Explanipedia

On the Meixner–Pollaczek polynomials and the Sturm–Liouville problems Open

Mourad E. H. Ismail, Nasser Saad · 2025

Orthogonal polynomials with periodic recurrence coefficients Open

Dan Dai, Mourad E. H. Ismail, Xiangsheng Wang · 2024

In this paper, we study a class of orthogonal polynomials defined by a three-term recurrence relation with periodic coefficients. We derive explicit formulas for the generating function, the associated continued fraction, the orthogonality…

Sharp Hardy’s inequalities in Hilbert spaces Open

Dimitar K. Dimitrov, Ivan Gadjev, Mourad E. H. Ismail · 2024

We study the behavior of the smallest possible constants d(a,b) and d_{n} in Hardy’s inequalities \int_{a}^{b}\biggl(\frac{1}{x}\int_{a}^{x}f(t)dt\biggr)^{2}\:dx\leq d(a,b){}\int_{a}^{b} [f(x)]^{2}\: dx and \sum_{k=1}^{n}\Big(\frac{1}{k}\s…

From continuous to discrete: weak limit of normalized Askey–Wilson measure Open

Dan Dai, Mourad E. H. Ismail, Xiangsheng Wang · 2024

$q$-Fractional Integral Operators With Two Parameters Open

Mourad E. H. Ismail, Keru Zhou · 2023

We use the Poisson kernel of the continuous $q$-Hermite polynomials to introduces families of integral operators, which are semigroups of linear operators. We describe the eigenvalues and eigenfunctions of one family of operators. The acti…

Solutions of the scattering problem in a complete set of Bessel functions with a discrete index Open

A. D. Alhaidari, Mourad E. H. Ismail · 2023

We use the tridiagonal representation approach to solve the radial\nSchr\\"odinger equation for the continuum scattering states of the Kratzer\npotential. We do the same for a radial power-law potential with inverse-square\nand inverse-cub…

Sharp Hardy's Inequalities in Hilbert Spaces Open

Dimitar K. Dimitrov, I. Gadjev, Mourad E. H. Ismail · 2023

We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy's inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^2\,dx\leq d(a,b)\,\int_a^b [f(x)]^2 dx $$ and $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}…

Preface Open

Krishnaswami Alladi, Howard S. Cohl, Frank Garvan, Mourad E. H. Ismail, Hjalmar Rosengren · 2023

Hypoglycemia-Induced Basal Ganglia Infarct: A Rare Case of Metformin Toxicity in a Hemodialysis Patient Open

Rovena Collins, Ariana R. Tagliaferri, Gabrielle LoBue, William Meng, Mourad E. H. Ismail · 2022

Orthogonal Polynomials of Askey-Wilson Type Open

Mourad E. H. Ismail, Ruiming Zhang, Keru Zhou · 2022

We study two families of orthogonal polynomials. The first is a finite family related to the Askey-Wilson polynomials but the orthogonality is on the real line. A limiting case of this family is an infinite system of orthogonal polynomials…

The Legacy of Dick Askey (1933–2019) Open

Howard S. Cohl, Mourad E. H. Ismail, Hung-Hsi Wu, Paul M Terwilliger, George E. Andrews , et al. · 2021

feelpp/feelpp: Feel++ V109 Open

Christophe Prud'Homme, Vincent Chabannes, StephaneVeys, Alexandre Ancel, Thibaut Métivet , et al. · 2021

Features Remeshing support including in fluid, cfpdes and heat toolboxes and added some quality indicators/metrics Support for rigid and elastic bodies in fluid toolbox Support for magnetostatic in cfpdes with Hcurl FE Homogeneize postproc…

feelpp/feelpp: Feel++ V109 RC 2 Open

Christophe Prud'Homme, Vincent Chabannes, StephaneVeys, Alexandre Ancel, Thibaut Métivet , et al. · 2021

What's Changed Added cmake Presets support by @prudhomm Added pyfeelpp-all component (only available in Preset) as a meta component to compile only feelpp python wrappers by @prudhomm Improved python wrappers for. toolboxes added tests for…

$q$-type Lidstone expansions and an interpolation problem for entire functions Open

Mourad E. H. Ismail, Zeinab S. I. Mansour · 2021

In this paper, we expand functions of specific $q$-exponential growth in terms of its even (odd) Askey- Wilson $q$-derivatives at $0$ and $η=(q^{1/4}+q^{-1/4})/2$. This expansion is a $q$-version of the celebrated Lidstone expansion theore…

q-Fractional Askey-Wilson Integrals and Related Semigroups of Operators Open

Mourad E. H. Ismail, Ruiming Zhang, Keru Zhou · 2020

We introduce three one-parameter semigroups of operators and determine their spectra. Two of them are fractional integrals associated with the Askey-Wilson operator. We also study these families as families of positive linear approximation…

A necessary and sufficient condition for sequences to be minimal completely monotonic Open

Xifeng Wang, Mourad E. H. Ismail, Necdet Batır, Senlin Guo · 2020

In this article, we present a necessary and sufficient condition under which sequences are minimal completely monotonic.

Response to “Comment on ‘The asymptotic iteration method revisited’” [J. Math. Phys. 61, 064101 (2020)] Open

Mourad E. H. Ismail, Nasser Saad · 2020

First Page

A discrete and $q$ Asymptotic Iteration Method Open

Mourad E. H. Ismail, Nasser Saad · 2020

We introduce a finite difference and $q$-difference analogues of the Asymptotic Iteration Method of Ciftci, Hall, and Saad. We give necessary, and sufficient condition for the existence of a polynomial solution to a general linear second-o…

On Entry II.16.12: A continued fraction of Ramanujan Open

Gaurav Bhatnagar, Mourad E. H. Ismail · 2019

We study a continued fraction due to Ramanujan, that he recorded as Entry 12 in Chapter 16 of his second notebook. It is presented in Part III of Berndt's volumes on Ramanujan's notebooks. We give two alternate approaches to proving Ramanu…

Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice Open

Mourad E. H. Ismail, Erik Koelink, Pablo Román · 2019

Orthogonal polynomials associated with a continued fraction of Hirschhorn Open

Gaurav Bhatnagar, Mourad E. H. Ismail · 2019

We study orthogonal polynomials associated with a continued fraction due to Hirschhorn. Hirschhorn's continued fraction contains as special cases the famous Rogers--Ramanujan continued fraction and two of Ramanujan's generalizations. The o…

Spectral analysis of two doubly infinite Jacobi matrices with exponential entries Open

Mourad E. H. Ismail, František Štampach · 2018

Matrix valued Hermite polynomials, Burchnall formulas and non-abelian\n Toda lattice Open

Mourad E. H. Ismail, Erik Koelink, Pablo Román · 2018

A general family of matrix valued Hermite type orthogonal polynomials is\nintroduced and studied in detail by deriving Pearson equations for the weight\nand matrix valued differential equations for these matrix polynomials. This is\nused t…

R type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle Open

Mourad E. H. Ismail, A. Sri Ranga · 2018

Asymptotics of partition functions in a fermionic matrix model and of related <i>q</i>‐polynomials Open

Dan Dai, Mourad E. H. Ismail, Xiangsheng Wang · 2018

In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix‐like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the St…

Asymptotics of partition functions in a fermionic matrix model and of related $q$-polynomials Open

Dan Dai, Mourad E. H. Ismail, Xiangsheng Wang · 2018

In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix-like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the St…

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions Open

Mourad E. H. Ismail, Erik Koelink, Pablo Román · 2018

Burchnall's method to invert the Feldheim-Watson linearization formula for\nthe Hermite polynomials is extended to all polynomial families in the\nAskey-scheme and its $q$-analogue. The resulting expansion formulas are made\nexplicit for s…

𝑞-Bessel functions and Rogers-Ramanujan type identities Open

Mourad E. H. Ismail, Ruiming Zhang · 2018

We evaluate-Bessel functions at an infinite sequence of points and introduce a generalization of the Ramanujan function and give an extension of the-version of the Rogers-Ramanujan identities. We also prove several generating functions for…

Diophantine properties of orthogonal polynomials and rational functions Open

Mourad E. H. Ismail, Mizan Rahman · 2017

Calogero and his collaborators recently observed that some hypergeometric polynomials can be factored as a product of two polynomials, one of which is factored into a product of linear terms. Chen and Ismail showed that this property preva…

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