Differential Equations for Ultraspherical Jacobian Polynomials - Aliyev, Sahib; Aliyev, Jeyhun (Karshi Multidisciplinary International Scientific Journal, 2025)
The aim of this article is to show a method of constructing Jacobi’s system of polynomials and Differential equations for ultraspherical Jacobian polynomials.Generalized Jacobian polynomials, which are one of the classical orthogonal polynomials, and orthonormal Jacobian polynomials with respect to the weight function have been determined. Fourier series for this polynomial has been investigated. Now we will look at the problem with some applications of Jacobian polynomials, including the application of differential equations.For any weight function h(x) there is a unique sequence of many terms Pn(x) that have a positive leading coefficient and satisfy the orthonormality condition. Given the weight function h(x) = (1 − x)α (1 + x)β, x ∈ (−1, 1), α > −1 , β > −1 using the Leibniz formula, a polynomial of degree n Pn(α,β)(x) is defined. It is clear that the polynomial Pn(α,β)(x) is a Jacobi polynomial with a high degree coefficient and it is proved that orthogonal with respect to the weight function in the interval (−1, 1).By defining the norm, the corresponding orthonormal polynomial is determined. Using the orthogonality and orthonormality conditions, a zero order polynomial P¯0(α,β)(x)is defined and constant. C0, C1 coefficients are determined using the corresponding lemma.P¯1(α,β)(x) is defined and other P¯2(α,β)(x) polynomials etc. can be found.Taking all this into account, it can be applied to differential equations and their solution methods for ultraspheric Jacobian polynomials.