An algebraic setting for the Roman-Rota umbra1 calculus is introduced. It is shown how many of th... more An algebraic setting for the Roman-Rota umbra1 calculus is introduced. It is shown how many of the umbra1 calculus results follow simply by introducing a comultiplication map and requiring it to be an algebra map. The same approach is used to construct a q-umbra1 calculus. Our umbra1 calculus yields some of Andrews recent results on Eulerian families of polynomials as corollaries. The homogeneous Eulerian families are studied. Operator and functional expansions are also included.
Proceedings of the American Mathematical Society, Feb 20, 2017
Calogero and his collaborators recently observed that some hypergeometric polynomials can be fact... more 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 prevails through all polynomials in the Askey scheme. We show that this factorization property is also shared by the associated Wilson and Askey-Wilson polynomials and some biorthogonal rational functions. This is applied to a specific model of an isochronous system of particles with small oscillations around the equilibrium position
In [1], Richard Askey analysed the LP convergence of the Lagrange interpolation polynomials when ... more In [1], Richard Askey analysed the LP convergence of the Lagrange interpolation polynomials when the zeros of the classical Jacobi polynomials, P n (α,β) (z), are used as the points of interpolation. His analysis was complete, except for some results concerning the positivity of the Cesaro means of some order γ, (C, γ), for the Poisson Kernel, $$ \begin{gathered}{\operatorname{P} _r}(x,y) = \sum\limits_{n = 0}^\infty {{r^n}P_n^{(\alpha ,\beta )}(x)P_n^{(\alpha ,\beta )}(y)K_n^{ - 1}} , \hfill \\{K_n} = \int\limits_{ - 1}^1 {{{[P_n^{(\alpha ,\beta )}(x)]}^2}{{(1 - x)}^\alpha }{{(1 + x)}^\beta }} \hfill \\\alpha ,\beta > - \frac{1}{2},\quad 0 < r < 1 \hfill \\\end{gathered} $$ .
Journal of Difference Equations and Applications, 2016
In this paper, we solve dual and triple sequences involving q-orthogonal polynomials. We also int... more In this paper, we solve dual and triple sequences involving q-orthogonal polynomials. We also introduce and solve a system of dual series equations when the kernel is the q-Laguerre polynomials. Examples are included.
By using asymptotic methods and fractional integration, it is shown that \[ {}_1 F_2 \left( {\beg... more By using asymptotic methods and fractional integration, it is shown that \[ {}_1 F_2 \left( {\begin{array}{*{20}c} {\lambda - a} \\ {\rho \lambda + b,\rho \lambda + c} \\ \end{array} | {\frac{{ - \mu ^2 }}{4}} } \right) \geqq 0,\quad \mu {\text{ real}},\]$0 \leqq a \leqq \lambda $, $0 \leqq b$, $1 \leqq 2c$, and either $2\rho \geqq 3$, $\lambda \geqq 1$ or $\rho \geqq 2$, $\lambda \geqq 0$. From this, it is deduced that for $x > 0$, $x^{2\lambda - 2\rho \lambda - b} (1 - x^2 )^{ - \lambda } $ is completely monotonic for $b \geqq 0$ and either $2\rho \geqq 3$, $\lambda \geqq 1$, or $\rho \geqq 2$, $\lambda \geqq 0$. This extends the results of [3] and proves some conjectures of Askey [1].
An algebraic setting for the Roman-Rota umbra1 calculus is introduced. It is shown how many of th... more An algebraic setting for the Roman-Rota umbra1 calculus is introduced. It is shown how many of the umbra1 calculus results follow simply by introducing a comultiplication map and requiring it to be an algebra map. The same approach is used to construct a q-umbra1 calculus. Our umbra1 calculus yields some of Andrews recent results on Eulerian families of polynomials as corollaries. The homogeneous Eulerian families are studied. Operator and functional expansions are also included.
Proceedings of the American Mathematical Society, Feb 20, 2017
Calogero and his collaborators recently observed that some hypergeometric polynomials can be fact... more 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 prevails through all polynomials in the Askey scheme. We show that this factorization property is also shared by the associated Wilson and Askey-Wilson polynomials and some biorthogonal rational functions. This is applied to a specific model of an isochronous system of particles with small oscillations around the equilibrium position
In [1], Richard Askey analysed the LP convergence of the Lagrange interpolation polynomials when ... more In [1], Richard Askey analysed the LP convergence of the Lagrange interpolation polynomials when the zeros of the classical Jacobi polynomials, P n (α,β) (z), are used as the points of interpolation. His analysis was complete, except for some results concerning the positivity of the Cesaro means of some order γ, (C, γ), for the Poisson Kernel, $$ \begin{gathered}{\operatorname{P} _r}(x,y) = \sum\limits_{n = 0}^\infty {{r^n}P_n^{(\alpha ,\beta )}(x)P_n^{(\alpha ,\beta )}(y)K_n^{ - 1}} , \hfill \\{K_n} = \int\limits_{ - 1}^1 {{{[P_n^{(\alpha ,\beta )}(x)]}^2}{{(1 - x)}^\alpha }{{(1 + x)}^\beta }} \hfill \\\alpha ,\beta > - \frac{1}{2},\quad 0 < r < 1 \hfill \\\end{gathered} $$ .
Journal of Difference Equations and Applications, 2016
In this paper, we solve dual and triple sequences involving q-orthogonal polynomials. We also int... more In this paper, we solve dual and triple sequences involving q-orthogonal polynomials. We also introduce and solve a system of dual series equations when the kernel is the q-Laguerre polynomials. Examples are included.
By using asymptotic methods and fractional integration, it is shown that \[ {}_1 F_2 \left( {\beg... more By using asymptotic methods and fractional integration, it is shown that \[ {}_1 F_2 \left( {\begin{array}{*{20}c} {\lambda - a} \\ {\rho \lambda + b,\rho \lambda + c} \\ \end{array} | {\frac{{ - \mu ^2 }}{4}} } \right) \geqq 0,\quad \mu {\text{ real}},\]$0 \leqq a \leqq \lambda $, $0 \leqq b$, $1 \leqq 2c$, and either $2\rho \geqq 3$, $\lambda \geqq 1$ or $\rho \geqq 2$, $\lambda \geqq 0$. From this, it is deduced that for $x > 0$, $x^{2\lambda - 2\rho \lambda - b} (1 - x^2 )^{ - \lambda } $ is completely monotonic for $b \geqq 0$ and either $2\rho \geqq 3$, $\lambda \geqq 1$, or $\rho \geqq 2$, $\lambda \geqq 0$. This extends the results of [3] and proves some conjectures of Askey [1].
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Papers by Mourad Ismail