Darboux Transformation of the Laguerre Operator

Abstract

Let ${\mathcal {L}}_{\alpha }$ be a Laguerre operator with $\alpha \not \in {\mathbb {Z}}_{-}=\{-1,-2,\ldots \}$. Constructing the pairs of the factorization operators $\mathrm {L},\mathrm {R}$ or ${\mathfrak {L}},{\mathfrak {R}}$ such that ${\mathcal {L}}_{\alpha }=\mathrm {LR}$ or ${\mathcal {L}}_{\alpha }={\mathfrak {L}}{\mathfrak {R}}$. In the first case, the Darboux transformation ${\mathbf {D}}^{\alpha +}$ of ${\mathcal {L}}_{\alpha }$ is the Laguerre operator ${\mathcal {L}}_{\alpha +1}=\mathrm {RL}$ and the eigenfunctions transform by a Christoffel formula. In the second case, the Darboux transformation ${\mathbf {D}}^{\alpha -}$ of ${\mathcal {L}}_{\alpha }$ is the Laguerre operator ${\mathcal {L}}_{\alpha -1}={\mathfrak {R}}{\mathfrak {L}}$ and the eigenfunctions transform by a Geronimus formula. The Darboux transformations ${\mathbf {D}}^{\alpha +}$ and ${\mathbf {D}}^{\alpha -}$ establish the relations between classical and non-classical Laguerre operators.

Appendix: Discrete Analog of the Darboux Transformation

As well known (see [13, 14]), the sequence of the monic Laguerre polynomials $\{\widetilde{L}_{n}(x,\alpha )\}_{n=0}^{\infty }$ satisfies a three-term recurrence relation

$$\begin{aligned} x\widetilde{L}_{n}(x,\alpha )= & {} \widetilde{L}_{n+1}(x,\alpha )+(2n+\alpha +1) \widetilde{L}_{n}(x,\alpha )\nonumber \\&+ (n+\alpha )n\widetilde{L}_{n-1}(x,\alpha )\end{aligned}$$

(5.1)

for all $n\in {\mathbb {Z}}_{+} \text{ and } \alpha \not \in {\mathbb {Z}}_{-}$, with the initial condition $\widetilde{L}_{-1}(x,\alpha )=0$.

The matrix

$$\begin{aligned} J=\left( \begin{array}{ccccc} 1+\alpha &{}\quad 1 &{} &{} &{} \\ 1+\alpha &{}\quad 3+\alpha &{}\quad 1 &{} &{} \\ &{}\quad 2(2+\alpha ) &{}\quad 5+\alpha &{}\quad 1 &{} \\ &{} &{}\quad 3(3+\alpha ) &{}\quad 7+\alpha &{}\quad \ddots \\ &{} &{} &{}\quad \ddots &{}\quad \ddots \\ \end{array}\right) \end{aligned}$$

(5.2)

is called a monic Jacobi matrix associated with the three-term recurrence relation (5.1), i.e. we mean the matrix J as a discrete Laguerre operator. On the other hand, we can rewrite the relation (5.1) as follows

$$\begin{aligned} J\mathbf{P }(x)=x\mathbf{P }(x),\end{aligned}$$

(5.3)

where $\mathbf{P }(x)=\left( \begin{array}{cccc} \widetilde{L}_{0}(x,\alpha ), &{} \widetilde{L}_{1}(x,\alpha ), &{} \widetilde{L}_{2}(x,\alpha ), &{}\ldots \\ \end{array}\right) ^{T}.$

1.1 Discrete Analog of the Darboux Transformation ${\mathbf {D}}^{\alpha +}$

Theorem 5.1

Let $\{\widetilde{L}_{n}(x,\alpha )\}_{n=0}^{\infty }$ be the sequence of the monic Laguerre polynomials with $\alpha \not \in {\mathbb {Z}}_{-}$ and let J be the monic Jacobi matrix associated with the sequence $\{\widetilde{L}_{n}(x,\alpha )\}_{n=0}^{\infty }$. Then the monic Jacobi matrix J admits the LU-factorization

$$\begin{aligned} J=LU,\end{aligned}$$

(5.4)

where

$$\begin{aligned} L=\left( \begin{array}{ccccc} 1 &{} &{} &{} &{} \\ 1 &{}\quad 1 &{} &{} &{} \\ &{}\quad 2 &{}\quad 1 &{} &{} \\ &{} &{}\quad 3 &{}\quad 1 &{} \\ &{} &{} &{} \quad \ddots &{}\quad \ddots \\ \end{array}\right) \text{ and } U=\left( \begin{array}{ccccc} 1+\alpha &{}\quad 1 &{} &{} &{} \\ &{} \quad 2+\alpha &{}\quad 1&{} &{} \\ &{} &{}\quad 3+\alpha &{}\quad 1 &{} \\ &{} &{} &{}\quad 4+\alpha &{} \quad \ddots \\ &{} &{} &{} &{}\quad \ddots \\ \end{array}\right) .\qquad \end{aligned}$$

(5.5)

Proof

The all monic Laguerre polynomials $\widetilde{L}_{n}(x,\alpha )$ do not vanish at 0, [see (1.11), (2.43)]. Then by Lemma 2.1 in [3] (or Theorem 3.5 in [6]) the monic Jacobi matrix J admits the LU-factorization of the form (5.4)–(5.5). This completes the proof.$\square$

Theorem 5.2

Let J be the monic Jacobi matrix associated with the sequence of the monic Laguerre polynomials $\{\widetilde{L}_{n}(x,\alpha )\}_{n=0}^{\infty }$ with $\alpha \not \in \mathbb {Z_{-}}$ and let $J=LU$ be its LU-factorization of the form (5.4)–(5.5). Then the matrix

$$\begin{aligned} J^{(p)}=UL\end{aligned}$$

(5.6)

is the monic Jacobi matrix associated with the sequence $\{\widetilde{L}_{n}(x,\alpha +1)\}_{n=0}^{\infty }$.

Proof

By Lemma 3.1 in [3] (or [6, Theorem 3.10]) the matrix

$$\begin{aligned} J^{(p)}=UL=\left( \begin{array}{ccccc} 2+\alpha &{}\quad 1 &{} &{} &{} \\ 2+\alpha &{}\quad 4+\alpha &{}\quad 1 &{} &{} \\ &{} \quad 2(3+\alpha ) &{} \quad 6+\alpha &{}\quad 1 &{} \\ &{} &{} \quad 3(4+\alpha ) &{}\quad 8+\alpha &{}\quad \ddots \\ &{} &{} &{}\quad \ddots &{}\quad \ddots \\ \end{array}\right) \end{aligned}$$

(5.7)

is the monic Jacobi matrix and by [6, Theorem 3.19]

$$\begin{aligned} J^{(p)}\mathbf{P }^{(p)}(x)=x\mathbf{P }^{(p)}(x), \end{aligned}$$

(5.8)

where

$$\begin{aligned} \mathbf{P }^{(p)}(x)= & {} \frac{1}{x}U\mathbf{P }(x)\\= & {} \frac{1}{x}\left( \widetilde{L}_{1}(x,\alpha )+(1+\alpha )\widetilde{L}_{0}(x,\alpha ),\right. \\&\left. \widetilde{L}_{2}(x,\alpha )+(2+\alpha )\widetilde{L}_{1}(x,\alpha ), \ldots \right) ^{T}. \end{aligned}$$

On the other hand, by Theorem 2.12

$$\begin{aligned} \mathbf{P }^{(p)}(x)= {\widetilde{L}_{0}(x,\alpha +1),\quad \widetilde{L}_{1}(x,\alpha +1), \quad \ldots }^{T}. \end{aligned}$$

(5.9)

Hence, the monic Jacobi matrix $J^{(p)}=UL$ is associated with the sequence of the monic Laguerre polynomials $\{\widetilde{L}_{n}(x,\alpha +1)\}_{n=0}^{\infty }$, i.e the Darboux transformation without parameter of the monic Jacobi matrix J of the form (5.2) transforms the monic Laguerre polynomials by the Christoffel formula (2.44). This completes the proof. $\square$

1.2 m-Function Transform for $\mathbf{D }^{\alpha +} (\alpha >-1)$

The m-function of the monic Jacobi matrix J can be found by

$$\begin{aligned} m_{\alpha }(\lambda )=\int \limits _{0}^{+\infty }\frac{\omega _{\alpha }(x)}{x-\lambda }dx, \end{aligned}$$

(5.10)

where $\omega _{\alpha }$ is the weight function defined by (1.2).

Lemma 5.3

Let $J=LU$ and $J^{(p)}=UL$ be the monic Jacobi matrices. Let $m_{\alpha }$ and $m_{\alpha +1}$ be the m-functions of J and $J^{(p)}$. Then

$$\begin{aligned} m_{\alpha +1}(\lambda )=\lambda m_{\alpha }(\lambda )+\Gamma (\alpha +1). \end{aligned}$$

(5.11)

Proof

Calculating the m-function of $J^{(p)}$

$$\begin{aligned} m_{\alpha +1}(\lambda )= & {} \int \limits _{0}^{+\infty }\frac{x^{\alpha +1}e^{-x}}{x-\lambda }dx= \int \limits _{0}^{+\infty }\frac{x^{\alpha }(x-\lambda +\lambda )e^{-x}}{x-\lambda }dx\nonumber \\= & {} \int \limits _{0}^{+\infty }\frac{x^{\alpha }(x-\lambda )e^{-x}}{x-\lambda }dx+\lambda \int \limits _{0}^{+\infty }\frac{x^{\alpha }e^{-x}}{x-\lambda }dx=\Gamma (\alpha +1)+\lambda m_{\alpha }(\lambda ).\nonumber \\ \end{aligned}$$

(5.12)

$\square$

1.3 Discrete Analog of the Darboux Transformation ${\mathbf {D}}^{\alpha -}$

As was shown in [3, 7] that the all monic Jacobi matrices admit an UL-factorization of the form

$$\begin{aligned} J=UL,\end{aligned}$$

(5.13)

where L and U are lower and upper triangular matrices defined by

$$\begin{aligned} L=\left( \begin{array}{cccc} 1 &{} &{} &{} \\ b_{1}+S_{0}&{}\quad 1 &{} &{} \\ &{} \quad b_{2}+S_{1} &{}\quad 1 &{} \\ &{} &{}\quad \ddots &{}\quad \ddots \\ \end{array}\right) \quad \text{ and } \quad U=\left( \begin{array}{cccc} -S_{0} &{}\quad 1 &{} &{} \\ &{}\quad -S_{1} &{}\quad 1 &{} \\ &{} &{}\quad -S_{2} &{}\quad \ddots \\ &{} &{} &{} \quad \ddots \\ \end{array}\right) \nonumber \\ \end{aligned}$$

(5.14)

and $S_{0}$ is the some parameter.

Lemma 5.4

Let J be the monic Jacobi matrix associated with the sequence of the monic Laguerre polynomials $\{\widetilde{L}_{n}(x,\alpha )\}_{n=0}^{\infty }$ and let $S_{0}=\alpha$ and $\alpha \not \in \mathbb {Z_{-}}$. Then the matrix J admits the UL-factorization

$$\begin{aligned} J=UL,\end{aligned}$$

(5.15)

where the matrices L and U are defined by (5.14) with

$$\begin{aligned} S_{n}=n+\alpha \quad \text{ and } \quad b_{n+1}+S_{n}=n+1\qquad n\in {\mathbb {Z}}_{+}.\end{aligned}$$

(5.16)

Proof

By [3, Proposition 2.4] (or [7, Theorem 3.6]) the matrix J admits the UL-factorization with parameter $S_{0}=\alpha$. Moreover, the matrices L and U are defined by

$$\begin{aligned} L=\left( \begin{array}{ccccc} 1 &{} &{} &{} &{} \\ 1&{}\quad 1 &{} &{} &{} \\ &{}\quad 2 &{}\quad 1 &{} &{} \\ &{} &{}\quad 3&{}\quad 1 &{} \\ &{}&{} &{}\quad \ddots &{}\quad \ddots \\ \end{array}\right) \text{ and } U=\left( \begin{array}{ccccc} \alpha &{}\quad 1 &{} &{} &{} \\ &{}\quad 1+\alpha &{}\quad 1 &{} &{} \\ &{}&{}\quad 2+\alpha &{}\quad 1 &{} \\ &{} &{} &{}\quad 3+\alpha &{}\quad \ddots \\ &{} &{} &{} &{}\quad \ddots \\ \end{array}\right) .\nonumber \\ \end{aligned}$$

(5.17)

This completes the proof. $\square$

Theorem 5.5

Let J be the monic Jacobi matrix associated with the sequence of the monic Laguerre polynomials $\{\widetilde{L}_{n}(x,\alpha )\}_{n=0}^{\infty }$ with $\alpha \not \in \mathbb {Z_{-}}$ and let $J=LU$ be its LU-factorization of the form (5.4)–(5.5). Then the matrix

$$\begin{aligned} J^{(d)}=LU\end{aligned}$$

(5.18)

is the monic Jacobi matrix associated with the sequence $\{\widetilde{L}_{n}(x,\alpha -1)\}_{n=0}^{\infty }$.

Proof

Consider the product LU of the matrices L and U

$$\begin{aligned} LU=\left( \begin{array}{ccccc} \alpha &{}\quad 1 &{} &{} &{} \\ \alpha &{}\quad 2+\alpha &{}\quad 1 &{} &{} \\ &{} \quad 2(1+\alpha ) &{}\quad 4+\alpha &{}\quad 1 &{} \\ &{} &{}\quad 3(2+\alpha ) &{}\quad 6+\alpha &{}\quad \ddots \\ &{} &{} &{}\quad \ddots &{}\quad \ddots \\ \end{array} \right) \qquad \alpha \not \in \mathbb {Z_{-}}.\end{aligned}$$

(5.19)

Therefore the matrices $J^{(d)}=LU$ is the monic Jacobi matrix and by [7, Theorem 3.8]

$$\begin{aligned} J^{(d)}\mathbf{P }^{(d)}(x)=x\mathbf{P }^{(d)}(x),\end{aligned}$$

(5.20)

where

$$\begin{aligned} \mathbf{P }^{(d)}(x)=L\mathbf{P }(x)=(\begin{array}{ccccc} \widetilde{L}_{0}(x,\alpha ), &{} \widetilde{L}_{1}(x,\alpha )+ \widetilde{L}_{0}(x,\alpha ), &{} \widetilde{L}_{2}(x,\alpha )+ 2\widetilde{L}_{1}(x,\alpha ),&{} \ldots \\ \end{array})^{T}. \end{aligned}$$

By Theorem 3.12 the relation (5.3) is rewritten as follows

$$\begin{aligned} \mathbf{P }^{(d)}(x)=(\begin{array}{ccccc} \widetilde{L}_{0}(x,\alpha -1), &{} \widetilde{L}_{1}(x,\alpha -1), &{} \widetilde{L}_{2}(x,\alpha -1),&{} \ldots \\ \end{array})^{T}, \end{aligned}$$

(5.21)

i.e the monic Jacobi matrix $J^{(d)}=LU$ is associated with the sequence of the monic Laguerre polynomials $\{\widetilde{L}_{n}(x,\alpha -1)\}_{n=0}^{\infty }$. This completes the proof. $\square$

1.4 m-Function Transform for $\mathbf{D }^{\alpha -} (\alpha >-1)$

Lemma 5.6

Let J and $J^{(d)}$ be the monic Jacobi matrices. Let $m_{\alpha }$ and $m_{\alpha +1}$ be the m-functions of J and $J^{(d)}$. Then

$$\begin{aligned} m_{\alpha }(\lambda )=\lambda m_{\alpha -1}(\lambda )+\Gamma (\alpha ). \end{aligned}$$

(5.22)

Proof

Calculating $m_{\alpha }$

$$\begin{aligned} m_{\alpha }(\lambda )= & {} \int \limits _{0}^{+\infty }\frac{x^{\alpha }e^{-x}}{x-\lambda }dx=\int \limits _{0}^{+\infty }\frac{x^{\alpha -1}(x-\lambda +\lambda )e^{-x}}{x-\lambda }dx\\= & {} \int \limits _{0}^{+\infty }\frac{x^{\alpha -1}(x-\lambda )e^{-x}}{x-\lambda }dx +\lambda \int \limits _{0}^{+\infty }\frac{x^{\alpha -1}e^{-x}}{x-\lambda }dx =\Gamma (\alpha )+\lambda m_{\alpha -1}(\lambda ). \end{aligned}$$

$\square$