Mathematics and Computational Sciences

Integral of Legendre polynomials and its properties

Document Type : Original Article

Author

Abdelhamid Rehouma

Department of Mathematics, Faculty of Exact Sciences, University of Hama Lakhdar of Eloued, Algeria

https://doi.org/10.30511/mcs.2025.2066416.1415
Abstract
This paper is concerned with deriving a new system of orthogonal polynomials whose inflection points coincide with their interior roots, primitives of Legendre polynomials. We show a connection between these orthogonal polynomials and two special cases of Jacobi polynomials, we demonstrate
certain identities and extremal properties involving both the integral Legendre polynomials. We use mathematical induction to establish the relation between them. We also present some results for these orthogonal polynomials by using some properties of Jacobi polynomials. General expressions are found for the kernels polynomials associated to integral Legendre polynomials, particularly by using the Jacobi polynomials we prove some results and comparisons and applications with kernel integral Legendre
polynomials. These kernel polynomials can be used to describe the approximation of continuous functions by integral Legendre polynomials. The results are then applied to find the minimum value and the minimizing function for various definite integrals. We conclude the paper with some results by using polynomials integrals of the kernels polynomials.

Keywords

Legendre polynomials
Kernels polynomials
Christoffel-Darboux formula
Three-term recursive relation

Subjects

[1] Abramowitz, M. & Stegun, I.A. (Eds.), Handbook of Mathematical Functions, 10th Edition, Dover, New York, 1972. 
[2] Belinsky, R. Integrals of Legendre polynomials and solution of some partial differential equations, Journal of Applied Analysis, Vol. 6, No. 2 (2000), pp. 259–282. 
[3] Chihara, T. S. An introduction to orthogonal polynomials, Gordon and Breach,New York, 1978. 
[4] Dehghan, R. (2024) Chelyshkov polynomials approximation for solving a class of linear and nonlinear equations arising in astrophysics,Mathematics and Computational Sciences, 5(3), Pages 19-37. 
[5] Foupouagnigni, M. On difference and differential equations for modifications of classical orthogonal polynomials. Habilitation Thesis, Department of Mathematics and Computer Science. University of Kassel, Kassel, January 2006. 
[6] Esmaeelzade Aghdam, Y. & Farnam, B., (2022) A numerical process of the mobile-immobile advection-dispersion model arising in solute transport Mathematics and Computational Sciences, 3(3), 1-10.
[7] Mason, J. C. & Handscomb, D. C. Chebyshev polynomials, (Chapman&Hall, London, 2003). 
[8] Noei Khorshidi, M. & Arab Firoozjaee, M. (2024) Legendre Ritz-Least squares method for the numerical solution of delay differential equations of the multi-pantograph type Mathematics and Computational Sciences, Vol. 5(1) 20-29. 
[9] H.Pijeira.Cabrera and Y.José Bello Cruz and W.Urbina, (2008), On Polar Legendre polynomials,Rocky Mountain Journal of Mathematics, vol 38,1-10.
[10] Rehouma, A., & Atia, M. J. (2022). Study of an example of Markov chains involving Chebyshev polynomials. Integral Transforms and Special Functions, 34(2), 195. 
[11] J. Shen, T. Tang, L.-L. Wang, Spectral methods: algorithms, Analysis and applications, (Springer,2011). 
[12] Szego, G. Orthogonal Polynomials, American Mathematical Society , New York, 1939. 
[13] Zaboli, M.& Tajadodi, H. (2025) A spectral collocation method for solving stochastic fractional integro-differential equation, Mathematics and Computational Sciences, 6(2) 1-31.
Mathematics and Computational Sciences
Volume 6, Issue 3
Summer 2025
Pages 61-76

  • Receive Date 20 July 2025
  • Revise Date 28 August 2025
  • Accept Date 13 September 2025

Home

Submit Manuscript

Reviewers

Contact Us