Interlacing of zeros of quasi-orthogonal Meixner polynomials

We consider the interlacing of zeros of polynomials within the sequences of quasi-orthogonal order one Meixner polynomials characterised by −β, c ∈ (0, 1). The interlacing of zeros of quasi-orthogonal Meixner polynomials Mn(x; β, c) with the zeros of their nearest orthogonal counterparts Ml(x; β + k, c), l, n ∈ ℕ, k ∈ {1, 2}, is also discussed.The DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) under grant number FA2016/008.http://www.tandfonline.com/loi/tqma202018-04-10hj2017Mathematics and Applied Mathematic

Quasi-Orthogonality of Some Hypergeometric and qq-Hypergeometric Polynomials

We show how to obtain linear combinations of polynomials in an orthogonal sequence $\{P_n\}_{n\geq 0}$, such as $Q_{n,k}(x)=\sum\limits_{i=0}^k a_{n,i}P_{n-i}(x)$, $a_{n,0}a_{n,k}\neq0$, that characterize quasi-orthogonal polynomials of order $k\le n-1$. The polynomials in the sequence $\{Q_{n,k}\}_{n\geq 0}$ are obtained from $P_{n}$, by making use of parameter shifts. We use an algorithmic approach to find these linear combinations for each family applicable and these equations are used to prove quasi-orthogonality of order $k$. We also determine the location of the extreme zeros of the quasi-orthogonal polynomials with respect to the end points of the interval of orthogonality of the sequence $\{P_n\}_{n\geq 0}$, where possible

Bounds for zeros of Meixner and Kravchuk polynomials

The zeros of certain different sequences of orthogonal polynomials interlace in a well-defined way. The study of this phenomenon and the conditions under which it holds lead to a set of points that can be applied as bounds for the extreme zeros of the polynomials. We consider different sequences of the discrete orthogonal Meixner and Kravchuk polynomials and use mixed three-term recurrence relations, satisfied by the polynomials under consideration, to identify bounds for the extreme zeros of Meixner and Kravchuk polynomials.National Research Foundation of South Africahttp://journals.cambridge.org/action/displayJournal?jid=JCMhb201

Inner bounds for the extreme zeros of 3F2 hypergeometric polynomials

Zeilberger’s celebrated algorithm finds pure recurrence relations (w. r. t. a single variable) for hypergeometric sums automatically. However, in the theory of orthogonal polynomials and special functions, contiguous relations w. r. t. several variables exist in abundance. We modify Zeilberger’s algorithm to generate unknown contiguous relations that are necessary to obtain inner bounds for the extreme zeros of orthogonal polynomial sequences with 3F2 hypergeometric representations. Using this method, we improve previously obtained upper bounds for the smallest and lower bounds for the largest zeros of the Hahn polynomials and we identify inner bounds for the extreme zeros of the Continuous Hahn and Continuous Dual Hahn polynomials. Numerical examples are provided to illustrate the quality of the new bounds. Without the use of computer algebra such results are not accessible. We expect our algorithm to be useful to compute useful and new contiguous relations for other hypergeometric functions.The first author would like to thank Alexander von Humboldt Foundation and TWAS for rewarding an AGNES Grant for Junior Researchers 2014, as well as TWAS and DFG for sponsoring a research visit at the Institute of Mathematics of the University of Kassel in 2016 (reference KO 1122/12-1). The second author would like to thank TWAS and DFG for sponsoring a research visit at the Institute of Mathematics of the University of Kassel in 2015 (reference 3240278140).http://www.tandfonline.com/loi/gitr202018-09-30hb2017Mathematics and Applied Mathematic

Zeros of Jacobi, Meixner and Krawtchouk Polynomials

Read abstract in the documentDissertation (PhD)--University of Pretoria, 2012.Mathematics and Applied MathematicsUnrestricte

Mixed recurrence equations and interlacing properties for zeros of sequences of classical q-orthogonal polynomials

Using the q-version of Zeilberger's algorithm, we provide a procedure to find mixed recurrence equations satisfied by classical q-orthogonal polynomials with shifted parameters. These equations are used to investigate interlacing properties of zeros of sequences of q-orthogonal polynomials. In the cases where zeros do not interlace, we give some numerical examples to illustrate this.A TWAS/DFG fellowship for A.S. Jooste and the Institute of Mathematics of the University of Kassel (Germany) for D.D. Tcheutia.http://www.elsevier.com/locate/apnum2019-03-01hj2018Mathematics and Applied Mathematic

On the zeros of Meixner polynomials

Please read abstract in the article.http://www.springerlink.com/content/100497

Quasi-orthogonality of some hypergeometric polynomials

The zeros of quasi-orthogonal polynomials play a key role in applications in areas such as interpolation theory, Gauss-type quadrature formulas, rational approximation and electrostatics. We extend previous results on the quasi-orthogonality of Jacobi polynomials and discuss the quasi-orthogonality of Meixner–Pollaczek, Hahn, Dual- Hahn and Continuous Dual-Hahn polynomials using a characterization of quasi-orthogonality due to Shohat. Of particular interest are the Meixner–Pollaczek polynomials whose linear combinations only exhibit quasi-orthogonality of even order. In some cases, we also investigate the location of the zeros of these polynomials for quasiorthogonality of order 1 and 2 with respect to the end points of the interval of orthogonality, as well as with respect to the zeros of different polynomials in the same orthogonal sequence.National Research Foundation of South Africa [grant number 2054423].http://www.tandfonline.com/loi/gitr202017-04-30hb201

On certain properties and applications of the perturbed Meixner–Pollaczek weight

This paper deals with monic orthogonal polynomials orthogonal with a perturbation of classical Meixner–Pollaczek measure. These polynomials, called Perturbed Meixner–Pollaczek polynomials, are described by their weight function emanating from an exponential deformation of the classical Meixner–Pollaczek measure. In this contribution, we investigate certain properties such as moments of finite order, some new recursive relations, concise formulations, differentialrecurrence relations, integral representation and some properties of the zeros (quasi-orthogonality, monotonicity and convexity of the extreme zeros) of the corresponding perturbed polynomials. Some auxiliary results for Meixner–Pollaczek polynomials are revisited. Some applications such as Fisher’s information, Toda-type relations associated with these polynomials, Gauss–Meixner– Pollaczek quadrature as well as their role in quantum oscillators are also reproduced.The NMU Council postdoctoral fellowshiphttps://www.mdpi.com/journal/mathematicsam2022Mathematics and Applied Mathematic