Bivariate Generalized Shifted Gegenbauer Orthogonal System

For K 0 ,K 1 ≥ 0, λ > − ( 1/2 ) , we examine C ∗ ( λ ,K 0 ,K 1 ) r ( x ) , generalized shifted Gegenbauer orthogonal polynomials, with reference to the weight W ( λ ,K 0 ,K 1 ) ( x ) � (( 2 λ Γ ( 2 λ )) / ( Γ ( λ + ( 1/2 )) 2 ))( x − x 2 ) λ − ( 1/2 ) I ( x ∈ ( 0 , 1 )) d x + K 0 δ 0 + K 1 δ 1 , where the indicator function is denoted by I ( x ∈ ( 0 , 1 )) and δ x indicates the Dirac δ − measure. Then, we construct a bivariate generalized shifted Gegenbauer orthogonal system C ∗ ( λ ,K 0 ,K 1 ) n,r,d ( u,v,w ) over a triangular domain T , with reference to a bivariate measure W ( λ , c ,K 0 ,K 1 ) ( u, v,w ) � (( Γ ( 2 λ + 1 )) / Γ ( λ + ( 1/2 )) 2 ) u λ − ( 1/2 ) ( 1 − v ) λ − ( 1/2 ) ( 1 − w ) c − 1 I ( u ∈ ( 0 , 1 − w )) I ( w ∈ ( 0 , 1 )) d u d w + K 0 δ 0 ( u ) + K 1 δ w − 1 ( u ) , which is given explicitly in the B´ezier form as C ∗ ( λ ,K 0 ,K 1 ) n,r,d ( u,v, w ) � 􏽐 i + j + k � n a n,r,di,j,k B ni,j,k ( u,v, w ) . In addition, for d � 0 , . .. , k , r � 0 , 1 , ... , n , and n ∈ 0 { } ∪ N , we write the coeﬃcients a n,r,di,j,k in closed form and produce

Finding new orthogonal systems which comprise as eigenfunctions of a differential equation is a motivating task. One of such eigenfunctions was developed by Koornwinder [2] which is the generalized Jacobi polynomials for constants K 0 , K 1 ≥ 0 with parameters α, β ∈ 0 { } ∪ N illustrating orthogonality on the interval [− 1, 1], where the corresponding differential equation of order 2(α + β + 3) is given to make the corresponding operator widely suitable for applications. en, Koekoek and Koekoek [3] established an explicit representation of the generalized Jacobi polynomials, of higher order linear differential equations satisfied by P (α,β,K 0 ,K 1 ) k (x) k∈ 0 { } ∪ N ; α, β > − 1; K 0 , K 1 ≥ 0. ese systems were studied by Koornwinder [2] as orthogonal polynomials with reference to a linear combination of W (α,β,K 0 ,K 1 ) (x), the Jacobi weight, and two delta functions δ − 1 , δ 1 at the endpoints of the interval On a square, inducing bivariate generalized orthogonal polynomials is straightforward [4], where tensor product can be utilized. However, the construction over triangular domains is more challenging, where some preliminary abstract was presented in ICNAAM [5]. Now, let i, j, k denote a triple of non-negative integers. In this paper, a construction of orthogonal system of bivariate generalized shifted (BGS) Gegenbauer polynomials in Bézier form over a triangular domain is expressed as C

Generalized Shifted Gegenbauer Polynomials (GSGP).
Generalized orthogonal polynomials were first introduced by Koornwinder [2] and developed by Koekoek and Koekoek in [3]. e univariate shifted Gegenbauer polynomial of degree j in x with parameter λ > − (1/2) denoted by C * (λ) j (x) is achieved by replacing x with 2x − 1 in the definition of the classical Gegenbauer polynomial, For K 0 , K 1 ≥ 0, the generalized shifted Gegenbauer polynomials (GSGP) of degree r have been characterized by AlQudah in [6] and are given by ese polynomials are orthogonal on the interval [0, 1] with reference to the weight [2].
where δ x is the Dirac δ− measure support at x.
and I(x ∈ (0, 1)) denotes the indicator function, which is equal to 1 if x ∈ (0, 1), and 0 otherwise. As the Bernstein polynomials are stable, these polynomials have been expressed in Bernstein form (see [7]). Now, next theorem provides an explicit form for GSGP of degree r in x symbolized by C * (λ,K 0 ,K 1 ) r (x), written in terms of Bernstein polynomials of degree r in x defined as It is worth mentioning that the polynomials B r j (x) describe a basis for Π r , the space of all polynomials of degree ≤r. In addition, for integers i, j, k ≥ 0, we can define the generalized Bernstein polynomials of degree r by B r ijk (x, y, z) � ((r!)/(i!j!k!))x i y j z k where i + j + k � r.
Theorem 2 (see [8]). For r, s � 0, 1, . . . , n and Λ k defined in equation (3), the entries of the transformation matrix of the GS Gegenbauer basis into Bernstein basis of degree n denoted by A n s,r are defined as where μ λ,n,k j,s � (k!(2λ − 1)!)/ Since Bézier curves are vital to numerous applications in "approximation theory" and "numerical analysis," with the demand of higher and/or different degree of Bézier curve [9,10], Corollary 1 presented by AlQudah and AlMheidat in [11] uses Bézier degree elevation developed by Farouki and Rajan [10] to write C * (λ,K 0 ,K 1 ) j (x) of degree j in terms of Bernstein basis of higher and fixed degree n, which helps in numerical stability and computation efficiency.
Corollary 1 (see [8]). GS Gegenbauer polynomials C * (λ, j K 0 , K 1 )(x) of degree j ≤ n are expressed using Bernstein basis of fixed degree n as where [A n s,r ] t is the transpose of the matrix [A n s,r ], for which entries are given in equation (7).

Bivariate Generalized Shifted Gegenbauer Polynomials on Triangular Domains
In this section, we introduce an explicit representation of the bivariate generalized shifted (BGS) Gegenbauer polynomials, denoted by C * (λ,K 0 ,K 1 ) n,r,d (u, v, w), in terms of Bernstein basis over a triangular domain T. With respect to the generalized weight W (λ,c,K 0 ,K 1 ) (u, v, w), these polynomials create an orthogonal system over T.

e Triangular Domain
T. Let p 1 , p 2 , p 3 be noncollinear vertices for a triangle T. en, any point p inside the triangle can be written uniquely in terms of the vertices p i , i � 1, 2, 3 utilizing a triple of non-negative coordinates, called "barycentric coordinates," defined as u � (area(p, We only have two degrees of freedom, despite the fact that we have three "barycentric coordinates." us, any point is exclusively defined by any two of the barycentric coordinates, and thus the triangular domain is For linearly independent and mutually orthogonal polynomials of some degree n, employing the barycentric coordinates (u, v, w), we can illustrate the basis in a triangular structure where the ith row contains i + 1 basis totaling n + 2 2 .
It is worthwhile to mention that generalized Bernstein polynomials, B n ijk (u, v, w), are non-negative and form a partition of unity, 0≤i, j,k≤n i+j+k � n (n!/(i!j!k!))u i v j w k � 1, and are equally weighted as basis functions with equal integrals over the domain T.
Lemma 1 (see [12]). e generalized Bernstein polynomials (3), η λ i,r defined by equation (6), where 2λ + c � 1; and analogues to the construction in [13] using barycentric coordinates u, v, w, the explicit representation of degree-ordered system of C * (λ, , BGS Gegenbauer polynomials in Bernstein form over the triangular domain T is written as these polynomials create an orthogonal system with respect to the weight function W (λ,c,K 0 ,K 1 ) (u, v, w) given as e idea of the construction of this system is to let the bivariate C * (λ,K 0 ,K 1 ) n,r,d (u, v, w) concur with the univariate shifted Gegenbauer polynomial C * (λ) n (x), along one side of T, and to make its variation along each chord parallel to that edge a scaled version of C * (λ,K 0 ,K 1 ) n,r,d (u, v, w). e variation of C * (λ,K 0 ,K 1 ) n,r,d (u, v, w) with w can be placed in a way that on the triangular domain, these polynomials are orthogonal with every polynomial of degree <n, and with other nth degree polynomials C * (λ,K 0 ,K 1 ) n,r,d (u, v, w) for r ≠ s.
Two steps are needed to prove that the polynomials C * (λ,K 0 ,K 1 ) n,r,d (u, v, w) form an orthogonal system over the triangular domain with respect to the generalized Gegenbauer weight function.
(1) Step 1: we need to show that C * (λ,K 0 ,K 1 ) n,r,d (u, v, w) ∈ L n , n ≥ 1, r � 0, 1, . . . , n, where for n ≥ 1, the space L n � p ∈ Π n : p⊥Π n− 1 is defined to be the set of nth degree polynomials that are orthogonal to all polynomials of degree ≤n over a triangular domain T, which will be given in eorem 3. (2) Step 2: for r ≠ s and d ≠ m, we need to show that C * (λ,K 0 ,K 1 ) n,r,d ⊥C * (λ,K 0 ,K 1 ) n,s,m .
Let r � 0, . . . , n and T be a triangular domain; we show that the system of polynomials C * (λ,K 0 ,K 1 ) n,r,d (u, v, w) is orthogonal to all polynomials of degree less than n as in the following theorem.
Note that using the generalized weight function ((Γ(2λ + 1))/(Γ(λ+ (1/2)) 2 )) enables us to separate the integrand, and taking the constrained 2λ + c � 1 facilitates the usage of the orthogonality property given in equation (12). Now we introduce eorem 4 which will be proved to finalize the construction of the BGS Gegenbauer polynomials orthogonal system over T, C * (λ,K 0 ,K 1 ) n,r,d which can be written as Note that for n ∈ 0 { } ∪ N; r, k � 0, . . . , n; and d � 0, 1, . . . , k, the BGS GP defined in equation (14) can be expressed in terms of q n,r (w) as

Recurrence Relation
Bernstein-Bézier basis has remarkable geometric properties, so they can be used in surfaces and curves which will facilitate numerical computations (see [9,14]). For n, i, j, k ∈ 0 { } ∪ N, such that i + j + k � n, let the BGS Gegenbauer orthogonal polynomials C * (λ,K 0 ,K 1 ) n,r,d (u, v, w), r � 0, 1, . . . , n, d � 0, . . . , k be written in terms of Bernstein-Bézier form as follows: Our goal is to obtain a closed formula for coefficients a n,r,d i,j,k and produce a recurrence relation equation to calculate coefficients efficiently.