We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials—the Chebyshev, Hermite, and Laguerre poly...

We discuss and compare upper and lower bounds obtained by two different methods for the positive zero of the ultraspherical polynomial C n that is greater than 1 when −3/2 < λ <−1/2. Our first approach uses mixed three term recurrence relations and interlacing of zeros while the second approach uses a method going back to Euler and Rayleigh and already applied to Bessel functions and Laguerre a...

The present paper deals with some geometric classes of analytic functions, such as starlike, convex and bounded turning property in a complex domain. These classes are defined by a new linear operator in the normalized space of analytic functions. The linear operator is introduced by a convolution of Szego function involving parametric coefficients type Laguerre polynomial, with the normalized ...

New solutions of the homogeneous wave equation of the type usually referred to as relatively undistorted waves are presented. Such solutions relate to the so-called “splash modes”, from which indeed they can be generated by applying the Laguerre polynomial operator. Accordingly, the solutions here presented resort to the relativistic Laguerre polynomials — introduced about one decade ago within...

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 ...

It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs (a, z) one has the large-n asymptotic behavior L(−a) n (−z) ∼ C(a, z)n−a/2−1/4e2 √ . We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations, yet also leads to the complete asymptoti...

Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval [0,∞) with respect to a weight function of the form w(x) = xαe−Q(x), Q(x) = m ∑ k=0 qkx , α > −1, qm > 0. The classical Laguerre polynomials correspond to Q(x) = x. The computation of higher-order terms of the asymptotic expansions of these polynomials for large degree becomes quite complicated, and a full descrip...

L (↵) n (x) for arbitrary real ↵. Such results are well-known in the case ↵ > 1. In the case 2 < ↵ < 1, we use a mixed 3-term recurrence relation to show, for example, that, apart from a single value of ↵, the (all real) zeros of (x + ↵ + 1)L n (x) interlace with those of xL n (x). By studying the changes in interlacing that occur when ↵ decreases through the negative integer values 1, 2, . . ....

We consider a fourth-order eigenvalue problem on a semi-infinite strip which arises in the study of viscoelastic shear flow. The eigenvalues and eigenfunctions are computed by a spectral method involving Laguerre functions and Legendre polynomials.