Juan J. Moreno-Balcázar

Asymptotic properties of generalized Laguerre orthogonal polynomials

Abstract In the present paper we deal with the polynomials L n ( α , M , N ) ( x ) orthogonal with respect to the Sobolev inner product (p,q) = 1 Г(α+1) ∫ 0 ∞ p(x)q(x)x α e −x dx + Mp(O)q(O) + Np′(O)q′(O), N, M ≥ O, α > −I , firstly introduced by Koekoek and Meijer in 1993 and extensively studied in the last years. We present some new asymptotic properties of these polynomials and also a limit relation between the zeros of these polynomials and the zeros of Bessel function J α ( x ). The results are illustrated with numerical examples. Also, some general asymptotic formulas for generalizations of these polynomials are conjectured.

Strong asymptotics for Gegenbauer-Sobolev orthogonal polynomials

We study the asymptotic behaviour of the manic orthogonal polynomials with respect to the Gegenbauer-Sobolev inner product (f, s)s = (f, g) + AU’, 9’) where (f, g) = s:, f(x)g(x)(l - x~)‘-“~ d x with CI> -i and 1>0. The asymptotics of the zeros and norms of these polynomials are also established.

Full length article: A new approach to the asymptotics of Sobolev type orthogonal polynomials

This paper deals with Mehler-Heine type asymptotic formulas for the so-called discrete Sobolev orthogonal polynomials whose continuous part is given by Laguerre and generalized Hermite measures. We use a new approach which allows to solve the problem when the discrete part contains an arbitrary (finite) number of mass points.

Ratio and Plancherel-Rotach asymptotics for Meixner-Sobolev orthogonal polynomials

Abstract We study the analytic properties of the monic Meixner–Sobolev polynomials {Qn} orthogonal with respect to the inner product involving differences (p,q) S = ∑ i=0 ∞ [p(i)q(i)+λ Δ p(i) Δ q(i)] μ i (γ) i i! , γ>0, 0 where λ⩾0, Δ is the forward difference operator (Δf(x)=f(x+1)−f(x)) and (γ)n denotes the Pochhammer symbol. Relative asymptotics for Meixner–Sobolev polynomials with respect to Meixner polynomials is obtained. This relative asymptotics is also given for the scaled polynomials. Moreover, a zero distribution for the scaled Meixner–Sobolev polynomials and Plancherel–Rotach asymptotics for {Qn} are deduced.

Asymptotics of Sobolev orthogonal polynomials for symmetrically coherent pairs of measures with compact support

We study the strong asymptotics for the sequence of manic polynomials Q&c), orthogonal with respect to the inner product U-3 9)s = s f(xMx) h(x) + 1 s f’(x)s’(x> 44X), A> 0, with x outside of the support of the measure ~2. We assume that ~1 and ~2 are symmetric and compactly supported measures on lR satisfying a coherence condition. As a consequence, the asymptotic behaviour of (Q”,Q”)s and of the zeros of Qn is obtained.

Asymptotics of Sobolev orthogonal polynomials for Hermite coherent pairs

Abstract Let Qn be the polynomials orthogonal with respect to the Sobolev inner product (f,g) S =∫ fg d μ 0 +∫ f′g′ d μ 1 , being (μ0,μ1) a coherent pair where one of the measures is the Hermite measure. The outer relative asymptotics for Qn with respect to Hermite polynomials are found. On the other hand, we consider the Sobolev scaled polynomials and we obtain the Plancherel–Rotach asymptotics for those as well as a consequence about their zeros.

Asymptotics for a generalization of Hermite polynomials

We consider a generalization of the classical Hermite polynomials by the addition of terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain Mehler{Heine type formulas for these polynomials and, as a consequence, we prove that there exists an acceleration of the convergence of the smallest positive zeros of these generalized Hermite polynomials towards the origin.

Asymptotic behavior of varying discrete Jacobi-Sobolev orthogonal polynomials

In this contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are interested in Mehler-Heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these Sobolev orthogonal polynomials and the Jacobi ones. Moreover, this asymptotic behavior provides an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other well-known special functions. We generalize some results appeared in the literature very recently.

Inner products involving q -differences: the little q Laguerre-Sobolev polynomials

In this paper, polynomials which are orthogonal with respect to the inner product 〈p,r〉S=∑k=0∞p(qk)r(qk)(aq)k(aq;q)∞(q;q)k+λ∑k=0∞(Dqp)(qk)(Dqr)(qk)(aq)k(aq;q)∞(q;q)k, where Dq is the q-difference operator, λ⩾0,0<q<1 and 0<aq<1 are studied. For these polynomials, algebraic properties and q-difference equations are obtained as well as their relation with the monic little q-Laguerre polynomials. Some properties about the zeros of these polynomials are also deduced. Finally, the relative asymptotics {Qn(x)/pn(x;a|q)}n on compact subsets of C⧹[0,1] is given, where Qn(x) is the nth degree monic orthogonal polynomial with respect to the above inner product and pn(x;a|q) denotes the monic little q-Laguerre polynomial of degree n.

Δ -Meixner-Sobolev orthogonal polynomials

We provide a Mehler-Heine type formula for a nonstandard family of discrete orthogonal polynomials. Concretely, we consider the Δ -Meixner-Sobolev polynomials which are orthogonal with respect to an inner product involving the Pascal distribution and the forward difference operator. Consequences on the zeros of these polynomials are analyzed and illustrated numerically.