TY - JOUR
T1 - Recursion Formulas for Integrated Products of Jacobi Polynomials
AU - Beuchler, Sven
AU - Haubold, Tim
AU - Pillwein, Veronika
N1 - Publisher Copyright:
© 2023, The Author(s).
PY - 2024/6
Y1 - 2024/6
N2 - 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 series. With these contiguous relations one can prove several recursion formulas of those series. This theoretical result allows to compute integrals over products of Jacobi polynomials in a very efficient recursive way. Moreover, the authors present an application to numerical analysis where it can be used in algorithms which compute the approximate solution of boundary value problem of partial differential equations by means of the finite elements method. With the aid of the contiguous relations, the approximate solution can be computed much faster than using numerical integration. A numerical example illustrates this effect.
AB - 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 series. With these contiguous relations one can prove several recursion formulas of those series. This theoretical result allows to compute integrals over products of Jacobi polynomials in a very efficient recursive way. Moreover, the authors present an application to numerical analysis where it can be used in algorithms which compute the approximate solution of boundary value problem of partial differential equations by means of the finite elements method. With the aid of the contiguous relations, the approximate solution can be computed much faster than using numerical integration. A numerical example illustrates this effect.
KW - High order finite element methods
KW - Hypergeometric function
KW - Orthogonal polynomials
KW - Recurrence equations
KW - 33C45
KW - 33C70
KW - 65N30
UR - http://www.scopus.com/inward/record.url?scp=85160232260&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2105.08989
DO - 10.48550/arXiv.2105.08989
M3 - Article
AN - SCOPUS:85160232260
VL - 59
SP - 583
EP - 618
JO - Constructive approximation
JF - Constructive approximation
SN - 0176-4276
IS - 3
ER -