Krall–Laguerre Collocation Methods

• Krall–Laguerre polynomial collocation is a spectral technique that uses modified Laguerre polynomials with Dirac masses to address differential and integral equations on [0,∞).
• It exploits determinantal structures, recurrence relations, and optimal collocation nodes to achieve high algebraic stability and efficiency in the discretization process.
• The method produces well-conditioned, high-order schemes suitable for handling boundary singularities and higher-order operator equations in applied mathematics and physics.

Krall–Laguerre polynomial collocation refers to a class of spectral and pseudospectral numerical methods that leverage the bispectral properties of Krall–Laguerre polynomials for the approximation of differential, integral, and related operator equations on the half-line $[0, \infty)$. These methods exploit the higher-order eigenstructure, orthogonality with respect to singular measures, and strong asymptotics of the polynomials, resulting in well-conditioned, high-order collocation schemes for equations with boundary singularities, convolutional operators, or higher-order derivatives.

1. Definition and Structural Properties of Krall–Laguerre Polynomials

Let $m \ge 1$ and $\alpha \in \mathbb{N}$ with $\alpha \ge m$. Krall–Laguerre polynomials $\{L_n^{\alpha, \beta}(x)\}$ are obtained by augmenting the classical Laguerre weight $x^\alpha e^{-x}$ on $[0, \infty)$ with $m$ Dirac masses (and derivatives up to order $m-1$) at $x=0$. Explicitly, the generalized measure is

$m \ge 1$0

with $m \ge 1$1 the $m \ge 1$2-th derivative of the Dirac mass at $m \ge 1$3 and $m \ge 1$4 determined from the construction parameters.

The polynomials possess a determinantal structure in terms of an $m \ge 1$5 Casorati determinant involving blocks of classical Laguerre polynomials $m \ge 1$6 and explicitly defined auxiliary polynomials $m \ge 1$7,

$m \ge 1$8

Alternatively, for each $m \ge 1$9,

$\alpha \in \mathbb{N}$0

with connection coefficients given by minors of the above determinant (Durán et al., 2019).

Krall–Laguerre polynomials are the unique polynomial systems on the real line that are orthogonal with respect to such measures involving a finite number of Dirac masses at the origin.

2. Spectral Properties: Differential Equations and Recurrence Relations

Krall–Laguerre polynomials are eigenfunctions of an even-order, $\alpha \in \mathbb{N}$1 linear differential operator,

$\alpha \in \mathbb{N}$2

where each $\alpha \in \mathbb{N}$3 is a polynomial of degree at most $\alpha \in \mathbb{N}$4, and

$\alpha \in \mathbb{N}$5

The spectrum $\alpha \in \mathbb{N}$6 is a degree-$\alpha \in \mathbb{N}$7 polynomial in $\alpha \in \mathbb{N}$8. Such operators arise via Darboux transformations of classical Laguerre operators and admit factorized forms as products of lower-order differential operators (Markett, 2017).

Recurrence relations for Krall–Laguerre polynomials take the form: $\alpha \in \mathbb{N}$9 where $\alpha \ge m$0 is any polynomial divisible by $\alpha \ge m$1 (typically $\alpha \ge m$2), and boundary coefficients $\alpha \ge m$3 are nonzero. This bispectrality underpins the high algebraic stability and enables interpretation using difference operators in the polynomial degree (Durán et al., 2019).

3. Collocation Nodes, Matrix Formulation, and Interlacing Properties

For each $\alpha \ge m$4, Krall–Laguerre polynomial $\alpha \ge m$5 has $\alpha \ge m$6 simple, positive real zeros $\alpha \ge m$7 that interlace strictly with those of $\alpha \ge m$8, and asymptotically exhibit the Bessel or scaled Laguerre zero distribution (Durán et al., 2019, Bihun, 2016, Deaño et al., 2013).

Collocation schemes discretize operator equations by enforcing them at these zeros. The collocation matrices arise as follows:

• Differentiation matrix $\alpha \ge m$9, with $\{L_n^{\alpha, \beta}(x)\}$0 a zero of $\{L_n^{\alpha, \beta}(x)\}$1
• Mass matrix $\{L_n^{\alpha, \beta}(x)\}$2

The collocation system for an operator equation $\{L_n^{\alpha, \beta}(x)\}$3 and the truncated series $\{L_n^{\alpha, \beta}(x)\}$4 is

$\{L_n^{\alpha, \beta}(x)\}$5

Boundary or side conditions at $\{L_n^{\alpha, \beta}(x)\}$6 are represented via additional rows enforcing conditions of the form $\{L_n^{\alpha, \beta}(x)\}$7 for $\{L_n^{\alpha, \beta}(x)\}$8, yielding a square system of size $\{L_n^{\alpha, \beta}(x)\}$9 (Durán et al., 2019, Markett, 2017, Bihun, 2016).

A crucial aspect is that the collocation matrix is similar to the diagonal spectral matrix of eigenvalues via an explicit similarity transformation. This links the pseudospectral differentiation representation to the classical spectral one and guarantees favorable conditioning (Bihun, 2016).

4. Algorithmic Implementation and Computational Procedures

The practical procedure for Krall–Laguerre polynomial collocation is as follows:

1. Node selection: Compute the $x^\alpha e^{-x}$0 zeros $x^\alpha e^{-x}$1 of $x^\alpha e^{-x}$2, typically via root finding on the Jacobi matrix with recurrence coefficients or Newton iteration.
2. Basis evaluation: Calculate $x^\alpha e^{-x}$3 for $x^\alpha e^{-x}$4, using either the explicit determinant formula, three-term recurrence, or high-precision asymptotic expansions (e.g., Buchholz representation in terms of Bessel functions (Deaño et al., 2013)).
3. Differentiation matrix: Assemble $x^\alpha e^{-x}$5 using the action of $x^\alpha e^{-x}$6 on basis polynomials, evaluated at node $x^\alpha e^{-x}$7.
4. Collocation system: Enforce the target operator equation at the zeros, fill the system as described above.
5. Boundary conditions: Add additional constraints corresponding to physical or mathematical requirements at the singular endpoint $x^\alpha e^{-x}$8.
6. Solution and back-transformation: Solve for the nodal or spectral coefficients. Optionally, reconstruct the solution as a sum over the spectral basis or in terms of Lagrange interpolants at the collocation nodes (Bihun, 2016, Durán et al., 2019).

In the context of integral equations (e.g., Volterra equations of the third kind), collocation can be adapted by approximating the unknown function as a finite Krall–Laguerre expansion, evaluating integrals via quadrature at the collocation points, and assembling the discrete linear system as documented in (Jami et al., 2021). For differential operators of order $x^\alpha e^{-x}$9, collocation matrices exploit the explicit differentiation of the Lagrange basis at the zeros.

5. Error Estimates, Conditioning, and Asymptotic Tools

Convergence and stability of Krall–Laguerre collocation are supported by strong asymptotic theory for the basis. For analytic input data and suitable quadrature/integration schemes, the error decays (super-)algebraically or even exponentially. For functions in weighted Sobolev spaces $[0, \infty)$0 with Laguerre-type weight, error bounds for best $[0, \infty)$1-term approximation satisfy

$[0, \infty)$2

for $[0, \infty)$3, where $[0, \infty)$4 is the degree-$[0, \infty)$5 Krall–Laguerre expansion (Jami et al., 2021, Deaño et al., 2013).

Condition numbers of collocation matrices can be estimated using Bessel asymptotics. For the classical Laguerre case,

$[0, \infty)$6

with the truncation error in collocation entries controlled via Buchholz expansions and symbolic computation of expansion coefficients (Deaño et al., 2013).

6. Applications and Extensions

Krall–Laguerre polynomial collocation is particularly suited for problems on $[0, \infty)$7 involving singularities at the origin, higher-order differential operators, and equations that naturally respect the orthogonality and boundary structure of the basis. Applications include:

• Spectral approximation of boundary-value and eigenvalue problems with singular endpoints and higher-order boundary conditions (Durán et al., 2019, Markett, 2017).
• Numerical solution of Volterra integral equations of infinite or finite type, including equations of the third kind with algebraic kernels (Jami et al., 2021).
• Analysis and construction of quadrature schemes and boundary–value matrix solvers for operators with generalized weights, point masses, or spatially varying coefficients.

Current research emphasizes bispectrality, self-adjointness, zeros distribution, explicit block structure for implementation, and the extension to other families such as Krall–Legendre and Krall–Jacobi polynomials (Durán et al., 2019, Bihun, 2016).

7. Zeros, Interlacing, and Advanced Properties

The zeros of Krall–Laguerre polynomials serve as optimal collocation nodes due to their real, simple, strictly interlacing structure and their asymptotic distribution governed by scaled zeros of Bessel functions. These zeros satisfy remarkable algebraic relations generalizing classical Christoffel–Darboux-type identities, which facilitate explicit construction of differentiation matrices, spectral–collocation similarity transforms, and high-order quadrature (Bihun, 2016).

Recent work provides symbolic-computation recipes for simultaneous expansion coefficients and matrix entries, enabling fully explicit, high-precision spectral and pseudospectral implementation in symbolic computing environments (Deaño et al., 2013).

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Key references:

• "Bispectral Laguerre type polynomials" (Durán et al., 2019)
• "A Method for Numerical Solution of Third-Kind Volterra Integral Equations Using Krall-Laguerre Polynomials" (Jami et al., 2021)
• "On the higher-order differential equations for the generalized Laguerre polynomials and Bessel functions" (Markett, 2017)
• "New Properties of the Zeros of Krall Polynomials" (Bihun, 2016)
• "Strong and ratio asymptotics for Laguerre polynomials revisited" (Deaño et al., 2013)