac{S_0+b_{i-1}}{a_i}·P_{i-1}(\lambda ),i\in \mathbb{N},& (3.6)\end{array}$

where P_i and $P_i^{(d)}$ are polynomials of the first kind associated with the matrix J and J^(d), respectively.

Proof. Let J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let J^(d) = 𝔏𝔘 be its Darboux transformation with parameter. Then

$\begin{array}{l}JP(\lambda )=\lambda P(\lambda )\Rightarrow 𝔘𝔏P(\lambda )=\lambda P(\lambda )\Rightarrow 𝔏𝔘𝔏P(\lambda )=\lambda 𝔏P(\lambda )\Rightarrow \\ \Rightarrow J^{(d)}{P}^{(d)}(\lambda )=\lambda P^{(d)}(\lambda ),\end{array}$

where

$\begin{array}{c}{P}^{(d)}(\lambda )=𝔏P(\lambda )=\left(\begin{array}{cccc}1& & & \\ \frac{S_0+b_0}{a_1}& 1& & \\ & \frac{S_0+b_1}{a_2}& 1& \\ & & \ddots & \ddots \end{array}\right)\left(\begin{array}{c}{P}_0(\lambda )\\ P_1(\lambda )\\ P_2(\lambda )\\ P_3(\lambda )\\ ⋮\end{array}\right)\\ =\left(\begin{array}{c}{P}_0(\lambda )\\ P_1(\lambda )+\frac{S_0+b_0}{a_1}{P}_0(\lambda )\\ P_2(\lambda )+\frac{S_1+b_1}{a_2}{P}_1(\lambda )\\ P_3(\lambda )+\frac{S_2+b_2}{a_3}{P}_2(\lambda )\\ ⋮\end{array}\right)=\left(\begin{array}{c}{P}_0^{(d)}(\lambda )\\ P_1^{(d)}(\lambda )\\ P_2^{(d)}(\lambda )\\ P_3^{(d)}(\lambda )\\ ⋮\end{array}\right).\end{array}$

So, the polynomials of the first kind are transformed by the Geronimus formula and Equation (3.6) holds. This completes the proof.

Proposition 3.5. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let the symmetric Jacobi matrix J^(d) = 𝔏𝔘 be the Darboux transformation with parameter of J. Let $\text{s}={\left\{s_n\right\}}_{n=0}^{\infty }$ and ${\text{s}}^{(d)}={\left\{s_n^{(d)}\right\}}_{n=0}^{\infty }$ be the moment sequences associated with the matrices J and J^(d), respectively. Then the moment sequence ${\text{s}}^{(d)}={\left\{s_n^{(d)}\right\}}_{n=0}^{\infty }$ can be found by

$\begin{array}{ll}{s}_0^{(d)}=1\text{and }{s}_n^{(d)}=-S_0{s}_{n-1}\text{ for all }n\in \mathbb{N}.& (3.7)\end{array}$

Proof. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let the symmetric Jacobi matrix J^(d) = 𝔏𝔘 be its Darboux transformation with parameter. By Equation (1.5), we obtain

$s_0^{(d)}=(e_0,{(J^{(d)})}^0{e}_0)=(e_0,e_0)=1$

and

$\begin{array}{c}{s}_n^{(d)}=(e_0,{(J^{(d)})}^n{e}_0)=(e_0,{𝔏𝔘}^n{e}_0)=(e_0,𝔏{(𝔏𝔘)}^{n-1}𝔘e_0)\\ =({𝔏}^T{e}_0,{(J)}^{n-1}(-S_0)e_0)=-S_0(e_0,{(J)}^{n-1}{e}_0)=-S_0{s}_{n-1}\\ \text{ for all }n\in \mathbb{N}.\end{array}$

Hence, Equation (3.7) holds. This completes the proof.

Corollary 3.6. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let the symmetric Jacobi matrix J^(d) = 𝔏𝔘 be the Darboux transformation with parameter of J. Let $\text{s}={\left\{s_n\right\}}_{n=0}^{\infty }$ and ${\text{s}}^{(d)}={\left\{s_n^{(d)}\right\}}_{n=0}^{\infty }$ be the moment sequences associated with the matrices J and J^(d), respectively. Then

$\begin{array}{ll}{s}_1^{(d)}=-S_0.& (3.8)\end{array}$

Proof. By Equation (3.7) and s₀ = 1, we obtain

$s_1^{(d)}=-S_0{s}_0\Rightarrow s_1^{(d)}=-S_0.$

So, Equation (3.8) holds. This completes the proof.

Corollary 3.7. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let the symmetric Jacobi matrix J^(d) = 𝔏𝔘 be the Darboux transformation with parameter of J. Let 𝔖 and 𝔖^(d) be the linear functionals associated with the matrices J and J^(d), respectively. Then

$\begin{array}{ll}{𝔖}^{(d)}(p(\lambda ))=-S_0𝔖\left(\frac{p(\lambda )-p(0)}{\lambda }\right)+p(0),p(\lambda )\in ℂ\left[\lambda \right].& (3.9)\end{array}$

Proof. Let 𝔖 and 𝔖^(d) be the linear functionals associated with the symmetric Jacobi matrices J = 𝔘𝔏 and J^(d) = 𝔏𝔘, respectively, where 𝔏 and 𝔘 are defined by Equations (3.2, 3.3). By Equation (1.1), we obtain

${𝔖}^{(d)}({\lambda }^n)=s_n^{(d)}=-S_0{s}_{n-1}=-S_0𝔖({\lambda }^{n-1}),\text{ for all }n\in \mathbb{N}.$

Consequently, Equation (3.9) holds. This completes the proof.

Corollary 3.8. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3). and let the symmetric Jacobi matrix J^(d) = 𝔏𝔘 be the Darboux transformation with parameter of J. Let dμ and dμ^(d) be the measures associated with the matrices J and J^(d), respectively. Then

$\begin{array}{ll}d\mu (\lambda )=-\frac{\lambda }{S_0}d{\mu }^{(d)}(\lambda ).& (3.10)\end{array}$

Proof. Let J = 𝔘𝔏 and J^(d) = 𝔏𝔘, where the matrices 𝔏 and 𝔘 are defined by Equations (3.2, 3.3, 3.5). The measures dμ and dμ^(d) are associated with the matrices J and J^(d), respectively. Then

$-S_0{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}{\lambda }^{n-1}d\mu (\lambda )=-S_0{s}_{n-1}=s_n^{(d)}={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}{\lambda }^{n}d{\mu }^{(d)}(\lambda ).$

Consequently,

${\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}{\lambda }^{n-1}d\mu (\lambda )=-{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}{\lambda }^{n-1}\frac{\lambda }{S_0}d{\mu }^{(d)}(\lambda ).$

Hence, Equation (3.10) holds. This completes the proof.

Proposition 3.9. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let the symmetric Jacobi matrix J^(d) = 𝔏𝔘 be the Darboux transformation with parameter of J. Let m and m^(d) be m–functions associated with the matrices J and J^(d), respectively. Then

$\begin{array}{ll}{m}^{(d)}(z)=\frac{1}{z}+\frac{S_{0}m(z)}{z}.& (3.11)\end{array}$

Proof. Let J = 𝔘𝔏 and J^(d) = 𝔏𝔘, where the matrices 𝔏 and 𝔘 are defined by Equations (3.2, 3.3, 3.5). Then m–functions of the matrices J and J^(d) are related by

$\begin{array}{l}m(z)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}\frac{d\mu (\lambda )}{\lambda -z}=-\frac{1}{S_0}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}\frac{\lambda d{\mu }^{(d)}(\lambda )}{\lambda -z}\\ =-\frac{1}{S_0}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}\frac{\lambda -z}{\lambda -z}d{\mu }^{(d)(\lambda )}+\frac{1}{S_0}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}\frac{zd{\mu }^{(d)}(\lambda )}{\lambda -z}\\ =-\frac{s_0^{(d)}}{S_0}+\frac{z{m}^{(d)}(z)}{S_0}.\end{array}$

On the other hand

$\frac{z{m}^{(d)}(z)}{S_0}=m(z)+\frac{s_0^{(d)}}{S_0}\Rightarrow m^{(d)}(z)=\frac{s_0^{(d)}}{z}+\frac{S_{0}m(z)}{z}.$

By Equation (3.7), $s_0^{(d)}=1$ and Equation (3.11) holds. This completes the proof.

Toda latice. There is the last target of our investigation.

Theorem 3.10. Let the symmetric Jacobi matrix J admit 𝔘𝔏–factorization of the form (3.1–3.3) and J be associated with the Toda lattice (1.9–1.11). Let the symmetric Jacobi matrix J^(d) = 𝔏𝔘 be the Darboux transformation without parameter of J. Then J^(d) is associated with the following Toda lattice

$\begin{array}{ll}{x}_k^{″}(t)=e^{x_{k-1}-x_k}-e^{x_k-x_{k+1}},& (3.12)\end{array}$

$\begin{array}{ll}{a}_k=\frac{1}{2}{e}^{\frac{x_{k-1}-x_k}{2}},S_0=\frac{1}{2}{x}_0^{\prime }\text{and }{b}_{k-1}=-\frac{1}{2}{x}_k^{\prime }.& (3.13)\end{array}$

$\begin{array}{ll}\begin{array}{c}{a}_0=0,a_1^{\prime }=a_1(b_0+S_0),a_k^{\prime }=a_k(b_{k-1}-b_{k-2}),\\ -S_0^{\prime }=2(a_1^2-a_0^2)\text{and }{b}_{k-1}^{\prime }=2(a_{k+1}^2-a_k^2),k\in \mathbb{N}.\end{array}& (3.14)\end{array}$

Furthermore, the matrix A does not change.

Proof. Let the symmetric Jacobi matrix J be associated with the Toda lattice (1.9–1.11) and let J = 𝔘𝔏, where 𝔏 and 𝔘 are defined by Equations (3.2, 3.3, 3.5). Consequently, the symmetric Jacobi matrix J^(d) = 𝔏𝔘 is the Darboux transformation with parameter of J. By Equation (3.4), we obtain $J_{+}=J_{+}^{(d)}$, $J_{-}=J_{-}^{(d)}$ and the matrix A does not change in the Lax pair, i.e.

$A=J_{+}-J_{-}=J_{+}^{(d)}-J_{-}^{(d)}.$

Moreover, similar to Equations (1.9–1.11), the symmetric Jacobi matrix J^(d) = 𝔏𝔘 is associated with the Toda lattice (3.12–3.14). This completes the proof.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

IK: Investigation, Writing – original draft. OL: Investigation, Writing – original draft.

Funding

The author(s) declare financial support was received for the research, authorship, and/or publication of this article. IK gratefully acknowledges financial support by the German Research Foundation (DFG, grant 520952250).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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