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Maurice Kenfack Nangho

Publications and source records attributed to Maurice Kenfack Nangho.

4 recordsLinked to original sources
arXiv2026

Characterization of classical orthogonal polynomials in two continuous variables

For a family of polynomials in two continuous variables, orthogonal with respect to a weight function, we prove, under suitable conditions, the equivalence of the following properties: the matrix Pearson equation of the weight, the second order linear partial differential equation, the orthogonality of the gradients, the matrix Rodrigues formula involving tensor products of matrices, and the so-called first structure relation. We then introduce a notion of classical orthogonal polynomials in two variables and relate the corresponding theory for weight functions and moment functionals. Finally, we present a nontrivial example that illustrates and delineates our contribution to the field.

math.CA↗
arXiv2022

Structure Relations of Classical Orthogonal Polynomials in the Quadratic and $q$-Quadratic Variable

We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty}$, the orthogonality of the second derivatives $\{\mathbb{D}_{x}^2P_n\}_{n= 2}^{\infty}$ and a generalized Sturm-Liouville type equation. Our treatment of the generalized Bochner theorem leads to explicit solutions of the difference equation [Vinet L., Zhedanov A., J. Comput. Appl. Math. 211 (2008), 45-56], which proves that the only monic orthogonal polynomials that satisfy the first structure relation are Wilson polynomials, continuous dual Hahn polynomials, Askey-Wilson polynomials and their special or limiting cases as one or more parameters tend to $\infty$. This work extends our previous result [arXiv:1711.03349] concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying the first structure relation.

math.CA↗
arXiv2018

A Characterization of Askey-Wilson polynomials

We show that the only monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty}$ that satisfy $$π(x)\mathcal{D}_{q}^2P_{n}(x)=\sum_{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\; x=\cosθ,\;~ a_{n,n-2}\neq 0,~ n=2,3,\dots,$$ where $π(x)$ is a polynomial of degree at most $4$ and $\mathcal{D}_{q}$ is the Askey-Wilson operator, are Askey-Wilson polynomials and their special or limiting cases. This completes and proves a conjecture by Ismail concerning a structure relation satisfied by Askey-Wilson polynomials. We use the structure relation to derive upper bounds for the smallest zero and lower bounds for the largest zero of Askey-Wilson polynomials and their special cases.

math.CA↗