Related papers: Jacobi series for general parameters and applicati…
Orthogonality of the Jacobi and of Laguerre polynomials, P_n^(a,b) and L_n^(a), is established for a,b complex (a,b not negative integers and a+b different from -2,-3,...) using the Hadamard finite part of the integral which gives their…
We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with point masses…
One may consider the generalization of Jacobi polynomials and the Jacobi function of the second kind to a general function where the index is allowed to be a complex number instead of a non-negative integer. These functions are referred to…
We look for spectral type differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with…
Classical Jacobi polynomials $P_{n}^{(\alpha,\beta)}$, with $\alpha, \beta>-1$, have a number of well-known properties, in particular the location of their zeros in the open interval $(-1,1)$. This property is no longer valid for other…
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…
From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric…
We investigate the differential equation for the Jacobi-type polynomials which are orthogonal on the interval $[-1,1]$ with respect to the classical Jacobi measure and an additional point mass at one endpoint. This scale of higher-order…
A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a…
A special class of generalized Jacobi operators which are self-adjoint in Krein spaces is presented. A description of the resolvent set of such operators in terms of solutions of the corresponding recurrence relations is given. In…
The Jacobi polynomial has been advocated by many authors as a useful tool to evolve non-singlet structure functions to higher $Q^2$. In this work, it is found that the convergence of the polynomial sum is not absolute, as there is always a…
The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of…
We give an analogy of Jacobi's formula, which relates the hypergeometric function with parameters $(1/4,1/4,1)$ and theta constants. By using this analogy and twice formulas of theta constants, we obtain a transformation formula for this…
We look for spectral type differential equations satisfied by the generalized Jacobi polynomials, which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with…
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…
Generalized Jacobi polynomials are orthogonal polynomials related to a weight function which is smooth and positive on the whole interval of orthogonality up to a finite number of points, where algebraic singularities occur. The influence…
We first present some identities involving the Pochhammer symbol (rising factorial). We also recall and present some new properties of the Jacobi polynomials. We use them to expand a general hypergeometric function in an orthogonal series…
A ladder structure of operators is presented for the Jacobi polynomials, J_n^(a,b)(x), with parameters n, a and b integers, showing that they are related to the unitary irreducible representation of SU(2,2) with quadratic Casimir…
We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi's canonical form of the hypergeometric differential equation. Analogy for $q$-hypergeometric…
The Jacobi set is a useful descriptor of mutual behavior of functions defined on a common domain. We introduce the piecewise linear Jacobi set for general vector fields on simplicial complexes. This definition generalizes the definition of…