We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to the classical weight function for the Jacobi polynomials together with point masses at both endpoints. In the special symmetric case that both parameters are equal and also the two point masses are equal we find all differential equations of spectral type satisfied by these symmetric generalized ultraspherical polynomials. We show that if the point masses are positive only for nonnegative integer values of the parameter there exists exactly one differential equation of spectral type which is of finite order. By using quadratic transformations we also obtain differential equations for some related sets of generalized Jacobi polynomials. In these cases we find finite order differential equations even though one of the parameters is not equal to an integer.
We find all spectral type differential equations satisfied by the symmetric generalized ultraspherical polynomials which are orthogonal on the interval [-1,1] with respect to the classical symmetric weight function for the Jacobi polynomials together with equal mass points at both ends of the interval of orthogonality. In order to find explicit formulas for the coefficients of these differential equations we have to solve systems of equations involving derivatives of the classical Jacobi polynomials. These systems of equations have a unique solution which is given explicitly. This is a consequ…
In a companion paper [On semiclassical orthogonal polynomials via polynomial mappings, J. Math. Anal. Appl. (2017)] we proved that the semiclassical class of orthogonal polynomials is stable under polynomial transformations. In this work we use this fact to derive in an unified way old and new properties concerning the sieved ultraspherical polynomials of the first and second kind. In particular we derive ordinary differential equations for these polynomials. As an application, we use the differential equation for sieved ultraspherical polynomials of the first kind to deduce that the zeros of …
In this research, by applying the extended Sturm-Liouville theorem for symmetric functions, a basic class of symmetric orthogonal polynomials (BCSOP) with four free parameters is introduced and all its standard properties, such as a generic second order differential equation along with its explicit polynomial solution, a generic orthogonality relation, a generic three term recurrence relation and so on, are presented. Then, it is shown that four main sequences of symmetric orthogonal polynomials can essentially be extracted from the introduced class. They are respectively the generalized ultra…
We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter $ν>0$. The LDU-decomposition of the weight is explicitly given in terms of Gegenbauer polynomials. We establish a matrix-valued Pearson equation for these matrix weights leading to explicit shift operators relating the weights for parameter $ν$ and $ν+1$. The matrix coefficients of the Pearson equation are obtained using a special matrix-valued differential operator in a commutative algebra of symmetric differential operato…
An explicit structure relation for Askey-Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey-Wilson inner product and which sends polynomials of degree n to polynomials of degree n+1. By specialization of parameters and by taking limits, similar structure relations, as well as lowering and raising relations, can be obtained for other families in the q-Askey scheme and the Askey scheme. This is explicitly discussed for Jacobi polynomials, continuous q-Jacobi polynomials, continuous q-ultraspherical polynomials, and for big…
The aim of this paper is to study $L^p$-boundedness property of the pseudo differential operator associated with a symbol, on rank one Riemannian symmetric spaces of noncompact type, where the symbol satisfies Hörmander-type conditions near infinity.
It is proven that second-order vectorial nonlinear differential systems y''=f(y) , possess a continuum of symmetric solutions. They are shown to possess a continuum of even solutions. If f(y) is an odd function of y , then y''=f(y) is shown also to possess a continuum of odd solutions. The results apply to a significant family of second-order vectorial nonlinear differential systems that are not dissipative. This family of differential equations includes the celebrated N-body problem of celestial mechanics and other central force problems.
We introduce the notion of Caputo-Fabrizio left and right derivatives. We present sufficient conditions for the existence of symmetric positive solutions for the following Caputo-Fabrizio fractional singular integro-differential boundary value problem \begin{align*} (2-μ)\,{}^{CF}D_{0}^{\,μ}x(t)+f(t,x(t))&=\left(\frac{μ-1}{2-μ}\right)^{2} \begin{cases} \int_{t}^{0}e^{-\frac{μ-1}{2-μ}(τ-t)}\,x(τ)dτ,\hspace{0.4cm}&t\in(-1,0],\\ \int_{0}^{t}e^{-\frac{μ-1}{2-μ}(t-τ)}\,x(τ)dτ,\hspace{0.4cm}&t\in[0,1), \end{cases}\\ x(\pm1)=x'(0^{\pm})&=0,\hspace{5.55cm}μ\in(1,2), \end{align*} where the nonlinearity…
Let $Ω$ be a symmetric cone and $V$ the corresponding simple Euclidean Jordan algebra. In \cite{ado,do,do04,doz2} we considered the family of generalized Laguerre functions on $Ω$ that generalize the classical Laguerre functions on $\mathbb{R}^+$. This family forms an orthogonal basis for the subspace of $L$-invariant functions in $L^2(Ω,dμ_ν)$, where $dμ_ν$ is a certain measure on the cone and where $L$ is the group of linear transformations on $V$ that leave the cone $Ω$ invariant and fix the identity in $Ω$. The space $L^2(Ω,dμ_ν)$ supports a highest weight representation of the group $G$ o…
In this paper we consider the formally symmetric differential expression $M[\cdot]$ of any order (odd or even) $\geq 2$. We characterise the dimension of the quotient space $D(T_{\max})/D(T_{\min})$ associated with $M[\cdot]$ in terms of the behaviour of the determinants {equation*} \det\limits_{r,s\in {\bf N}_{n}} [[f_{r}g_{s}](\infty)] {equation*} where $1\leq n\leq$ (order of the expression + 1); here $[fg](\infty) = \lim\limits_{x\to\infty}[fg](x)$, where $[fg](x)$ is the sesquilinear form in $f$ and $g$ associated with $M$. These results generalise the well-known theorem that $M$ is in th…
Formally symmetric differential operators on weighted Hardy-Hilbert spaces are analyzed, along with adjoint pairs of differential operators. Eigenvalue problems for such operators are rather special, but include many of the classical Riemann and Heun equations. Symmetric minimal operators are characterized. A regular class whose leading coefficients have no zeros on the unit circle are shown to be essentially self-adjoint. Eigenvalue asymptotics are established. Some extensions to non-self-adjoint operators are also considered.
In this paper, usual Sturm-Liouville problems are extended for symmetric functions so that the corresponding solutions preserve the orthogonality property. Two basic examples, which are special cases of a generalized Sturm-Liouville problem, are then introduced. First example generalizes the associated Legendre functions having extensive applications in physics and the second example introduces a generic differential equation with various sub-cases that have orthogonal solutions. For instance, this generic equation possesses a symmetric differential equation containing a basic solution of symm…
A symmetric function of $N$ variables can be given in terms of symmetric polynomials of these variables. We determine those symmetric polynomials in which the dual differential operators take the neatest form when expressed in terms of our original variables, producing a simple form for the associated Weyl algebra. Both our coordinates, and the form of our differential operators at total diagonal points, exhibit interesting properties, and are related to interesting objects like Bell polynomials. They can be modified to give a simple formula for the gradient of our symmetric function at any po…
We obtain an explicit classification of all $2\times 2$ real hypergeometric Bochner pairs, ie. pairs $(W(x),\mathfrak{D})$ consisting of a $2\times 2$ real hypergeometric differential operator $\mathfrak{D}$ and a $2\times 2$ weight matrix satisfying the property that $\mathfrak{D}$ is symmetric with respect to the matrix-valued inner product defined by W(x). Furthermore, we obtain a classifying space of hypergeometric Bochner pairs by describing a bijective correspondence between the collection of pairs and an open subset of a real algebraic set whose smooth paths correspond to isospectral de…
We prove that for every planar differential system with a period annulus there exists an involution $σ$ such that the system is $σ$-symmetric. We also prove that for for every planar differential system with a period annulus there exist infinitely many involutions $σ$ such that the system is $σ$-reversible.
