Abstract
The main aim of this paper is to develop four innovative linearization formulas for some nonsymmetric Jacobi polynomials. This means that we find the coefficients of the products of Jacobi polynomials of certain parameters. In general, these coefficients are expressed in terms of certain hypergeometric functions of the unit argument. We employ some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij for reducing such coefficients. Moreover, and based on a certain Whipple transformation, two new closed formulas for summing certain terminating hypergeometric functions of the unit argument are deduced. New formulas for some definite integrals are given with the aid of the derived linearization formulas.
Highlights
The general linearization problem consists of finding the coefficients Gi,j,k in the expansion of the product of two polynomials Ai(x) and Bj(x) in terms of an arbitrary sequence of orthogonal polynomials {Pk(x)}, that is, i+jAi(x)Bj(x) = Gi,j,kPk(x). ( ) k=An important particular case of problem ( ) is the standard linearization or ClebschGordan-type problem, which consists of finding the coefficients Li,j,k in the expansion of the product of two polynomials Ai(x) and Aj(x) in terms of the sequence {Ak(x)}k≥, i.e., i+jAi(x)Aj(x) = Li,j,kAk(x).Linearization problems appear in several applications
The basic idea behind obtaining such formula is obtaining a recurrence relation satisfied by the linearization coefficients, and solving it
The standard linearization problem associated to Jacobi polynomials and to establish the conditions of nonnegativity of the linearization coefficients has been extensively studied by many authors
Summary
The general linearization problem consists of finding the coefficients Gi,j,k in the expansion of the product of two polynomials Ai(x) and Bj(x) in terms of an arbitrary sequence of orthogonal polynomials {Pk(x)}, that is, i+j. The main objective of this article is to establish new linearization formulas for the product of two nonsymmetric Jacobi polynomials Another objective is to introduce some applications based on the developed linearization formulas and their related hypergeometric functions. The basic idea behind obtaining the new derived formulas is based on reducing the linearization coefficients of Jacobi polynomials which were obtained by Rahman [ ] with the aid of employing some symbolic algebraic algorithms such as the algorithms of Zeilberger, Petkovsek and van Hoeij. Rahman in [ ] has developed another linearization formula for the Jacobi polynomials This formula is expressed in terms of a terminating hypergeometric function of the type F ( ), and it is explicitly stated in the following theorem.
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