Some sharp two-sided Turán type inequalities for parabolic cylinder functions and Tricomi conflue... more Some sharp two-sided Turán type inequalities for parabolic cylinder functions and Tricomi confluent hypergeometric functions are deduced. The proofs are based on integral representations for quotients of parabolic cylinder functions and Tricomi confluent hypergeometric functions, which arise in the study of the infinite divisibility of the Fisher-Snedecor F distribution. Moroever, some complete monotonicity results are given concerning Turán determinants of Tricomi confluent hypergeometric functions. These complement and improve some of the results of Ismail and Laforgia [23].
The relation between the spectral decomposition of a self-adjoint operator which is realizable as... more The relation between the spectral decomposition of a self-adjoint operator which is realizable as a higher order recurrence operator and matrix-valued orthogonal polynomials is investigated. A general construction of such operators from scalar-valued orthogonal polynomials is presented. Two examples of matrix-valued orthogonal polynomials with explicit orthogonality relations and three-term recurrence relation are presented, which both can be considered as 2 × 2-matrix-valued analogues of subfamilies of Askey-Wilson polynomials.
By splitting the real line into intervals of unit length a doubly infinite integral of the form ∞... more By splitting the real line into intervals of unit length a doubly infinite integral of the form ∞ −∞ F (q x) dx, 0 < q < 1, can clearly be expressed as 1 0 ∞ n=−∞ F (q x+n) dx, provided F satisfies the appropriate conditions. This simple idea is used to prove Ramanujan's integral analogues of his 1 ψ 1 sum and give a new proof of Askey and Roy's extention of it. Integral analogues of the well-poised 2 ψ 2 sum as well as the very-well-poised 6 ψ 6 sum are also found in a straightforward manner. An extension to a very-well-poised and balanced 8 ψ 8 series is also given. A direct proof of a recent q-beta integral of Ismail and Masson is given.
We nd the adjoint of the Askey-Wilson divided dierence operator with respect to the inner product... more We nd the adjoint of the Askey-Wilson divided dierence operator with respect to the inner product on L 2 (01; 1; (1 0 x 2) 01=2 dx) dened as a Cauchy principal value and show that the Askey-Wilson polynomials are solutions of a q-Sturm-Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward way. We also provide an operator theoretic description of the Askey-Wilson operator.
We prove a generalization of the Kibble-Slepian formula (for Hermite polynomials) and its unitary... more We prove a generalization of the Kibble-Slepian formula (for Hermite polynomials) and its unitary analogue involving the 2D Hermite polynomials recently proved in [17]. We derive integral representations for the 2D Hermite polynomials which are of independent interest. Several new generating functions for 2D q-Hermite polynomials will also be given.
In this paper, we study asymptotics of the thermal partition function of a model of quantum mecha... more 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 Stieltjes-Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also develop a new asymptotic method for general q-polynomials.
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
We propose a new approach to the combinatorial interpretations of linearization coefficient probl... more We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of separation of variables. We illustrate our approach by applying it to determine the number of perfect matchings, derangements, and other weighted permutation problems. The separation of variables technique naturally leads to integral representations of combinatorial numbers where the integrand contains a product of one or more types of orthogonal polynomials. This also establishes the positivity of such integrals.
By applying an integral representation for q k 2 we systematically derive a large number of new F... more By applying an integral representation for q k 2 we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of q-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problems. These functions include q-Bessel functions, the Ramanujan function, Stieltjes-Wigert polynomials, q-Hermite and q −1-Hermite polynomials, and the q-exponential functions eq, Eq and Eq. Their representations are in turn used to derive many new identities involving q-functions and polynomials. In this work we also present contour integral representations for the above mentioned functions and polynomials.
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].
This book deals with the theory and gives 25 algorithms for solving computational problems. The a... more This book deals with the theory and gives 25 algorithms for solving computational problems. The areas covered include the solving of equations with particular emphasis on the solution of linear least squares problems. There are chapters and algorithms
Some sharp two-sided Turán type inequalities for parabolic cylinder functions and Tricomi conflue... more Some sharp two-sided Turán type inequalities for parabolic cylinder functions and Tricomi confluent hypergeometric functions are deduced. The proofs are based on integral representations for quotients of parabolic cylinder functions and Tricomi confluent hypergeometric functions, which arise in the study of the infinite divisibility of the Fisher-Snedecor F distribution. Moroever, some complete monotonicity results are given concerning Turán determinants of Tricomi confluent hypergeometric functions. These complement and improve some of the results of Ismail and Laforgia [23].
The relation between the spectral decomposition of a self-adjoint operator which is realizable as... more The relation between the spectral decomposition of a self-adjoint operator which is realizable as a higher order recurrence operator and matrix-valued orthogonal polynomials is investigated. A general construction of such operators from scalar-valued orthogonal polynomials is presented. Two examples of matrix-valued orthogonal polynomials with explicit orthogonality relations and three-term recurrence relation are presented, which both can be considered as 2 × 2-matrix-valued analogues of subfamilies of Askey-Wilson polynomials.
By splitting the real line into intervals of unit length a doubly infinite integral of the form ∞... more By splitting the real line into intervals of unit length a doubly infinite integral of the form ∞ −∞ F (q x) dx, 0 < q < 1, can clearly be expressed as 1 0 ∞ n=−∞ F (q x+n) dx, provided F satisfies the appropriate conditions. This simple idea is used to prove Ramanujan's integral analogues of his 1 ψ 1 sum and give a new proof of Askey and Roy's extention of it. Integral analogues of the well-poised 2 ψ 2 sum as well as the very-well-poised 6 ψ 6 sum are also found in a straightforward manner. An extension to a very-well-poised and balanced 8 ψ 8 series is also given. A direct proof of a recent q-beta integral of Ismail and Masson is given.
We nd the adjoint of the Askey-Wilson divided dierence operator with respect to the inner product... more We nd the adjoint of the Askey-Wilson divided dierence operator with respect to the inner product on L 2 (01; 1; (1 0 x 2) 01=2 dx) dened as a Cauchy principal value and show that the Askey-Wilson polynomials are solutions of a q-Sturm-Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward way. We also provide an operator theoretic description of the Askey-Wilson operator.
We prove a generalization of the Kibble-Slepian formula (for Hermite polynomials) and its unitary... more We prove a generalization of the Kibble-Slepian formula (for Hermite polynomials) and its unitary analogue involving the 2D Hermite polynomials recently proved in [17]. We derive integral representations for the 2D Hermite polynomials which are of independent interest. Several new generating functions for 2D q-Hermite polynomials will also be given.
In this paper, we study asymptotics of the thermal partition function of a model of quantum mecha... more 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 Stieltjes-Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also develop a new asymptotic method for general q-polynomials.
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
We propose a new approach to the combinatorial interpretations of linearization coefficient probl... more We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of separation of variables. We illustrate our approach by applying it to determine the number of perfect matchings, derangements, and other weighted permutation problems. The separation of variables technique naturally leads to integral representations of combinatorial numbers where the integrand contains a product of one or more types of orthogonal polynomials. This also establishes the positivity of such integrals.
By applying an integral representation for q k 2 we systematically derive a large number of new F... more By applying an integral representation for q k 2 we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of q-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problems. These functions include q-Bessel functions, the Ramanujan function, Stieltjes-Wigert polynomials, q-Hermite and q −1-Hermite polynomials, and the q-exponential functions eq, Eq and Eq. Their representations are in turn used to derive many new identities involving q-functions and polynomials. In this work we also present contour integral representations for the above mentioned functions and polynomials.
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].
This book deals with the theory and gives 25 algorithms for solving computational problems. The a... more This book deals with the theory and gives 25 algorithms for solving computational problems. The areas covered include the solving of equations with particular emphasis on the solution of linear least squares problems. There are chapters and algorithms
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