Matrix exceptional Laguerre polynomials

We give an analog of exceptional polynomials in the matrix-valued setting by considering suitable factorizations of a given second-order differential operator and performing Darboux transformations. Orthogonality and density of the exceptional sequence are discussed in detail. We give an example of matrix-valued exceptional Laguerre polynomials of arbitrary size.     

Laguerre polynomials

The square notation for proportions is used to investigate a question arising from the theory of statistics.     

Moment representations of the exceptional X1‐Laguerre orthogonal polynomials

Exceptional orthogonal Laguerre polynomials can be viewed as an extension of the classical Laguerre polynomials per excluding polynomials of certain order(s) from being eigenfunctions for the corresponding exceptional differential operator. We are interested in the (so‐called) Type I X₁‐Laguerre polynomial sequence , and , where the constant polynomial is omitted. We derive two representations for the polynomials in terms of moments by using determinants. The first representation in terms of the canonical moments is rather cumbersome. We introduce adjusted moments and find a second, more elegant formula. We deduce a recursion formula for the moments and the adjusted ones. The adjusted moments are also expressed via a generating function. We observe a certain detachedness of the first two moments from the others.     

On Darboux and singular Darboux transformations ofG-unitons

We present a unified method for constructing generalized flag transformations of unitons into a Lie groupG via singular Darboux transformations. These flag transformations provide the possibility to establish some factorizations forG-unitons.     

Two-sided Darboux transformations

We refine the notion of a Darboux transformation for differential operators of an order higher than two and consider commutative rings of differential operators that are invariant under the group of dilations of the independent variable.     

On Darboux and singular Darboux transformations of G-unitons

We present a unified method for constructing generalized flag transformations of unitons intoa Lie group G via singular Darboux transformations. These flag transformations provide the possibility toestablish some factorizations for G-unitons.     

Higher-order Darboux transformations with foreign auxiliary equations and equivalence with generalized Darboux transformations

We show that a recently developed modified Darboux transformation that uses foreign auxiliary equations, can be unified with the Darboux transformation for generalized Schrödinger equations. As a consequence of this unification, we obtain explicit Darboux transformations with foreign auxiliary equations of arbitrary order.     

On Darboux and singular Darboux transformations ofG-unitons

We present a unified method for constructing generalized flag transformations of unitons into a Lie groupG via singular Darboux transformations. These flag transformations provide the possibility to establish some factorizations forG-unitons.     

Widened derangements and generalized Laguerre polynomials

The Ramanujan Journal - Let $$D_h$$ , $$E_k$$ and $$F_{alpha }$$ be sets of size $$h,k,alpha $$ respectively, with $$k le h$$ . We define a strongly widened derangement to be a permutation of...     

Asymptotic estimates for Laguerre polynomials

We give a brief summary of recent results concerning the asymptotic behaviour of the Laguerre polynomialsL _n ⁽⁾ (x). First we summarize the results of a paper of Frenzen and Wong in whichn and >–1 is fixed. Two different expansions are needed in that case, one with aJ-Bessel function and one with an Airy function as main approximant. Second, three other forms are given in which is not necessarily fixed. These results follow from papers of Dunster and Olver, who considered the expansion of Whittaker functions. Again Bessel and Airy functions are used, and in another form the comparison function is a Hermite polynomial. A numerical verification of the new expansion in terms of the Hermite polynomial is given by comparing the zeros of the approximant with the related zeros of the Laguerre polynomial.     

Weighted permutation problems and Laguerre polynomials

The connection between integrals of products of Laguerre polynomials, power series coefficients of certain rational functions of several variables, and certain numbers of weighted permutation problems is investigated. A combinatorial proof of our main result would be very desirable since this could lead the way to more general result, q-analogs, and perhaps even a q-analog of MacMahon's Master Theorem.     

Differential equations for deformed Laguerre polynomials

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite size may be expressed in terms of a solution of the fifth Painlevé transcendent. The generating function of a certain discontinuous linear statistic of the Laguerre unitary ensemble can similarly be expressed in terms of a solution of the fifth Painlevé equation. The methodology used to derive these results rely on two theories regarding differential equations for orthogonal polynomial systems, one involving isomonodromic deformations and the other ladder operators. We compare the two theories by showing how either can be used to obtain a characterization of a more general Laguerre unitary ensemble average in terms of the Hamiltonian system for Painlevé V.