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On Laguerre-Sobolev matrix orthogonal polynomials

  • Edinson Fuentes EMAIL logo , Luis E. Garza and Martha L. Saiz
Published/Copyright: August 6, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this manuscript, we study some algebraic and differential properties of matrix orthogonal polynomials with respect to the Laguerre-Sobolev right sesquilinear form defined by

⟨ p , q ⟩ S ≔ ∫ 0 ∞ p * ( x ) W L A ( x ) q ( x ) d x + M ∫ 0 ∞ ( p ′ ( x ) ) * W ( x ) q ′ ( x ) d x ,

where W L A ( x ) = e − λ x x A is the Laguerre matrix weight, W is some matrix weight, p and q are the matrix polynomials, M and A are the matrices such that M is non-singular and A satisfies a spectral condition, and λ is a complex number with positive real part.

Keywords: matrix orthogonal polynomials; Laguerre-Sobolev polynomials
MSC 2010: 33C45; 42C05; 46E99; 34A30

1 Introduction

Throughout this work, the ring of m × m matrices with complex entries and the linear space of polynomials of real variable x with matrix coefficients will be denoted by M n and P ( m ) ≔ P ( m ) [ x ] , respectively. Also, let us denote the identity matrix by I ∈ M m , the zero matrix by 0 ∈ M m , and the conjugate transpose of A ∈ M m by A * .

A matrix weight of order m × m supported on E , an infinite subset of the real numbers, is a matrix function W such that every element of the sequence of matrix moments { Ω n ′ } n ≥ 0 is finite, i.e., for every n ∈ N 0 ≔ { 0 , 1 , … } , the n th matrix moment defined by

Ω n ′ ≔ ⟨ W , x n I ⟩ = ∫ E W ( x ) x n I d x ∈ M m

is finite. Any matrix weight W supported on E defines a right sesquilinear form on P ( m ) × P ( m ) by

⟨ p , q ⟩ W = ⟨ p * W , q ⟩ = ∫ E p * ( x ) W ( x ) q ( x ) d x ∈ M m , p , q ∈ P ( m ) .

The aforementioned sesquilinear form satisfies the standard properties, i.e., additivity in both terms and ⟨ p A , q B ⟩ W = A * ⟨ p , q ⟩ W B , for all A , B ∈ M m and p , q ∈ P ( m ) .

A sequence of matrix polynomials { p n } n ≥ 0 in P ( m ) is said to be right-orthogonal with respect to W if each p n has degree n ∈ N 0 , its leading coefficient is non-singular, ⟨ x m I , p n ⟩ W = 0 for m = 0 , 1 , … , n − 1 , and ⟨ x n I , p n ⟩ W ∈ M m is a non-singular matrix. Moreover, we say that a sequence of matrix orthogonal polynomials (MOP) with respect to W is monic if the leading coefficient of each polynomial is the identity matrix. In such a case, the sequence is unique and there exists a unique sequence of non-singular matrices { ⟨ x n I , p n ⟩ W = ⟨ p n , p n ⟩ W } n ≥ 0 in M m .

Jódar et al. [1] introduced the Laguerre matrix polynomials as finite series solutions of the system of second-order matrix differential equations of the form

x p ″ ( x ) + ( A + ( 1 − λ x ) I ) p ′ ( x ) + λ n p ( x ) = 0 , p ∈ P ( m ) ,

where λ is a complex number with positive real part and A is a matrix in M m satisfying the following spectral conditions:

(1) A + n I is non-singular for every n ∈ N 0 .

With these conditions on λ and A , the Laguerre matrix polynomials constitute a MOP with respect to the Laguerre matrix weight given by W L A ( x ) ≔ e − λ x x A supported on the interval E = ( 0 , ∞ ) , where x A = exp ( A ln x ) for x > 0 , i.e., the associated Laguerre right sesquilinear form is given by

(2) ⟨ p , q ⟩ W L A = ∫ 0 ∞ p * ( x ) W L A ( x ) q ( x ) d x , p , q ∈ P ( m ) .

Following [1], several researchers have studied algebraic and analytic properties of Laguerre matrix polynomials (see [2–6], among others). These matrix polynomials have found applications in different branches of pure and applied mathematics such as second-order differential equations [1,7], random matrix theory [8], Laplace transforms of matrix functions [9], and series expansions of matrix functions [10,11]. Moreover, some extensions to Laguerre matrix polynomials on two variables have been considered in [12,13].

On the other hand, the Laguerre-Sobolev right sesquilinear form will be defined in this document by

(3) ⟨ p , q ⟩ S ≔ ⟨ p , q ⟩ W L A + M ⟨ p ′ , q ′ ⟩ W = ∫ 0 ∞ p * ( x ) W L A ( x ) q ( x ) d x + M ∫ 0 ∞ ( p ′ ( x ) ) * W ( x ) q ′ ( x ) d x , p , q ∈ P ( m ) ,

where W is a matrix weight supported on ( 0 , ∞ ) , M and A are the matrices in M m such that M is non-singular and A satisfies the spectral condition (1), and λ is a complex number with positive real part. The sequence of MOP associated with (3) will be called a sequence of Laguerre-Sobolev MOP.

In the scalar case, the well-known Laguerre polynomials are orthogonal with respect to the scalar weight w ( x ) = x α e − x for x > 0 and α > − 1 (see [14] for a nice summary and several references). Likewise, the Laguerre-Sobolev polynomials associated with the scalar weight have been widely studied. For instance, the case α = 0 was considered in [15], and the general case ( α > − 1 ) was studied in [16]. Some other contributions considering generalizations or modifications of Laguerre-Sobolev polynomials can be found in [17–19] among others. Finally, Laguerre-Sobolev polynomials (see [20]) constitute an interesting example of the so-called coherent pairs that were introduced in [21].

Unlike the scalar case, properties of Laguerre-Sobolev MOP have not been studied. In fact, not even general Sobolev-type matrix orthogonality has been deeply studied, although a recent contribution on this topic can be found in [22]. The great deal of applications of Laguerre matrix polynomials motivates the study of Laguerre-Sobolev matrix polynomials. For this reason, the main goal of this contribution is to study properties of the MOP associated with the Laguerre-Sobolev right sesquilinear form (3).

The organization of this manuscript is as follows. In the next section, we define Laguerre orthogonality by using the notation given Jódar et al. [1], and then we summarize some properties of the monic Laguerre MOP that we will use in the sequel. The details of their proofs can be found in [1,2,4–6,10]. In Section 3, we introduce Laguerre-Sobolev matrix polynomials associated with the Sobolev-type sesquilinear form defined by (3). Moreover, we define an auxiliary sequence of monic matrix polynomials and deduce some of their properties. These polynomials will be useful to establish some connection formulas for the Laguerre-Sobolev matrix polynomials. In the scalar case, these auxiliary polynomials are known as limit polynomials [16]. In Section 4, we study the properties of Laguerre-Sobolev polynomials taking into account a coherence condition. Coherent pairs have been widely studied in the scalar case (see, for example, [20] for many references and historic remarks). In the matrix case, a recent contribution can be found in [22]. We also deduce some algebraic properties such as connection and structure formulas, recurrence relations, and some differential properties of Laguerre-Sobolev matrix polynomials. Finally, we characterize the left-hand-side second-order differential equations satisfied by Laguerre-Sobolev matrix polynomials. The main result is that if such polynomials satisfy a second-order differential equation, then we must have that the matrix weight W must be equal to e − λ x x A + I , i.e., W must be also a Laguerre-type weight.

2 Laguerre matrix polynomials and their properties

A matrix A in M m is called hermitian if A * = A , and the set of all eigenvalues of A is called the spectrum of A , and it will be denoted by σ ( A ) . If f ( z ) and g ( z ) are the holomorphic functions of the complex variable z , which are defined in an open set Ω of the complex plane, and A ∈ M m with σ ( A ) ⊂ Ω , then from the properties of the matrix functional calculus [23, p. 588], it follows that

(4) f ( A ) g ( A ) = g ( A ) f ( A ) .

Furthermore, if Re ( μ ) > 0 for every μ ∈ σ ( A ) , the gamma matrix function Γ ( A ) is defined by [24]

Γ ( A ) ≔ ∫ 0 ∞ e − x x A − I d x ,

where x A − I = exp ( ( A − I ) ln x ) for x > 0 . The reciprocal gamma function denoted by Γ − 1 ( z ) = 1 ∕ Γ ( z ) is an entire function of the complex variable z . Then, the image of Γ − 1 ( z ) acting on A , denoted by Γ − 1 ( A ) , is a well-defined matrix. Moreover, if A satisfies (1), then Γ ( A ) is non-singular, its inverse coincides with Γ − 1 ( A ) , and

A ( A + I ) … ( A + ( n − 1 ) I ) Γ − 1 ( A + n I ) = Γ − 1 ( A ) , n ≥ 1 ,

see [25, p. 253]. Under condition (1), by (4), the aforementioned equation can be written in the form

( A ) n = Γ ( A + n I ) Γ − 1 ( A ) , n ≥ 1 ,

where ( A ) n is the matrix analogue of the Pochhammer symbol or shifted factorial, defined by

(5) ( A ) 0 ≔ I and ( A ) n ≔ A ( A + I ) … ( A + ( n − 1 ) I ) , for n ≥ 1 .

If A is a matrix in M m satisfying (1) and λ is a complex number with positive real part, the n th degree Laguerre matrix polynomial L n ( A , λ ) and the n th degree monic Laguerre matrix polynomial L n ( A , λ ) are defined by [1]

(6) L n ( A , λ ) ( x ) = ∑ k = 0 n ( − 1 ) k λ k k ! ( n − k ) ! ( A + I ) n [ ( A + I ) k ] − 1 x k and L n ( A , λ ) ( x ) = ( − 1 ) n n ! λ n L n ( A , λ ) ( x ) ,

respectively, where ( A + I ) k is as in (5). Moreover, the matrix weight W L A ( x ) = e − λ x x A supported on ( 0 , ∞ ) is called the Laguerre matrix weight, and it satisfies the following properties.

Theorem 2.1

Let A be a matrix in M m with Re ( μ ) > − 1 for every μ ∈ σ ( A ) and let λ be a complex number with positive real part.

  1. For any fixed p ∈ P ( m )

    (7) lim x → 0 + x W L A ( x ) p ( x ) = 0 , and lim x → ∞ x W L A ( x ) p ( x ) = 0 .

  2. The sequence of Laguerre matrix moments { Ω n } n ≥ 0 is well defined, and for every n ∈ N 0 , we have

    (8) Ω n = ⟨ W L A , x n I ⟩ = ∫ 0 ∞ e − λ x x A + n I d x = λ − ( n + 1 ) − A Γ ( A + ( n + 1 ) I ) .

  3. Rodrigues’ formula. The nth monic Laguerre matrix polynomial L n ( A , λ ) is given by

    (9) ( x n W L A ( x ) ) ( n ) = ( − λ ) n W L A ( x ) L n ( A , λ ) ( x ) , n ≥ 0 .

Proof

See [1] Theorems 2.1 and 3.1.□

In particular, taking n = 1 in (9), we obtain a Pearson’s equations of matrix type of the form

( x W L A ( x ) ) ′ = W L A ( x ) [ A + ( 1 − λ x ) I ] ,

which is equivalent to

(10) x ( W L A ( x ) ) ′ = W L A ( x ) [ A − λ x I ] .

Lemma 2.2

Let A be a matrix in M m satisfying (1), and let λ be a complex number with positive real part. Then, for every n ∈ N 0 ,

W L A ( x ) L n ( A , λ ) ( x ) = L n ( A , λ ) ( x ) W L A ( x ) .

Proof

From the definition, x A = exp ( A ln x ) = lim m → ∞ ∑ k = 0 m 1 k ! ( A ln x ) k , and since for every i ∈ N 0 , A − k ( A + i I ) = ( A + i I ) A − k , then A k ( A + I ) i = ( A + I ) i A k and A k [ ( A + I ) i ] − 1 = [ ( A + I ) i ] − 1 A k . As a consequence, ( A ln x ) k L n ( A , λ ) ( x ) = L n ( A , λ ) ( x ) ( A ln x ) k . The required result follows taking the limit on both sides of

□ e − λ x ∑ j = 0 m ( A ln x ) j j ! L n ( A , λ ) ( x ) = L n ( A , λ ) ( x ) e − λ x ∑ j = 0 m ( A ln x ) j j ! .

Following a similar procedure as in [1], we show that Laguerre matrix polynomials are right-orthogonal with respect to the right sesquilinear form (2) and, under certain conditions on A and λ , we also have left orthogonality.

Proposition 2.3

Let A be a matrix in M m satisfying (1) with Re ( μ ) > − 1 for every  μ ∈ σ ( A ) and let λ be a complex number with positive real part. Then, { L n ( A , λ ) } n ≥ 0 is right-orthogonal with respect to (2), i.e., for all m , n ∈ N 0 with m < n , ⟨ x m I , L n ( A , λ ) ⟩ W L A = 0 and

(11) L n ( A , λ ) ≔ ⟨ x n I , L n ( A , λ ) ⟩ W L A = n ! λ − ( 2 n + 1 ) − A Γ ( A + ( n + 1 ) I ) = n ! λ − ( 2 n + 1 ) − A ( A ) n + 1 Γ ( A )

is non-singular, where λ − A = exp ( − A log ( λ ) ) and log denotes the principal branch of the complex logarithm. Moreover, if A is hermitian and λ is positive, then for all n , m ∈ N 0 ,

⟨ L m ( A , λ ) , L n ( A , λ ) ⟩ W L A = δ m , n L n ( A , λ ) ,

where δ m , n is the Kronecker delta.

Proof

Using the Rodrigues’ formula given in (9), for n , m ∈ N 0 , we obtain

⟨ x m I , L n ( A , λ ) ⟩ W L A = 1 ( − λ ) n ∫ 0 ∞ x m ( x n W L A ( x ) ) ( n ) d x .

Integrating by parts n times and using (7), we obtain

⟨ x m I , L n ( A , λ ) ⟩ W L A = 1 λ n ∫ 0 ∞ ( x m ) ( n ) x n W L A ( x ) d x .

As a consequence, for m = 0 , 1 , … , n − 1 , ⟨ x m I , L n ( A , λ ) ⟩ W L A = 0 and if m = n , the result follows from (8). Moreover, if A is hermitian and λ > 0 , so are W L A and L n ( A , λ ) . In such a case, the result follows using Lemma 2.2 and a similar procedure.□

As in the scalar case, Laguerre matrix polynomials have the following properties whose proof can be found in [1, Theorem 3.1].

Theorem 2.4

Let A be a matrix in M m satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) , and let λ be a complex number with positive real part. The Laguerre matrix polynomials verify the following properties with L 0 ( A , λ ) ( x ) = I :

  1. Three-term recurrence relation. For n ≥ 1 , we have

    ( n + 1 ) L n + 1 ( A , λ ) ( x ) + [ x λ I − ( A + ( 2 n + 1 ) I ) ] L n ( A , λ ) ( x ) + ( A + n I ) L n − 1 ( A , λ ) ( x ) = 0 ,

    and the monic version is

    (12) λ 2 L n + 1 ( A , λ ) ( x ) + λ [ A + ( 2 n + 1 ) I − λ x I ] L n ( A , λ ) ( x ) + n ( A + n I ) L n − 1 ( A , λ ) ( x ) = 0 .

  2. Second-order differential equation. The nth Laguerre matrix polynomial L n ( A , λ ) ( x ) is a solution of the second-order differential equation

    (13) x p ″ ( x ) + ( A + ( 1 − λ x ) I ) p ′ ( x ) + λ n p ( x ) = 0 , p ∈ P ( m ) .

Further properties of Laguerre matrix polynomials can be found in [1,2,4– 6,10], among others. We state some of them without proof in the following proposition.

Proposition 2.5

Let A be a matrix in M m satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) , and let λ be a complex number with positive real part. The Laguerre matrix polynomials verify the following properties:

  1. For n ∈ N 0 , we have L n ′ ( A , λ ) ( x ) = − λ L n − 1 ( A + I , λ ) ( x ) , and for monic polynomials

    (14) L n ′ ( A , λ ) ( x ) = n L n − 1 ( A + I , λ ) ( x ) ,

    where ′ denotes the differentiation with respect to x and L n ′ ( A , λ ) ( x ) ≔ ( L n ( A , λ ) ( x ) ) ′ .

  2. For n ∈ N 0 , λ L n ( A , λ ) ( x ) = L n ′ ( A , λ ) ( x ) − L n + 1 ′ ( A , λ ) ( x ) , and for monic polynomials

    (15) L n ( A , λ ) ( x ) = 1 n + 1 L n + 1 ′ ( A , λ ) ( x ) + 1 λ L n ′ ( A , λ ) ( x ) .

  3. For n ∈ N 0 , L n ( A , λ ) ( x ) = L n ( A + I , λ ) ( x ) + L n − 1 ( A + I , λ ) ( x ) , and for monic polynomials

    (16) L n ( A , λ ) ( x ) = L n ( A + I , λ ) ( x ) + n λ L n − 1 ( A + I , λ ) ( x ) .

From (6), it is possible to obtain the value of the Laguerre matrix polynomial at x = 0 . Indeed, we have

(17) L n ( A , λ ) ( 0 ) = ( − 1 ) n λ n ( A + I ) n = ( − 1 ) n λ n Γ ( A + ( n + 1 ) I ) ( Γ ( A + I ) ) − 1 .

Note that L n ( A , λ ) ( 0 ) is non-singular since A satisfies (1). Moreover, from (11) and (17), we have the following relation:

(18) L n − 1 ( A , λ ) L n − 1 − ( A , λ ) ( 0 ) = − λ n L n ( A , λ ) L n − ( A , λ ) ( 0 ) ,

which is equivalent to

(19) L n − 1 − ( A , λ ) L n ( A , λ ) = n λ 2 ( A + n I ) .

The kernel associated with the Laguerre matrix polynomials of degree n for x , y ∈ R is defined as

K n ( x , y ) = ∑ k = 0 n L k ( A , λ ) ( x ) L k − ( A , λ ) L k * ( A , λ ) ( y ) ,

where L k − ( A , λ ) ≔ ( L k ( A , λ ) ) − 1 and L k * ( A , λ ) ( y ) ≔ ( L k ( A , λ ) ( y ) ) * . As in the scalar case, it is possible to prove the following result.

Lemma 2.6

Let K n ( 0 , 1 ) ( x , y ) = ∂ ∂ y K n ( x , y ) , and assume p is any matrix polynomial in P ( m ) with degree less or equal to n. If A is hermitian satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) and λ > 0 , then

(20) ⟨ p ( x ) , K n ( 0 , 1 ) ( x , y ) ⟩ W L A = ( p * ( y ) ) ′ , ⟨ K n ( 0 , 1 ) ( x , y ) , p ( x ) ⟩ W L A = p ′ ( y ) .

As a consequence, the sequence { K n ( x , 0 ) } n ≥ 0 is orthogonal with respect to W L A + I = x W L A . In fact, from (20), for m = 1 , 2 … , n ,

⟨ x m − 1 I , K n ( x , 0 ) ⟩ W L A + I = 0 .

Moreover, comparing coefficients

(21) K n ( x , 0 ) = L n ( A + I , λ ) ( x ) L n − ( A , λ ) L n ( A , λ ) ( 0 ) ,

when A is hermitian and λ > 0 .

3 Laguerre-Sobolev matrix polynomials

From now on, we assume that the matrix A ∈ M m satisfies condition (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) and λ is a complex number with positive real part, unless stated otherwise. The functions W L A and W are the Laguerre matrix weight and any matrix weight supported on ( 0 , ∞ ) , with associated monic MOPs { L n ( A , λ ) } n ≥ 0 and { T n } n ≥ 0 , respectively. We also write

L n ( A , λ ) = ⟨ x n I , L n ( A , λ ) ⟩ W L A and F n = ⟨ x n I , T n ⟩ W ,

where L n ( A , λ ) and F n are non-singular matrices for all n ∈ N 0 , and { Ω n } n ≥ 0 , { Ω n ′ } n ≥ 0 will denote the corresponding sequences of matrix moments.

The pair { W L A , W } defines a right sesquilinear form as in (3). This sesquilinear form generates the double sequence { s n , m } n , m ≥ 0 in M m , where for every n , m ∈ N 0 , we have

s m , n = ⟨ x m I , x n I ⟩ S = ⟨ x m I , x n I ⟩ W L A + n m M ⟨ x m I , x n I ⟩ W = Ω n + m + n m M Ω n + m − 2 ′ .

It was proved in [22] that the block matrix [ s i , j ] i , j = 0 n − 1 ∈ M m n is non-singular for every n ∈ N if and only if there exists a sequence of MOP associated with (3). In such a case, the sequence of right MOP associated with (3) is called a sequence of Laguerre-Sobolev matrix orthogonal polynomials (LSMOP). Throughout this document, we assume that such condition is satisfied. The determination of necessary and sufficient conditions on M and W that guarantee the existence of the LSMOP is still an open problem. The monic LSMOP will be denoted by { S n ( M , A , λ ) } n ≥ 0 , i.e., for m = 0 , 1 , … , n , we have

⟨ x m I , S n ( M , A , λ ) ⟩ S = δ n , m S n ,

where S n is a non-singular matrix for every n ∈ N 0 . Moreover, if we write S 1 ( M , A , λ ) ( x ) = L 1 ( A , λ ) ( x ) + K , with K ∈ M m , then

0 = ⟨ I , S 1 ( M , A , λ ) ⟩ S = ⟨ I , L 1 ( A , λ ) + K ⟩ W L A = Ω 0 K ,

and thus, K = 0 since Ω 0 is non-singular. As a consequence, S 1 ( M , A , λ ) = L 1 ( A , λ ) and S 0 ( M , A , λ ) = L 0 ( A , λ ) = I , because the polynomials are monic.

Now, we introduce an auxiliary sequence of monic matrix polynomials { R n ( A , λ ) } n ≥ 0 as the Schur complement of x n I in the matrix

Q n ( x ) = lim M − 1 → 0 M − 1 T n ( x ) = Ω 0 Ω 1 … Ω n − 1 Ω n 0 Ω 0 ′ … ( n − 1 ) Ω n − 2 ′ n Ω n − 1 ′ ⋮ ⋮ ⋱ ⋮ ⋮ 0 ( n − 1 ) Ω n − 2 ′ … ( n − 1 ) 2 Ω 2 n − 4 ′ n ( n − 1 ) Ω 2 n − 3 ′ I x I … x n − 1 I x n I ,

where lim M − 1 → 0 means that every entry of the matrix M − 1 converges to zero, and

T n ( x ) = s 0 , 0 s 1 , 0 … s n − 1 , 0 s n , 0 s 0 , 1 s 1 , 1 … s n − 1 , 1 s n , 1 ⋮ ⋮ ⋱ ⋮ ⋮ s 0 , n − 1 s 1 , n − 1 … s n − 1 , n − 1 s n , n − 1 I x I … x n − 1 I x n I , M = I 0 M ⋱ M 0 I ,

i.e.,

R n ( A , λ ) ( x ) = x n I − I x I … x n − 1 I Ξ n − 1 − 1 Ω n n Ω n − 1 ′ ⋮ n ( n − 1 ) Ω 2 n − 3 ′ ,

where

Ξ n − 1 = Ω 0 Ω 1 … Ω n − 1 0 Ω 0 ′ … ( n − 1 ) Ω n − 2 ′ ⋮ ⋮ ⋱ ⋮ 0 ( n − 1 ) Ω n − 1 ′ … ( n − 1 ) 2 Ω 2 n − 4 ′ .

The non-singularity of Ξ n − 1 follows from the existence of LSMOP.

The sequence { R n ( A , λ ) } n ≥ 0 of monic matrix polynomials satisfies the following properties.

Lemma 3.1

Let A be a matrix in M m satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) , and let λ be a complex number with positive real part. The matrix polynomial R n ( A , λ ) satisfies for n ≥ 2

  1. ⟨ I , R n ( A , λ ) ⟩ W L A = 0 .

  2. ⟨ x m I , R n ′ ( A , λ ) ⟩ W = 0 , for 0 ≤ m ≤ n − 2 , where R n ′ ( A , λ ) = ( R n ( A , λ ) ) ′ .

  3. R n ′ ( A , λ ) ( x ) = n T n − 1 ( x ) .

Proof

For the first equality, we obtain

⟨ I , R n ( A , λ ) ⟩ W L A = Ω n − [ Ω 0 Ω 1 … Ω n − 1 ] Ξ n − 1 − 1 Ω n n Ω n − 1 ′ ⋮ n ( n − 1 ) Ω 2 n − 3 ′ = Ω n − [ I 0 … 0 ] Ω n n Ω n − 1 ′ ⋮ n ( n − 1 ) Ω 2 n − 3 ′ = Ω n − Ω n = 0 .

For the second property, we have for m = 0 , 1 , … , n − 2 ,

⟨ x m I , R n ′ ( A , λ ) ⟩ W = n Ω n + m − 1 ′ − 0 Ω m ′ … ( n − 1 ) Ω n + m − 1 ′ Ξ n − 1 − 1 Ω n n Ω n − 1 ′ ⋮ n ( n − 1 ) Ω 2 n − 3 ′ = n Ω n + m − 1 ′ − 1 m + 1 0 δ 1 , m + 1 I … δ n − 1 , m + 1 I Ω n n Ω n − 1 ′ ⋮ n ( n − 1 ) Ω 2 n − 3 ′ = n Ω n + m − 1 ′ − n Ω n + m − 1 ′ = 0 .

Finally, since R n ( A , λ ) is a monic matrix polynomial of degree n , the aforementioned orthogonality condition implies that R n ′ ( A , λ ) ( x ) = n T n − 1 ( x ) .□

Note that expanding the monic polynomial R n ( A , λ ) (for n ≥ 2 ) of degree n in terms of the Laguerre-Sobolev polynomials, i.e.,

(22) R n ( A , λ ) ( x ) = S n ( M , A , λ ) ( x ) + ∑ k = 0 n − 1 S k ( M , A , λ ) ( x ) b k n , b k n ∈ M m ,

we obtain that b 0 n = 0 . In fact, using the first equality from Lemma 3.1 and (22), we obtain

0 = ⟨ I , R n ( A , λ ) ⟩ W L A = ⟨ I , R n ( A , λ ) ⟩ S = ⟨ I , S n ( M , A , λ ) ⟩ S + ∑ k = 1 n − 1 ⟨ I , S k ( M , A , λ ) ⟩ S b k n + ⟨ I , I ⟩ S b 0 n = ⟨ I , I ⟩ S b 0 n ,

which implies b 0 n = 0 since ⟨ I , I ⟩ S is non-singular. Moreover, using the third equality from Lemma (3.1) we obtain

n T n − 1 ( x ) = S n ′ ( M , A , λ ) ( x ) + ∑ k = 1 n − 1 S k ′ ( M , A , λ ) ( x ) b k n .

4 Algebraic and differential properties of Laguerre-Sobolev MOP

We begin this section with a remark about the coherence properties of Laguerre matrix polynomials. Following the definition of coherence in the scalar case [20], we will say that the matrix weights W L A and W constitute a coherent pair if their associated sequences { L n ( A , λ ) } n ≥ 0 and { T n } n ≥ 0 satisfy the following coherence relation:

(23) T n ( x ) = 1 n + 1 L n + 1 ′ ( A , λ ) ( x ) + 1 λ L n ′ ( A , λ ) ( x ) , n ≥ 1 .

Note that, since monic Laguerre-Sobolev matrix polynomials satisfy (15), it follows that (23) is satisfied if { T n } n ≥ 0 = { L n ( A , λ ) } n ≥ 0 , i.e., W ( x ) = W L A ( x ) = e − λ x x A . In such case, since R n ′ ( A , λ ) ( x ) = n T n − 1 ( x ) , from (23), we can obtain

R n ′ ( A , λ ) ( x ) = L n ′ ( A , λ ) ( x ) + n λ L n − 1 ′ ( A , λ ) ( x ) , n ≥ 2 ,

which is equivalent to

(24) R n ( A , λ ) ( x ) = L n ( A , λ ) ( x ) + n λ L n − 1 ( A , λ ) ( x ) , n ≥ 2 .

In other words, we conclude that the matrix weights W L A and W = W L A constitute an example of a matrix coherent pair. We also say that the Laguerre matrix weight is self-coherent.

4.1 Connection formulas and structure relations

Here, we present some connection and structure formulas between Laguerre-Sobolev and Laguerre polynomials, as well as some recurrence relations for the Laguerre-Sobolev polynomials. In what follows, we assume that W = W L A and { S n ( M , A , λ ) } n ≥ 0 is the LSMOP associated with (3).

Lemma 4.1

Let A be a matrix in M m satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) , and let λ be a complex number with positive real part. Then, for m = 0 , … , n and n ≥ 2 , we have

(25) ⟨ x m I , R n ( A , λ ) ⟩ S = 0 , if 0 ≤ m ≤ n − 2 , n λ L n − 1 ( A , λ ) , if m = n − 1 , ∑ k = 0 n 1 λ n − k n ! k ! L k ( A + ( n − k ) I , λ ) + n 2 M L n − 1 ( A , λ ) , if m = n .

Proof

From Lemma 3.1 with W = W L A together with (24), we obtain

⟨ x m I , R n ( A , λ ) ⟩ S = ⟨ x m I , R n ( A , λ ) ⟩ W L A + m M ⟨ x m − 1 I , R n ′ ( A , λ ) ⟩ W L A = ⟨ x m I , L n ( A , λ ) ⟩ W L A + n λ ⟨ x m I , L n − 1 ( A , λ ) ⟩ W L A + m n M ⟨ x m − 1 I , L n − 1 ( A , λ ) ⟩ W L A .

From the orthogonality of { L n ( A , λ ) } n ≥ 0 with respect to W L A , the result is immediate for m = 0 , … , n − 1 . For m = n ,

(26) ⟨ x n I , R n ( A , λ ) ⟩ S = L n ( A , λ ) + n λ ⟨ x n I , L n − 1 ( A , λ ) ⟩ W L A + n 2 M L n − 1 ( A , λ ) .

Since x W L A = W L A + I , using (16),

⟨ x n I , L n − 1 ( A , λ ) ⟩ W L A = L n − 1 ( A + I , λ ) + n − 1 λ ⟨ x n − 1 I , L n − 2 ( A + I , λ ) ⟩ W L A + I ⋮ = ∑ k = 0 n − 1 1 λ n − k − 1 ( n − 1 ) ! k ! L k ( A + ( n − k ) I , λ ) ,

and the result follows at once from (26).□

Using the previous lemma, we can present an important relation that describes the structure of Laguerre-Sobolev matrix polynomials.

Theorem 4.2

Let A be a matrix in M m satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) , and let λ be a complex number with positive real part. Then, S 0 ( M , A , λ ) = L 0 ( A , λ ) = I , S 1 ( M , A , λ ) = L 1 ( A , λ ) , and for n ≥ 2 ,

(27) R n ( A , λ ) ( x ) = S n ( M , A , λ ) ( x ) + S n − 1 ( M , A , λ ) ( x ) b n − 1 n ,

(28) L n ( A , λ ) ( x ) + n λ L n − 1 ( A , λ ) ( x ) = S n ( M , A , λ ) ( x ) + S n − 1 ( M , A , λ ) ( x ) b n − 1 n ,

(29) n L n − 1 ( A , λ ) ( x ) = S n ′ ( M , A , λ ) ( x ) + S n − 1 ′ ( M , A , λ ) ( x ) b n − 1 n ,

where

(30) b n − 1 n = n λ S n − 1 − 1 L n − 1 ( A , λ ) ∈ M m .

Proof

From (22) R n ( A , λ ) ( x ) = S n ( M , A , λ ) ( x ) + ∑ k = 1 n − 1 S k ( M , A , λ ) ( x ) b k n and using (25), we obtain

0 = ⟨ x I , R n ( A , λ ) ⟩ S = S 1 b 1 n , ⋮ 0 = ⟨ x n − 2 I , R n ( A , λ ) ⟩ S = S n − 2 b n − 2 n , n λ L n − 1 ( A , λ ) = ⟨ x n − 1 I , R n ( A , λ ) ⟩ S = S n − 1 b n − 1 n .

This implies b k n = 0 for 1 ≤ k ≤ n − 2 and b n − 1 n = n λ S n − 1 − 1 L n − 1 ( A , λ ) . Thus, (27) and (28) follow immediately from (24). Since R n ′ ( A , λ ) = n T n − 1 = n L n − 1 ( A , λ ) , we obtain (29).□

Next, we establish a relation between the corresponding norms.

Proposition 4.3

Let A be a hermitian matrix in M m satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) , and let λ > 0 . Then, for n ≥ 0 ,

(31) S n = L n ( A , λ ) − λ M L n ( A , λ ) A n ( M ) = L n ( A , λ ) + ( − λ ) n + 1 M L n ( A , λ ) [ ( A + I ) n ] − 1 S n ′ ( M , A , λ ) ( 0 ) ,

where A n ( M ) ≔ L n − ( A , λ ) ( 0 ) S n ′ ( M , A , λ ) ( 0 ) .

Proof

Using (14), (21), and the orthogonality of { S n ( M , A , λ ) } n ≥ 0 , we obtain

⟨ S n ( M , A , λ ) , S n ( M , A , λ ) ⟩ S = ⟨ L n ( A , λ ) , S n ( M , A , λ ) ⟩ S = L n ( A , λ ) + n M ⟨ L n − 1 ( A + I , λ ) , S n ′ ( M , A , λ ) ⟩ W L A = L n ( A , λ ) + n M ⟨ K n − 1 ( x , 0 ) L n − 1 − ( A , λ ) ( 0 ) L n − 1 − ( A , λ ) , S n ′ ( M , A , λ ) ( x ) ⟩ W L A ,

from (20), we obtain S n = L n ( A , λ ) + n M L n − 1 ( A , λ ) L n − 1 − ( A , λ ) ( 0 ) S n ′ ( M , A , λ ) ( 0 ) . From (18), we obtain the first equality, and from (11), the second one.□

Note that b n − 1 n (as in (30)), according to the above proposition, becomes

(32) b n − 1 n = n λ [ I − λ L n − 1 − ( A , λ ) M L n − 1 ( A , λ ) A n − 1 ( M ) ] − 1 .

The matrix polynomial x L n ( A , λ ) can be expressed in terms of Laguerre-Sobolev matrix polynomials.

Proposition 4.4

Let A be a matrix in M m satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) , and let λ be a complex number with positive real part. Then, for n ≥ 2 ,

(33) x L n ( A , λ ) ( x ) = R n + 1 ( M , A , λ ) ( x ) + R n ( M , A , λ ) ( x ) a n = S n + 1 ( M , A , λ ) ( x ) + S n ( M , A , λ ) ( x ) A ˜ n + S n − 1 ( M , A , λ ) ( x ) B ˜ n − 1 ,

where A ˜ n = b n n + 1 + a n , B ˜ n − 1 = b n − 1 n a n , and a n = 1 λ [ A + n I ] . Moreover, if A is hermitian and λ > 0 , then

A ˜ n = 1 λ ( ( n + 1 ) [ I − λ L n − ( A , λ ) M L n ( A , λ ) A n ( M ) ] − 1 + ( A + n I ) ) , B ˜ n − 1 = n λ 2 [ I − λ L n − 1 − ( A , λ ) M L n − 1 ( A , λ ) A n − 1 ( M ) ] − 1 ( A + n I ) .

Proof

Since for every n ∈ N 0 , we have A L n ( A , λ ) = L n ( A , λ ) A , then from (12),

λ 2 x L n ( A , λ ) ( x ) = λ 2 L n + 1 ( A , λ ) ( x ) + λ L n ( A , λ ) ( x ) ( A + ( 2 n + 1 ) I ) + n L n − 1 ( A , λ ) ( x ) ( A + n I ) = λ 2 L n + 1 ( A , λ ) ( x ) + n + 1 λ L n ( A , λ ) ( x ) + λ L n ( A , λ ) ( x ) + n λ L n − 1 ( A , λ ) ( x ) ( A + n I ) .

The required results follow from (24) and (27), respectively.□

As a consequence of the aforementioned proposition, the sequence of Laguerre-Sobolev satisfies the following recurrence relation.

Corollary 4.5

If A and λ satisfy the same hypotheses of Proposition 4.4, then the Laguerre-Sobolev matrix polynomials satisfy for n ≥ 3

x S n ( M , A , λ ) ( x ) + x S n − 1 ( M , A , λ ) ( x ) b n − 1 n = S n + 1 ( M , A , λ ) ( x ) + S n ( M , A , λ ) ( x ) E ˜ n + S n − 1 ( M , A , λ ) ( x ) F ˜ n − 1 + n λ S n − 2 ( M , A , λ ) ( x ) G ˜ n − 2 ,

where E ˜ n = b n n + 1 + a 2 n , F ˜ n − 1 = b n − 1 n a 2 n + n λ a n − 1 , and G ˜ n − 2 = b n − 2 n − 1 a n − 1 . Moreover, if A is hermitian and λ > 0 , then b n − 1 n is as in (32) and

E ˜ n = 1 λ ( ( n + 1 ) [ I − λ L n − ( A , λ ) M L n ( A , λ ) A n ( M ) ] − 1 + ( A + 2 n I ) ) , F ˜ n − 1 = n λ ( [ I − λ L n − 1 − ( A , λ ) M L n − 1 ( A , λ ) A n − 1 ( M ) ] − 1 ( A + 2 n I ) + ( A + ( n − 1 ) I ) ) , G ˜ n − 2 = n − 1 λ 2 [ I − λ L n − 2 − ( A , λ ) M L n − 2 ( A , λ ) A n − 2 ( M ) ] − 1 ( A + ( n − 1 ) I ) .

Proof

From (28) and the aforementioned proposition,

x S n ( M , A , λ ) ( x ) + x S n − 1 ( M , A , λ ) ( x ) b n − 1 n = x L n ( A , λ ) ( x ) + n λ x L n − 1 ( A , λ ) ( x ) = R n + 1 ( A , λ ) ( x ) + R n ( A , λ ) ( x ) a n + n λ I + n λ R n − 1 ( A , λ ) ( x ) a n − 1 .

Since a 2 n = a n + n λ I , we obtain the result from (27).□

4.2 Linear operator and differential properties

In this section, we again assume that W = W L A and { S n ( M , A , λ ) } n ≥ 0 is the LSMOP associated with (3) with the additional assumption M = m I , m > 0 . We will study differential properties of Laguerre-Sobolev matrix polynomials. To that end, we consider the following second-order differential operator F [ ⋅ ] defined by

F [ ⋅ ] ( x ) : P ( m ) ⟶ P ( m ) q ⟶ F [ q ] ( x ) = x q ( x ) − m F 0 [ q ] ( x ) ,

where F 0 [ ⋅ ] is an auxiliary differential operator defined by

F 0 [ ⋅ ] ( x ) : P ( m ) ⟶ P ( m ) q ⟶ F 0 [ q ] ( x ) = ( A − λ x I ) q ′ ( x ) + x q ″ ( x ) .

Note that F [ ⋅ ] is a right-hand linear operator, since for every H, K ∈ M m and p , q ∈ P ( m ) , we have F [ p H + q K ] = F [ p ] H + F [ q ] K . The same is true for F 0 [ ⋅ ] . There is a close relation between the Laguerre matrix weight and F 0 [ ⋅ ] , as the next lemma shows.

Lemma 4.6

Let A be a matrix in M m with Re ( μ ) > − 1 for every μ ∈ σ ( A ) , and let λ be a complex number with positive real part. Then, for every p , q ∈ P ( m ) , we have

(34) x [ W L A ( x ) q ′ ( x ) ] ′ = W L A ( x ) F 0 [ q ] ( x ) , x [ ( p * ) ′ ( x ) W L A ( x ) ] ′ = [ x p ″ ( x ) + ( A − λ x I ) * p ′ ( x ) ] * W L A ( x ) ,

and

(35) ⟨ ( x p ) ′ , q ′ ⟩ W L A = − ⟨ p , F 0 [ q ] ⟩ W L A , ⟨ p ′ , ( x q ) ′ ⟩ W L A = − ∫ 0 ∞ [ x p ″ ( x ) + ( A − λ x I ) * p ′ ( x ) ] * W L A ( x ) q ( x ) d x .

Moreover, if A is hermitian and λ > 0 , then

(36) x [ ( p * ( x ) ) ′ W L A ( x ) ] ′ = F 0 * [ p ] ( x ) W L A ( x ) , ⟨ p ′ , ( x q ) ′ ⟩ W L A = − ⟨ F 0 [ p ] , q ⟩ W L A .

Proof

Substituting (10) into x ( W L A q ′ ) ′ = x W L A q ″ + x ( W L A ) ′ q ′ , we obtain

x [ W L A q ′ ] ′ = x W L A q ″ + W L A ( A − λ x ) q ′ = W L A F 0 [ q ] ,

so the first equation of (34) is immediate. Furthermore, using integration by parts

⟨ ( x p ) ′ , q ′ ⟩ W L A = − ∫ 0 ∞ p * ( x ) x [ W L A ( x ) q ′ ( x ) ] ′ d x ,

the first equation follows from (35) by substituting the first equation of (34). Finally, since A − λ x I commutes with W L A (see Lemma (2.2)) the other equations from (34), (35), and (36) can be deduced in a similar way.□

Lemma 4.7

Let A be a matrix in M m with Re ( μ ) > − 1 for every μ ∈ σ ( A ) , and let λ be a complex number with positive real part. Then, for every p , q ∈ P ( m ) ,

⟨ x p , q ⟩ S = ⟨ p , F [ q ] ⟩ W L A .

Proof

Using integration by parts and the asymptotic properties (7), we obtain

⟨ x p , q ⟩ S = ∫ 0 ∞ x p * ( x ) W L A ( x ) q ( x ) d x − m ∫ 0 ∞ x p * ( x ) ( W L A ( x ) q ′ ( x ) ) ′ d x .

The required result follows from (34).□

Now, we present a differential relation between Laguerre and Laguerre-Sobolev matrix polynomials.

Proposition 4.8

Let A be a matrix in M m satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) , and λ be a complex number with positive real part. Then, for every n ∈ N 0 ,

(37) F [ S n ( m , A , λ ) ] ( x ) = L n + 1 ( A , λ ) ( x ) + L n ( A , λ ) ( x ) c n n + L n − 1 ( A , λ ) ( x ) c n − 1 n ,

where c − 1 0 = 0 and

c n − 1 n = L n − 1 − ( A , λ ) S n , c n n = L n − ( A , λ ) [ ⟨ S n + 1 ( m , A , λ ) , S n ( m , A , λ ) ⟩ S + ( b n n + 1 + a n ) * S n − ⟨ L n ( A , λ ) , L n − 1 ( A , λ ) ⟩ W L A c n − 1 n ] ,

with a n = 1 λ ( A + n I ) and b n n + 1 is as in (30). Moreover, if A is hermitian and λ > 0 , then

c n − 1 n = n λ 2 [ A + n I ] [ I − λ m A n ( m ) ] , c n n = L n − ( A , λ ) 1 λ ( ( n + 1 ) [ I − λ m ( A n ( m ) ) * ] − 1 + A + n I ) L n ( A , λ ) [ I − λ m A n ( m ) ] ,

where A n ( m ) ≔ L n − ( A , λ ) ( 0 ) S n ′ ( m , A , λ ) ( 0 ) .

Proof

We expand the n + 1 degree polynomial F [ S n ( m , A , λ ) ] as

F [ S n ( m , A , λ ) ] ( x ) = L n + 1 ( A , λ ) ( x ) + ∑ k = 0 n L k ( A , λ ) ( x ) c k n , c k n ∈ M m .

From Lemma (4.7) and the orthogonality, we obtain

c 0 n = L 0 − ( A , λ ) ⟨ L 0 ( A , λ ) , F [ S n ( m , A , λ ) ] ⟩ W L A = L 0 − ( A , λ ) ⟨ x L 0 ( A , λ ) , S n ( m , A , λ ) ⟩ s = 0 , ⋮ c n − 2 n = L n − 2 − ( A , λ ) ⟨ L n − 2 ( A , λ ) , F [ S n ( m , A , λ ) ] ⟩ W L A = L n − 2 − ( A , λ ) ⟨ x L n − 2 ( A , λ ) , S n ( m , A , λ ) ⟩ s = 0 , c n − 1 n = L n − 1 − ( A , λ ) ⟨ L n − 1 ( A , λ ) , F [ S n ( m , A , λ ) ] ⟩ W L A = L n − 1 − ( A , λ ) ⟨ x L n − 1 ( A , λ ) , S n ( m , A , λ ) ⟩ s = L n − 1 − ( A , λ ) S n , c n n = L n − ( A , λ ) [ ⟨ L n ( A , λ ) , F [ S n ( m , A , λ ) ] ⟩ W L A − ⟨ L n ( A , λ ) , L n − 1 ( A , λ ) ⟩ W L A c n − 1 n ] , = L n − ( A , λ ) [ ⟨ x L n ( A , λ ) , S n ( m , A , λ ) ⟩ S − ⟨ L n ( A , λ ) , L n − 1 ( A , λ ) ⟩ W L A c n − 1 n ] .

Hence, using (33)

⟨ x L n ( A , λ ) , S n ( m , A , λ ) ⟩ S = ⟨ S n + 1 ( m , A , λ ) , S n ( m , A , λ ) ⟩ S + A ˜ n * S n ,

substituting in the expression for c n n , we obtain the result. On the other hand, if A is hermitian and λ > 0 , then using (19) and (31), the matrix coefficient c n − 1 n becomes n λ 2 [ A + n I ] [ I − λ m A n ( m ) ] . Similarly, taking into account the orthogonality of the Laguerre and Laguerre-Sobolev matrix polynomials, we obtain c n n = L n − ( A , λ ) A ˜ n * S n ; substituting (31) and A ˜ n = b n n + 1 + a n as in Proposition 4.4, the result is immediate.□

With the notation a n = 1 λ ( A + n I ) and taking into account that a n commutes with L n + 1 ( A , λ ) for every n ∈ N 0 , the three-term recurrence relation (12) becomes

L n + 1 ( A , λ ) ( x ) = L n ( A , λ ) ( x ) ( x I − a 2 n + 1 ) − n λ L n − 1 ( A , λ ) ( x ) a n .

Substituting into (37) and assuming A and λ satisfy the same hypotheses of Proposition 4.8, we obtain

F [ S n ( m , A , λ ) ] ( x ) = L n ( A , λ ) ( x ) [ x I + c n n − a 2 n + 1 ] + L n − 1 ( A , λ ) ( x ) c n − 1 n − n λ a n .

Finally, we will express F [ S n ( m , A , λ ) ] in terms of the Laguerre-Sobolev polynomials, i.e., we present a second-order differential for the Laguerre-Sobolev matrix polynomials. To do this, we first state some results. In what follows, we denote F 0 ′ [ p ] ≔ ( F 0 [ p ] ) ′ .

Lemma 4.9

Let A be a matrix in M m satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) , and let λ be a complex number with positive real part. Then,

(38) ⟨ F 0 ′ [ p ] , q ′ ⟩ W L A = ⟨ p ′ , F 0 ′ [ q ] ⟩ W L A ,

for all p , q ∈ P ( m ) if and only if A is hermitian and λ > 0 .

Proof

Using integration by parts, we obtain for all p , q ∈ P ( m )

⟨ F 0 ′ [ p ] , q ′ ⟩ W L A = − ∫ 0 ∞ F 0 * [ p ] ( W L A q ′ ) ′ d x = − ∫ 0 ∞ [ ( p * ) ′ ( A − λ x I ) * + x ( p * ) ″ ] [ ( W L A ) ′ q ′ + W L A q ″ ] d x

and

⟨ p ′ , F 0 ′ [ q ] ⟩ W L A = − ∫ 0 ∞ ( ( p * ) ′ W L A ) ′ F 0 [ q ] d x = − ∫ 0 ∞ [ ( p * ) ′ ( W L A ) ′ + ( p * ) ″ W L A ] [ ( A − λ x I ) q ′ + x q ″ ] d x .

Taking into account that A commutes with W L A and using (10), for all p , q ∈ P ( m ) , we obtain

⟨ F 0 ′ [ p ] , q ′ ⟩ W L A − ⟨ p ′ , F 0 ′ [ q ] ⟩ W L A = ∫ 0 ∞ ( p * ) ′ [ ( A − λ x I ) − ( A − λ x I ) * ] ( W L A ) ′ q ′ d x + ∫ 0 ∞ ( p * ) ′ [ ( A − λ x I ) − ( A − λ x I ) * ] W L A q ″ d x .

Suppose (38) holds. First, taking p = q = x I and using integration by parts, we obtain

0 = ∫ 0 ∞ [ ( A − λ x I ) − ( A − λ x I ) * ] ( W L A ) ′ d x = ( λ ¯ − λ ) Ω 0 .

Since Ω 0 is non-singular, we obtain λ ∈ R , and therefore, λ > 0 . In similar way, taking p = x 2 2 I , q = x I and using integration by parts, we obtain

0 = ∫ 0 ∞ [ ( A − A * ) ] x ( W L A ) ′ d x = ( A − A * ) Ω 0 .

Thus, we conclude that A is hermitian. The converse is immediate.□

Lemma 4.10

Let A be a matrix in M m satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) and let λ be a complex number with positive real part. Then, F [ ⋅ ] is symmetric with respect to ⟨ ⋅ , ⋅ ⟩ s , i.e., for every p , q ∈ P ( m ) ,

⟨ F [ p ] , q ⟩ s = ⟨ p , F [ q ] ⟩ s ,

if and only if A is hermitian and λ > 0 .

Proof

From the definition of F [ ⋅ ] and the first equation of (35), we obtain

⟨ F [ p ] , q ⟩ S − ⟨ p , F [ q ] ⟩ S = − m ( ⟨ F 0 [ p ] , q ⟩ W L A + ⟨ p ′ , ( x q ) ′ ⟩ W L A ) + m 2 ( ⟨ F 0 ′ [ p ] , q ′ ⟩ W L A − ⟨ p ′ , F 0 ′ [ q ] ⟩ W L A ) .

Taking into account the second equality of (36), the symmetry of operator F [ ⋅ ] with respect to ⟨ ⋅ , ⋅ ⟩ s is equivalent to ⟨ F 0 ′ [ p ] , q ′ ⟩ W L A = ⟨ p ′ , F 0 ′ [ q ] ⟩ W L A for all p , q ∈ P ( m ) . As a consequence, the required result follows from Lemma 4.9.□

Proposition 4.11

Let A be a hermitian matrix in M m satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) , and let λ > 0 . Then, for every n ∈ N 0 ,

F [ S n ( m , A , λ ) ] ( x ) = S n + 1 ( m , A , λ ) ( x ) + S n ( m , A , λ ) ( x ) d n n + S n − 1 ( m , A , λ ) ( x ) d n − 1 n ,

where d − 1 0 = 0 and

d n − 1 n = n λ 2 [ I − λ m A n − 1 ( m ) ] − 1 [ A + n I ] [ I − λ m A n ( m ) ] , d n n = [ I − λ m A n ( m ) ] − 1 [ ( n + 1 ) m A n ( m ) + L n − ( A , λ ) ( c n n ) * L n − ( A , λ ) [ I − λ m A n ( m ) ] ] ,

where c n n is as in Proposition 4.8.

Proof

From Lemma 4.10, the operator F [ ⋅ ] is symmetric with respect to ⟨ ⋅ , ⋅ ⟩ s . Expanding the n + 1 degree polynomial F [ S n ( m , A , λ ) ] in terms of the Laguerre-Sobolev polynomials

F [ S n ( m , A , λ ) ] ( x ) = S n + 1 ( m , A , λ ) ( x ) + ∑ k = 0 n S k ( m , A , λ ) ( x ) d k n , d k n ∈ M m .

Proceeding as in Proposition 4.8 and taking into account that F [ ⋅ ] is symmetric, we obtain that d k n = 0 for k = 0 , 1 , … , n − 2 . Moreover,

d n − 1 n = S n − 1 − 1 ⟨ S n − 1 ( m , A , λ ) , F [ S n ( m , A , λ ) ] ⟩ S = S n − 1 − 1 S n ,

and thus, substituting (31) and using (19), the expression for d n − 1 n follows. In a similar way,

d n n = S n − 1 ⟨ S n ( m , A , λ ) , F [ S n ( m , A , λ ) ] ⟩ S = S n − 1 ⟨ F [ S n ( m , A , λ ) ] , S n ( m , A , λ ) ⟩ S .

Using (37), the orthogonality of the Laguerre-Sobolev matrix polynomials and (3), we obtain

(39) d n n = S n − 1 [ m ⟨ L n + 1 ′ ( A , λ ) , S n ′ ( m , A , λ ) ⟩ W L A + ( c n n ) * S n ] .

From (14), (21), and using property (20), we obtain

⟨ L n + 1 ′ ( A , λ ) , S n ′ ( m , A , λ ) ⟩ W L A = ( n + 1 ) ⟨ L n ( A + I , λ ) , S n ′ ( m , A , λ ) ⟩ W L A = ( n + 1 ) L n ( A , λ ) L n − ( A , λ ) ( 0 ) S n ′ ( m , A , λ ) ( 0 ) = ( n + 1 ) L n ( A , λ ) A n ( m ) .

Substituting into (39) and using (31), we obtain the required result.□

4.3 Second-order differential equations

In the existing literature on MOP on the real line, it has been possible to find several examples of MOP satisfying a left-hand-side second-order differential equations of the form

(40) A 2 ( x ) p ″ ( x ) + A 1 ( x ) p ′ ( x ) + A 0 p ( x ) = p ( x ) Γ n , p ∈ P ( m ) ,

where A 2 , A 1 , and A 0 are the matrix polynomials in P ( m ) (which do not depend on n ) of degrees not bigger than 2, 1 and 0, respectively, and Γ n is a matrix in M m , which depends on n (see [26–30], among others).

Thus, a natural question is to ask if there are sequences of LSMOP associated with the sesquilinear form (3) such that they satisfy a second-order differential equation of the form (40). In particular, taking { S n ( M , A , λ ) } n ≥ 0 = { L n ( A , λ ) } n ≥ 0 and since M is non-singular, we obtain for m = 0 , 1 , … , n − 1

⟨ x m − 1 W , L n ′ ( A , λ ) ⟩ = 0 .

As a consequence, { L n ′ ( A , λ ) } n ≥ 0 is an MOP with respect to W and since L n ′ ( A , λ ) = n L n − 1 ( A + I , λ ) , we can infer that W = x W L A = W L A + I , i.e., W must be a Laguerre matrix weight. We conclude that taking W = W L A + I , the sequence { L n ( A , λ ) } n ≥ 0 is an LSMOP associated with (3), and it satisfies a second-order differential equation of the form (40) with

(41) A 2 ( x ) = x I , A 1 ( x ) = A + ( 1 − λ x ) I , A 0 = 0 , and Γ n = − λ n I .

In fact, if A 0 is any matrix in M m such that A 0 L n ( A , λ ) = L n ( A , λ ) A 0 , for all n ∈ N 0 , then { L n ( A , λ ) } n ≥ 0 satisfies (40) with Γ n = A 0 − λ n I . Note that there exist many different choices for A 0 ; for instance, we can take A 0 = A n of any linear combination of { I , A , … , A n } with constant coefficients. On the other hand, the following result characterizes the uniqueness of a matrix polynomial sequence as a solution of a second-order differential equation in terms of the eigenvalues Γ n .

Proposition 4.12

[31, Proposition 3.6] Let { p n } n ≥ 0 be a sequence of monic matrix polynomials where p n is exactly of degree n and p n satisfies the second-order differential equation (40), such that Γ n p n = p n Γ n for every n ∈ N 0 . The following statements are equivalent:

  1. det ( Γ n − Γ k ) ≠ 0 , for n ≠ k , n , k ∈ N 0 .

  2. For each n ∈ N 0 , p n is the unique n-degree matrix polynomial satisfying (40).

For Laguerre MOP, we have Γ n = − λ n I , and thus,

det ( Γ n − Γ k ) = λ m ( k − n ) m .

As a consequence, from Proposition 4.12, L n ( A , λ ) is the unique matrix polynomial solution of (13) with A 2 , A 1 , A 0 , and Γ n as in (41).

Unfortunately, { L n ( A , λ ) } n ≥ 0 is the only sequence of LSMOP satisfying a second-order differential equation of the form (40), i.e., if a sequence of Laguerre-Sobolev matrix polynomials satisfies (40), then A 2 , A 1 , and A 0 must be as in (41). To prove this statement, we need the following auxiliary result, whose proof can be found in [28].

Lemma 4.13

[28, Lemma 2.3] Let W be a matrix weight. If the sequence of matrix polynomials orthogonal with respect to W satisfies a second-order differential equations of the form (40) with A 0 = 0 , then W satisfies

( A 2 W ) ′ = A 1 W , A 2 W = W A 2 * , a n d A 1 W = W A 1 * .

Proposition 4.14

Let A be a matrix in M m satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) , and let λ be a complex number with positive real part. If { L n ( A , λ ) } n ≥ 0 satisfies a second-order differential equation of the form (40) with A 0 = 0 , then A 2 , A 1 , and Γ n are as in (41).

Proof

Suppose { L n ( A , λ ) } n ≥ 0 satisfies a second-order differential equation of the form (40) and

A 2 ( x ) = A 22 x 2 + A 21 x + A 20 and A 1 ( x ) = A 11 x + A 10 .

With this notation, since { L n ( A , λ ) } n ≥ 0 is monic, then Γ n = n ( ( n − 1 ) A 22 + A 21 ) . Thus, using Lemma 4.13 and since x W L A = W L A + I , we have

A 22 ( x W L A + I ) ′ + A 21 ( x W L A ) ′ + A 20 ( x W L A − I ) ′ = x A 11 W L A + A 10 W L A .

Using (9) and multiplying for x ,

− λ ( A 22 x 2 W L A L 1 ( A + I , λ ) + A 21 x W L A L 1 ( A , λ ) + A 20 W L A L 1 ( A − I , λ ) ) = A 11 x 2 W L A + A 10 x W L A .

Comparing coefficients of x 3 and x 0 , we obtain that A 22 = A 20 = 0 . Moreover, comparing coefficients of x 2 and x , we obtain that A 11 = − λ A 21 and A 10 = A 21 ( A + I ) . As a consequence, (40) becomes

(42) A 21 x L n ″ ( A , λ ) ( x ) + ( A 11 x + A 10 ) L n ′ ( A , λ ) ( x ) = n L n ( A , λ ) ( x ) A 11 .

On the other hand, from (13), equation (42) becomes

− A 21 ( ( A + ( 1 − λ x ) I ) L n ′ ( A , λ ) ( x ) + λ n L n ( A , λ ) ( x ) ) + ( A 11 x + A 10 ) L n ′ ( A , λ ) ( x ) = n L n ( A , λ ) ( x ) A 11 ,

and comparing the leading coefficients, it follows that A 21 = I . Hence, the required result follows from (42).□

Finally, we use a more general result on differential properties of MOP associated with Sobolev-type sesquilinear forms to obtain more information about the Laguerre-Sobolev case considered here.

Proposition 4.15

[31, Proposition 4.11] Suppose that { S n ( M ) } n ≥ 0 satisfies a second-order differential equation of the form (40), and it is orthogonal with respect to the Laguerre-Sobolev sesquilinear form

⟨ p , q ⟩ S = ⟨ p , q ⟩ W 0 + M ⟨ p ′ , q ′ ⟩ W 1 , p , q ∈ P ( m ) ,

where W 0 and W 1 are the matrix weights and { P n } n ≥ 0 is the  MOP associated with W 0 . If A 2 W 0 = W 0 A 2 * , then P n satisfies a second-order differential equation of the form (40). Moreover, if P n is the unique n-degree matrix polynomial satisfying (40), then S n ( M ) = P n , for every n ∈ N 0 .

Assuming that { S n ( M , A , λ ) } n ≥ 0 satisfies a second-order differential equation of the form (40) with A 2 W L A = W L A A 2 * and it is orthogonal with respect to (3), then from Proposition 4.15, the n -degree Laguerre matrix polynomial satisfies (40). Moreover, from the second part of Propositions 4.15 and 4.14, it follows that S n ( M , A , λ ) = L n ( A , λ ) , for every n ∈ N 0 , Therefore, W = x W L A . We summarize these findings in the following proposition.

Proposition 4.16

Let A be a matrix in M m satisfying (1) with Re ( μ ) > − 1 for every μ ∈ σ ( A ) and let λ be a complex number with positive real part. Assume that { S n ( M , A , λ ) } n ≥ 0 is the LSMOP associated with (3). Then, { S n ( M , A , λ ) } n ≥ 0 satisfies a second-order differential equations of the form (40) with A 0 = 0 and A 2 W L A = W L A A 2 * if and only if S n ( M , A , λ ) = L n ( A , λ ) , for every n ∈ N 0 .

Acknowledgment

The authors are grateful for the reviewers’ valuable comments that improved our manuscript.

  1. Funding information: The work of the first author was supported by the Dirección General de Investigaciones de la Universidad de los Llanos con código de proyecto C03-F02-009-2022. The work of the second author was supported by Universidad de Colima, Fortalecimiento de la Investigación Científica 2023, and Conahcyt Grant CFB2023-2024-625.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. EF prepared the manuscript with contributions from all co-authors.

  3. Conflict of interest: The authors state no conflicts of interest.

References

[1] L. Jódar, R. Company, and E. Navarro, Laguerre matrix polynomials and systems of second-order differential equations, Appl. Numer. Math. 15 (1994), no.1, 53–63, DOI: https://doi.org/10.1016/0168-9274(94)00012-3. 10.1016/0168-9274(94)00012-3Search in Google Scholar

[2] R. S. Batahan and A. A. Bathanya, On generalized Laguerre matrix polynomials, Acta Univ. Sapientiae Math. 6 (2014), no. 2, 121–134, DOI: https://doi.org/10.2478/ausm-2018-0003. 10.2478/ausm-2018-0003Search in Google Scholar

[3] L. Jódar and J. Sastre, On Laguerre matrix polynomials, Util. Math. 53 (1998), 37–48, DOI: https://utilitasmathematica.com/index.php/Index/article/view/114. Search in Google Scholar

[4] L. Jódar and J. Sastre, Asymptotic expressions of normalized Laguerre matrix polynomials on bounded intervals, Util. Math. 65 (2004), 3–31. Search in Google Scholar

[5] J. Sastre and E. Defez, On the asymptotics of Laguerre matrix polynomials for large x and n, Appl. Math. Lett. 19 (2006), no. 8, 721–727, DOI: https://doi.org/10.1016/j.aml.2005.10.003. 10.1016/j.aml.2005.10.003Search in Google Scholar

[6] A. Shehata, Some relations on Laguerre matrix polynomials, Malays. J. Math. Sci. 9 (2015), no. 3, 443–462. Search in Google Scholar

[7] L. Jódar, and R. Company, Hermite matrix polynomials and second-order matrix differential equations, Approx. Theory Appl. (N.S.) 12 (1996), no. 2, 20–30, DOI: https://doi.org/10.1007/BF02836202. 10.1007/BF02836202Search in Google Scholar

[8] M. Vanlessen, Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory, Constr. Approx. 25 (2007), no. 2, 125–175, DOI: https://doi.org/10.1007/s00365-005-0611-z. 10.1007/s00365-005-0611-zSearch in Google Scholar

[9] J. Sastre, E. Defez, and L. Jódar, Application of Laguerre matrix polynomials to the numerical inversion of Laplace transforms of matrix functions, Appl. Math. Lett. 24 (2011), no. 9, 1527–1532, DOI: https://doi.org/10.1016/j.aml.2011.03.039. 10.1016/j.aml.2011.03.039Search in Google Scholar

[10] J. Sastre, E. Defez, and L. Jódar, Laguerre matrix polynomial series expansion: theory and computer applications, Math. Comput. Modelling 44 (2006), no. 11–12, 1025–1043, DOI: https://doi.org/10.1016/j.mcm.2006.03.006. 10.1016/j.mcm.2006.03.006Search in Google Scholar

[11] J. Sastre and L. Jódar, On Laguerre matrix polynomial series, Util. Math. 71 (2006), 109–130. Search in Google Scholar

[12] M. G. Bin-Saad, A note on two variable Laguerre matrix polynomials, J. Assoc. Arab Univ. Basic Appl. Sci. 24 (2017), 271–276, DOI: https://doi.org/10.1016/j.jaubas.2016.09.001. 10.1016/j.jaubas.2016.09.001Search in Google Scholar

[13] S. Khan and N. A. M. Hassan, 2-variable Laguerre matrix polynomials and Lie-algebraic techniques, J. Phys. A 43 (2010), no. 23, 235204, DOI: https://doi.org/10.1088/1751-8113/43/23/235204. 10.1088/1751-8113/43/23/235204Search in Google Scholar

[14] T. S. Chihara, An introduction to orthogonal polynomials, Mathematical Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Search in Google Scholar

[15] J. Brenner, Über eine Erweiterung des Orthogonalitätsbegriffes bei Polynomen, Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory), Budapest, 1969, pp. 77–83. Search in Google Scholar

[16] F. Marcellán, T. E. Pérez, and M. A. Piñar, Laguerre-Sobolev orthogonal polynomials, J. Comput. Appl. Math. 71 (1996), no. 2, 245–265, DOI: https://doi.org/10.1016/0377-0427(95)00234-0. 10.1016/0377-0427(95)00234-0Search in Google Scholar

[17] H. Dueñas and F. Marcellán, The Laguerre-Sobolev-type orthogonal polynomials Holonomic equation and electrostatic interpretation, Rocky Mountain J. Math. 41 (2011), no. 1, 95–131, DOI: https://doi.org/10.1216/RMJ-2011-41-1-95. 10.1216/RMJ-2011-41-1-95Search in Google Scholar

[18] R. Koekoek and H. G. Meijer, A generalization of Laguerre polynomials, SIAM J. Math. Anal. 24 (1993), no. 3, 768–782, DOI: https://doi.org/10.1137/0524047. 10.1137/0524047Search in Google Scholar

[19] C. Markett, On the differential equation for the Laguerre-Sobolev polynomials, J. Approx. Theory 247 (2019), 48–67, DOI: https://doi.org/10.1016/j.jat.2019.07.007. 10.1016/j.jat.2019.07.007Search in Google Scholar

[20] F. Marcellán and Y. Xu, On Sobolev orthogonal polynomials, Expo. Math. 33 (2015), no. 3, 308–352, DOI: https://doi.org/10.1016/j.exmath.2014.10.002. 10.1016/j.exmath.2014.10.002Search in Google Scholar

[21] A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna, On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory 65 (1991), no. 2, 151–175, DOI: https://doi.org/10.1016/0021-9045(91)90100-O. 10.1016/0021-9045(91)90100-OSearch in Google Scholar

[22] E. Fuentes and L .E. Garza, Coherent pairs and Sobolev-type orthogonal polynomials on the real line: an extension to the matrix case, J. Math. Anal. Appl. 518 (2023), no. 1, 126674, DOI: https://doi.org/10.1016/j.jmaa.2022.126674. 10.1016/j.jmaa.2022.126674Search in Google Scholar

[23] N. Dunford and J. Schwartz, Linear Operators, 1st edition, Interscience, New York, 1963. Search in Google Scholar

[24] L. Jódar and J. C. Cortés, Some properties of gamma and beta matrix functions, Appl. Math. Lett. 11 (1998), no. 1, 89–93, DOI: https://doi.org/10.1016/S0893-9659(97)00139-0. 10.1016/S0893-9659(97)00139-0Search in Google Scholar

[25] E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, New York, 1969. Search in Google Scholar

[26] R. Álvarez-Nodarse, A. J. Durán, and A. M. de los Ríos, Orthogonal matrix polynomials satisfying second-order difference equations, J. Approx. Theory 169 (2013), 40–55, DOI: https://doi.org/10.1016/j.jat.2013.02.003. 10.1016/j.jat.2013.02.003Search in Google Scholar

[27] M. M. Castro and F. A. Grünbaum, Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples, J. Nonlinear Math. Phys. 12 (2005), 63–76, DOI: https://doi.org/10.2991/jnmp.2005.12.s2.6. 10.2991/jnmp.2005.12.s2.6Search in Google Scholar

[28] A. J. Durán, Matrix inner product having a matrix symmetric second-order differential operator, Rocky Mountain J. Math. 27 (1997), no. 2, 585–600, DOI: https://doi.org/10.1216/rmjm/1181071926. 10.1216/rmjm/1181071926Search in Google Scholar

[29] A. J. Durán and F. A. Grünbaum, Survey on orthogonal matrix polynomials satisfying second-order differential equations, J. Comput. Appl. Math. 178 (2005), no. 1–2, 169–190, DOI: https://doi.org/10.1016/j.cam.2004.05.023. 10.1016/j.cam.2004.05.023Search in Google Scholar

[30] A. J. Durán and F. A. Grünbaum, Orthogonal matrix polynomials satisfying second-order differential equations, Int. Math. Res. Not. 2004 (2004), no. 10, 461–484, DOI: https://doi.org/10.1155/S1073792804132583. 10.1155/S1073792804132583Search in Google Scholar

[31] E. Fuentes, L. E. Garza, and M. L. Saenz, On Sobolev-type matrix orthogonality and eigenvectors of second-order differential operators, submitted. Search in Google Scholar
Received: 2024-02-05
Accepted: 2024-06-09
Published Online: 2024-08-06

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2024-0029
Keywords for this article
matrix orthogonal polynomials; Laguerre-Sobolev polynomials
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  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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