Abstract
The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.
Highlights
The connection and linearization problems of polynomials in general and of orthogonal polynomials, in particular, are crucial in mathematical analysis and its applications.For example, the linearization coefficients are useful in the computation of physical and chemical properties of quantum-mechanical systems [1,2]
The linearization coefficients are given in terms of a certain hypergeometric function of the type 5 F4 (1), which can be summed for particular choices of the involved parameters by making use of some well-known formulas in the literature or via the application of some suitable symbolic computation
We have considered the linearization problem of Jacobi polynomials
Summary
The connection and linearization problems of polynomials in general and of orthogonal polynomials, in particular, are crucial in mathematical analysis and its applications. These coefficients are often expressed in terms of hypergeometric functions of certain arguments; see, for example, [6,13] It is well-known that the Jacobi polynomials Pn ( x ), and their particular polynomials play distinguished parts in mathematical analysis theoretically and practically (see, for example, [14,15,16,17]). The linearization coefficients are given in terms of a certain hypergeometric function of the type 5 F4 (1), which can be summed for particular choices of the involved parameters by making use of some well-known formulas in the literature or via the application of some suitable symbolic computation.
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