The aim of this paper are two folds. The first part is concerned with the associated and the so-called co-polynomials, i.e., new sequences obtained when finite perturbations of the recurrence coefficients are considered. Moreover, the second part deals with Darboux factorization of Jacobi matrices. Here the respective co-polynomials solutions are explicitly expressed in terms of the fundamental solutions of a (d+2)-term recurrence relation. New identities and formulas related to determinants with co-polynomials entries are obtained. Accordingly, further determinants bring out partial generalizations of Christoffel Darboux formula. Some of new sequences proved useful for determining the entries of matrices in LU and UL decomposition of Jacobi matrix. The last one gives rise of a d-analogue of kernel polynomials with quite a few properties, and further a new characterization of the d-quasiorthogonality. Kernel polynomials also appear in the (d+1)-decomposition of a d-symmetric sequence. Exploiting properties of d-symmetric sequences, reveal a simple proof of Darboux factorizations. It terms out that Jacobi matrix for d-OPS is a product of d lower bidiagonal matrices and one upper bidiagonal matrix and that each lower bidiagonal matrix is in fact a closed connection between two adjacent components for some d-(symmetric)OPS. Furthermore, we pointed out that if the first component is Hahn classical d-OPS then the corresponding d-symmetric sequence as well as all the components are Hahn classical d-OPS as well. Oscillation matrices assert that zeros of d-OPS are positive and simple whenever the recurrence coefficients are strict positive. Further interlacing properties are justified by the same approach.2010 Mathematics Subject Classification. Primary 42C05; Secondary 33C45.
Using the theory of the Riordan group and d-orthogonal polynomials, we shall show that sequences of d-orthogonal polynomials, can be also generated by the Riordan group. We interpret some families of d-orthogonal polynomials within the framework of the Riordan group.
ABSTRACT. The associated sequence of order r for a given d-OPS (i.e. a sequence of orthogonal polynomials satisfying a (d + 1)-order recurrence relation), is again a d-OPS. In this paper we are interested in the determination of the corresponding dual sequence. The explicit form of the dual sequence of the first associated sequence and the corresponding formal Stieltjes function are given. Indeed, we construct by recurrence the dual sequence of the r-associated sequence and we give some properties of the corresponding Stieltjes function. Second, we give the definition of co-recursive polynomials of dimension d and some relations in the particular cases d = 3 and d = 4. Some properties of the dual sequence as well as of the corresponding Stieltjes functions are given. Preliminaries and notationsLet P be the linear space of complex polynomials in one variable and P its topological dual space. We denote by u, p the action of u ∈ P on p ∈ P. In particular, we denote by (u) n = u, x n , n ≥ 0, the moments of u. Let {P n } n≥0 be a sequence of monic polynomials with deg P n = n, n ≥ 0. Its dual sequence {u n } n≥0 , u n ∈ P , is defined by u n , P m := δ n,m , n, m ≥ 0. Let us recall the following result [6][7][8] Ä ÑÑ 1º For any u ∈ P and any integer p ≥ 1, the following statements are equivalent:
We construct the linear differential equations of third order satisfied by the classical2-orthogonal polynomials. We show that these differential equations have the following form:R4,n(x)Pn+3(3)(x)+R3,n(x)P″n+3(x)+R2,n(x)P′n+3(x)+R1,n(x)Pn+3(x)=0, where the coefficients{Rk,n(x)}k=1,4are polynomials whose degrees are, respectively, less than or equal to4,3,2, and1. We also show that the coefficientR4,n(x)can be written asR4,n(x)=F1,n(x)S3(x), whereS3(x)is a polynomial of degree less than or equal to3with coefficients independent ofnanddeg(F1,n(x))≤1. We derive these equations in some cases and we also quote some classical2-orthogonal polynomials, which were the subject of a deep study.
We discuss the connections between the2-orthogonal polynomials and the generalized birth and death processes. Afterwards, we find the sufficient conditions to give an integral representation of the transition probabilities from these processes.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.