Orthogonal polynomials suggested by a queueing model

The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle

On the frequency mapping of modal parameters identification

Orthogonal matrix polynomials and higher-order recurrence relations

The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials whose properties give the structure and dynamics of the corresponding physical system. For a certain range of parameters, one of these polynomials has a mix of continuous and discrete spectra making it suitable for describing physical systems with both scattering and bound states. In this work, we define these polynomials by their recursion relations and highlight some of their properties using numerical means. Due to the prime significance of these polynomials in physics, we hope that our short expose will encourage experts in the field of orthogonal polynomials to study them and derive their properties (weight functions, generating functions, asymptotics, orthogonality relations, zeros, etc.) analytically.

We discuss connections between birth and death processes and orthogonal polynomials. We survey certain spectral properties of orthogonal polynomials and spectra of birth and death processes. Birth and death processes with linear birth and death rates are studied in some detail. We also mention some results concerning birth and death processes whose transition rates are rational functions.

In this paper, we present a Geom/Geom/1 queueing model with variable input probability. In this queueing model, an arriving customer, who sees many customers waiting for service in the queueing system, will consider whether to enter the system or not. We consider the possibility that the customer enters the system to receive service to be a probability called the "Input Probability". We derive the transition probability matrix of the birth and death chain of the queueing model. Using a birth and death process, we gain the probability distributions of the stationary queue length and the waiting time in the queueing model. Then we derive special cases of the considered model by applying different input probability distributions, which lead to several known specific queueing models. We also derive some performance measures of these specific queueing models. Therefore, this queueing model that we present in this paper has a certain extension, extending to the existing models. Finally, we compare the effect of the parameters on the stationary queue length and waiting time by using numerical results.

In this paper, we determine all pairs of orthogonal polynomial sequences $\{ p_n (x)\} $ and $\{ q_n (x)\} $, such that their convolution, \[ Q_n (x,y) = \sum_{k = 0}^n {p_k (x)q_{n - k} (y)} ,\quad n \geqq 0.\] defines $\{ Q_n (x,y)\} $ as an orthogonal polynomial sequence in x for all y. All such triples are determined explicitly in terms of their three-term recurrence formulas. Generating functions and “explicit” representation formulas are obtained. The resulting sequences are found to consist of a class of orthogonal polynomials characterized by J. Meixner (which class includes the Laguerre and Hermite polynomials) together with a new class of orthogonal polynomials which includes the orthogonal q-polynomials of Al-Salam and Carlitz. Explicit orthogonality relations are found for one new special case of this latter class.

Companion orthogonal polynomials: some applications

The weight functions, generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the Jacobi and the Gegenbauer polynomials

We consider the eigenvalue problem associated with the Dunkl-type differential operator (in which the reflection operator R is involved)in the context of supersymmetric quantum mechanical models. By solving this eigenvalue problem with the help of known exactly solvable potentials, we construct several classes of functions satisfying certain orthogonality relations. We call them the Dunkl-supersymmetric (Dunkl-SUSY) orthogonal functions. These functions can be expressed in terms of the classical orthogonal polynomials. The key feature of these functions is that they appear by pairs, i.e. Qn(x) and Qn(−x) are both the eigenfunctions of . A general formulation of the Dunkl-SUSY orthogonal polynomials is also presented.

On the orthogonality and convolution orthogonality via the Kontorovich–Lebedev transform

In his paper, ‘Random walks and orthogonal polynomials: some challenges’, F. A. Grunbaum gave the polynomials orthogonal with respect to the weight on explicitly as where and are, respectively, the Chebyshev polynomials of the first and second types. In this paper, similarly, we introduce the polynomials defined by where and are, respectively, the Chebyshev polynomials of the third and fourth types, then we give the three term recurrence relation of the polynomials . Second, we give the kernel where is the Christoffel–Darboux formula for the polynomials . Finally we give the integral of function of and we show how we deduce that is orthogonal with respect to the weight on .

We advocate the exploitation of symmetry (recurrence relation) techniques for the derivation of properties associated with families of basic hypergeometric functions, in analogy with the local Lie theory techniques for ordinary hypergeometric functions. Here these ideas are applied to the (continuous) Askey-Wilson polynomials, introduced by Askey and Wilson, to obtain a strikingly simple derivation of their orthogonality relations. 1. In t roduct ion . In [1] the authors and A.K. Agarwal introduced symmetry techniques for the study of families of basic hypergeometric functions, in analogy with the local Lie theory techniques for ordinary hypergeometric functions. The fundamental objects in this study are the recurrence relations obeyed by the families: generating functions and identities for each family are characterized in terms of the recurrence relations. The functions studied in [1] are of the form ^ / a i , . . . , a r _ y > (ai;q)k'(ar;q)k z k V 6 i , . . . , 6 8 ' ' / f^Q(bi;q)k'(b8;q)k{q;q)k and many-variable extension. Here , . _ J 1 if k = 0 ( f l ; g , ' ( l a ) ( l g a ) . . . ( l c * 1 o ) if fc = 1,2,... and \q . ( , , . ) ( . ( I ^ ) ( I ^ E.G. KALNINS AND W. MILLER, JR. 225 Here Eff(z) = f(qz). Relation (2.1A) follows from T(az;q)J-;q) = J ^ ( l q)(aqz;q)k.1 (^— ;q) \Z /k q' Z / k-l and (2.IB) follows from KaZ;q)k(±q)k = (1 a b q ^ ^ q ) ^ ^ q ) ^ The first relation was pointed out by Askey and Wilson [2]; we have not found (2. IB) in the literature. Consider relation (2.IB); it suggests that there should be an operator Aaq-^,bq-^,cq,dq) . Aa,b,c,d) T , , a A l± mapping Cn t o <f)n . Indeed we find l(cq ^,dq^iaq' ^ibq^)ji{aq-^ ,bq-^,cqAq) (2.3) ^(l-Ht-^) ,(aM

In applications of mathematics involving either the Laplace or the Helmholtz equation in spherical coordinates the associated Legendre equation occurs. Its solutions are called associated Legendre functions. They have some relations to classical Legendre orthogonal polynomials and classical Jacobi orthogonal polynomials. The classical Legendre polynomials and the associated Legendre functions have been used by some authors at the approximation of length of curves and cylindrical surfaces and the approximation of Earth shape in geosciences and engineering computations. In the present paper we investigate some relations between these functions pointing out possible applications in geosciences.

We show that in the large matrix limit, the eigenvalues of the normal matrix model for matrices with spectrum inside a compact domain with a special class of potentials homogeneously fill the interior of a polynomial curve uniquely defined by the area of its interior domain and its exterior harmonic moments which are all given as parameters of the potential. Then we consider the orthogonal polynomials corresponding to this matrix model and show that, under certain assumptions, the density of the zeros of the highest relevant orthogonal polynomial in the large matrix limit is (up to some constant factor) given by the discontinuity of the Schwarz function of this polynomial curve.