# Bivariate Generalized Shifted Gegenbauer Orthogonal System

Authors: Mohammad A. Alqudah; Maalee N. Almheidat; Tareq Hamadneh
Journal: Journal of Mathematics
 (2021)

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## Abstract

ForK0,K1≥0, λ>−1/2, we examine Cr∗λ,K0,K1x, generalized shifted Gegenbauer orthogonal polynomials, with reference to the weight Wλ,K0,K1x=2λΓ2λ/Γλ+1/22x−x2λ−1/2Ix∈0,1dx+K0δ0+K1δ1, where the indicator function is denoted by Ix∈0,1 and δx indicates the Dirac δ−measure. Then, we construct a bivariate generalized shifted Gegenbauer orthogonal system ℭn,r,d∗λ,K0,K1u,v,w over a triangular domain T, with reference to a bivariate measure Wλ,γ,K0,K1u,v,w=Γ2λ+1/Γλ+1/22uλ−1/21−vλ−1/21−wγ−1Iu∈0,1−wIw∈0,1dudw+K0δ0u+K1δw−1u, which is given explicitly in the Bézier form as ℭn,r,d∗λ,K0,K1u,v,w=∑i+j+k=nai,j,kn,r,dBi,j,knu,v,w. In addition, for d=0,…,k, r=0,1,…,n, and n∈0∪ℕ, we write the coefficients ai,j,kn,r,d in closed form and produce an equation that generates the coefficients recursively.


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## Body

## 1. Motivation and Introduction

The Gegenbauer polynomialsCkλx are solutions to the Gegenbauer differential equation 1−x2d2y/dx2−2λ+1xdy/dx+kk+2λy=0, k=0,1,…. They can be written using Pkα,βx, Jacobi polynomials defined by Szegö [1] where β=α=λ−1/2 as Ckλx=Γλ+1/2Γk+2λ/Γ2λΓk+λ+1/2Pkλ−1/2,λ−1/2x. Even those orthogonal polynomials are described on the interval −1,1; it is more suitable for analytical and numerical computations to use the interval 0,1.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 K0,K1≥0 with parameters α,β∈0∪ℕ illustrating orthogonality on the interval −1,1, where the corresponding differential equation of order 2α+β+3 is given to make the corresponding operator widely suitable for applications.Then, Koekoek and Koekoek [3] established an explicit representation of the generalized Jacobi polynomials, of higher order linear differential equations satisfied by Pkα,β,K0,K1xk∈0∪ℕ; α,β>−1; K0,K1≥0. These systems were studied by Koornwinder [2] as orthogonal polynomials with reference to a linear combination of Wα,β,K0,K1x, the Jacobi weight, and two delta functions δ−1,δ1 at the endpoints of the interval −1,1.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, leti,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 ℭn,r,d∗λ,K0,K1u,v,w=∑i+j+k=nai,j,kn,r,dBi,j,knu,v,w. For d=0,…,k, r=0,1,…,n, and n∈0∪ℕ, we obtain an explicit form for the coefficients ai,j,kn,r,d and produce a recursive formula to calculate the coefficients ai,j,kn,r,d efficiently.


### 1.1. Generalized Shifted Gegenbauer Polynomials (GSGP)

Generalized orthogonal polynomials were first introduced by Koornwinder [2] and developed by Koekoek and Koekoek in [3]. The univariate shifted Gegenbauer polynomial of degree j in x with parameter λ>−1/2 denoted by Cj∗λx is achieved by replacing x with 2x−1 in the definition of the classical Gegenbauer polynomial, Cjλx.ForK0,K1≥0, the generalized shifted Gegenbauer polynomials (GSGP) of degree r have been characterized by AlQudah in [6] and are given by(1)Cr∗λ,K0,K1x=r!2λ−1!r+2λ−1!Cr∗λx+∑j=0rΛjj!2λ−1!j+2λ−1!Cj∗λx.These polynomials are orthogonal on the interval0,1 with reference to the weight [2].(2)2λΓ2λΓλ+1/22x−x2λ−1/2Ix∈0,1dx+K0δ0+K1δ1,where δx is the Dirac δ−measure support at x.(3)Λj=K0+K1Γ2λ+jΓλ+j+1/2λ+1/2ΓjΓ2λ+1Γλ+j−1/2+K0K1Γ2λ+jΓ2λ+j+1λ+1/22ΓjΓ2λ+12Γj−1,and Ix∈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 Cr∗λ,K0,K1x, written in terms of Bernstein polynomials of degree r in x defined as(4)Bjrx=r!j!r−j!xj1−xr−j,j=0,1,…,r.It is worth mentioning that the polynomialsBjrx describe a basis for Πr, 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 Bijkrx,y,z=r!/i!j!k!xiyjzk where i+j+k=r.Theorem 1 (see [6]).
Forλ>−1/2, K0,K1≥0, and Λk defined in equation (3), the generalized shifted Gegenbauer polynomials Cr∗λ,K0,K1x of degree r have the following Bernstein representation:(5)Cr∗λ,K0,K1x=r!2λ−1!r+2λ−1!∑j=0rηj,rλBjrx+∑k=0rΛkk!2λ−1!k+2λ−1!∑i=0rηi,kλBikx,where(6)ηj,kλ=−1k−jk+λ−1/2jk+λ−1/2k−jki;j=0,1,…,k.In addition, Theorem2 introduced by AlQudah and AlMheidat [8] utilizes the geometric properties of Bernstein basis and proposes the transformation matrix of the GSG basis into Bernstein basis of degree n.Theorem 2 (see [8]).
Forr,s=0,1,…,n and Λk defined in equation (3), the entries of the transformation matrix of the GS Gegenbauer basis into Bernstein basis of degree n denoted by As,rn are defined as(7)As,rn=μk,sλ,n,r+∑k=0rΛkμj,sλ,n,k,where μj,sλ,n,k=k!2λ−1!/ksk+2λ−1!∑j=max0,s+k−nmins,k−1k−jn−ks−jk+λ−1/2jk+λ−1/2k−j.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 Cj∗λ,K0,K1x 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 polynomialsCj∗λ,K0,K1x of degree j≤n are expressed using Bernstein basis of fixed degree n as(8)Cj∗λ,K0,K1x=∑s=0nAs,rntBsnx,r=0,1,…,n,where As,rnt is the transpose of the matrix As,rn, for which entries are given in equation (7).


## 1.1. Generalized Shifted Gegenbauer Polynomials (GSGP)

Generalized orthogonal polynomials were first introduced by Koornwinder [2] and developed by Koekoek and Koekoek in [3]. The univariate shifted Gegenbauer polynomial of degree j in x with parameter λ>−1/2 denoted by Cj∗λx is achieved by replacing x with 2x−1 in the definition of the classical Gegenbauer polynomial, Cjλx.ForK0,K1≥0, the generalized shifted Gegenbauer polynomials (GSGP) of degree r have been characterized by AlQudah in [6] and are given by(1)Cr∗λ,K0,K1x=r!2λ−1!r+2λ−1!Cr∗λx+∑j=0rΛjj!2λ−1!j+2λ−1!Cj∗λx.These polynomials are orthogonal on the interval0,1 with reference to the weight [2].(2)2λΓ2λΓλ+1/22x−x2λ−1/2Ix∈0,1dx+K0δ0+K1δ1,where δx is the Dirac δ−measure support at x.(3)Λj=K0+K1Γ2λ+jΓλ+j+1/2λ+1/2ΓjΓ2λ+1Γλ+j−1/2+K0K1Γ2λ+jΓ2λ+j+1λ+1/22ΓjΓ2λ+12Γj−1,and Ix∈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 Cr∗λ,K0,K1x, written in terms of Bernstein polynomials of degree r in x defined as(4)Bjrx=r!j!r−j!xj1−xr−j,j=0,1,…,r.It is worth mentioning that the polynomialsBjrx describe a basis for Πr, 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 Bijkrx,y,z=r!/i!j!k!xiyjzk where i+j+k=r.Theorem 1 (see [6]).
Forλ>−1/2, K0,K1≥0, and Λk defined in equation (3), the generalized shifted Gegenbauer polynomials Cr∗λ,K0,K1x of degree r have the following Bernstein representation:(5)Cr∗λ,K0,K1x=r!2λ−1!r+2λ−1!∑j=0rηj,rλBjrx+∑k=0rΛkk!2λ−1!k+2λ−1!∑i=0rηi,kλBikx,where(6)ηj,kλ=−1k−jk+λ−1/2jk+λ−1/2k−jki;j=0,1,…,k.In addition, Theorem2 introduced by AlQudah and AlMheidat [8] utilizes the geometric properties of Bernstein basis and proposes the transformation matrix of the GSG basis into Bernstein basis of degree n.Theorem 2 (see [8]).
Forr,s=0,1,…,n and Λk defined in equation (3), the entries of the transformation matrix of the GS Gegenbauer basis into Bernstein basis of degree n denoted by As,rn are defined as(7)As,rn=μk,sλ,n,r+∑k=0rΛkμj,sλ,n,k,where μj,sλ,n,k=k!2λ−1!/ksk+2λ−1!∑j=max0,s+k−nmins,k−1k−jn−ks−jk+λ−1/2jk+λ−1/2k−j.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 Cj∗λ,K0,K1x 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 polynomialsCj∗λ,K0,K1x of degree j≤n are expressed using Bernstein basis of fixed degree n as(8)Cj∗λ,K0,K1x=∑s=0nAs,rntBsnx,r=0,1,…,n,where As,rnt is the transpose of the matrix As,rn, for which entries are given in equation (7).


## 2. 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 byℭn,r,d∗λ,K0,K1u,v,w, in terms of Bernstein basis over a triangular domain T. With respect to the generalized weight Wλ,γ,K0,K1u,v,w, these polynomials create an orthogonal system over T.


### 2.1. The Triangular DomainT

Letp1,p2,p3 be noncollinear vertices for a triangle T. Then, any point p inside the triangle can be written uniquely in terms of the vertices pi,i=1,2,3 utilizing a triple of non-negative coordinates, called “barycentric coordinates,” defined as u=areap,p2,p3/areap1,p2,p3, v=areap1,p,p3/areap1,p2,p3, w=areap1,p2,p/areap1,p2,p3, such that u+v=1−w and p=up1+vp2+wp3. We only have two degrees of freedom, despite the fact that we have three “barycentric coordinates.”Thus, any point is exclusively defined by any two of the barycentric coordinates, and thus the triangular domain is defined asT=u,v,w:u,v,w≥0;w=1−u+v.For linearly independent and mutually orthogonal polynomials of some degreen, 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+22.It is worthwhile to mention that generalized Bernstein polynomials,Bijknu,v,w, are non-negative and form a partition of unity, ∑0≤i,j,k≤ni+j+k=nn!/i!j!k!uivjwk=1, and are equally weighted as basis functions with equal integrals over the domain T.Lemma 1 (see [12]).
The generalized Bernstein polynomialsBijknu,v,w such that i+j+k=n on the triangular domain T satisfy ∬TBijknu,v,wdA=Δ/n+22,Δ=2∗areaT.


### 2.2. Construction

Forγ>−1, n∈0∪ℕ; k=0,…,n; r=0,1,…,n; d=0,1,…,k; Λk defined by equation (3), ηi,rλ defined by equation (6), where 2λ+γ=1; and analogues to the construction in [13] using barycentric coordinates u,v,w, the explicit representation of degree-ordered system of ℭn,r,d∗λ,K0,K1u,v,w, BGS Gegenbauer polynomials in Bernstein form over the triangular domain T is written as(9)ℭn,r,d∗λ,K0,K1u,v,w=∑i=0rηi,rλBiru,v∑j=0n−r−1jn+r+1jBjn−rw,u+v+∑k=0nΛk∑i=0dηi,dλBidu,v∑j=0k−d−1jk+d+1jBjk−dw,u+v,these polynomials create an orthogonal system with respect to the weight function Wλ,γ,K0,K1u,v,w given as(10)Γ2λ+1Γλ+1/22uλ−1/21−vλ−1/21−wγ−1Iu∈0,1−wIw∈0,1dudw+K0δ0u+K1δw−1u.The idea of the construction of this system is to let the bivariateℭn,r,d∗λ,K0,K1u,v,w concur with the univariate shifted Gegenbauer polynomial Cn∗λx, along one side of T, and to make its variation along each chord parallel to that edge a scaled version of ℭn,r,d∗λ,K0,K1u,v,w.The variation ofℭn,r,d∗λ,K0,K1u,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 ℭn,r,d∗λ,K0,K1u,v,w for r≠s.Fori=0,…,k, k∈0∪ℕ, employ the polynomial qk,iw introduced by Farouki et al. [12] defined as(11)qk,iw=∑j=0k−i−1jk+i+1jBjk−iw.Then, any polynomialpw of degree ≤k−i−1, qk,iw is orthogonal to 1−w2i+l+1 on 0,1 for l=0,…,k−i−1. Thus,(12)∫01qk,iw1−w2i+1pwdw=0.By choosingℭ0,0,0∗λ,K0,K1=1, for 0≤r≤n and n=0,1,2,…, the bivariate polynomials ℭn,r,d∗λ,K0,K1u,v,w form a degree-ordered orthogonal system over the triangular domain T.Using the fact thatu+vrBiru/1−w=Biru,v, we rephrase polynomials (9) as(13)ℭn,r,d∗λ,K0,K1u,v,w=∑i=0r−1r−ir+λ−1/2ir+λ−1/2r−iriBiru1−w1−wrqn,rw+∑k=0nΛk∑i=0d−1d−id+λ−1/2id+λ−1/2d−idiBidu1−w1−wdqk,dw.Using Theorem1, for d=0,…,k, r=0,…,n, we get(14)ℭn,r,d∗λ,K0,K1u,v,w=Cr∗λu1−w1−wrqn,rw+Cd∗λu1−w1−wd∑k=0nΛkqk,dw,where qk,dw is defined in equation (11) and Cr∗λt denotes the univariate shifted Gegenbauer polynomial of degree r.Two steps are needed to prove that the polynomialsℭn,r,d∗λ,K0,K1u,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 thatℭn,r,d∗λ,K0,K1u,v,w∈ℒn, n≥1, r=0,1,…,n, where for n≥1, the space ℒn=p∈Πn:p⊥Πn−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 Theorem 3.(2)
Step 2: forr≠s and d≠m, we need to show that ℭn,r,d∗λ,K0,K1⊥ℭn,s,m∗λ,K0,K1.Letr=0,…,n and T be a triangular domain; we show that the system of polynomials ℭn,r,d∗λ,K0,K1u,v,w is orthogonal to all polynomials of degree less than n as in the following theorem.Theorem 3.
Forr=0,1,…,n; n∈ℕ; K0,K1≥0; and λ>−1/2, γ>−1, such that 2λ+γ=1, we have ℭn,r,d∗λ,K0,K1u,v,w with respect to the weight function Wλ,γ,K0,K1u,v,w written as(15)Γ2λ+1Γλ+1/22uλ−1/21−vλ−1/21−wγ−1Iu∈0,1−wIw∈0,1dudw+K0δ0u+K1δw−1u.Proof.
Forl=0,…,m; m=0,…,n−1, construct a set of bivariate polynomials:(16)Hl,mλu,v,w=Cl∗λu1−w1−wmwn−m−1,where Cl∗λt denotes the univariate shifted Gegenbauer polynomial of degree l. Employ equation (6); then, Hl,mλu,v,w can be expressed as(17)Hl,mλu,v,w=1−wmwn−m−1∑j=0l−1l−jl+λ−1/2jl+λ−1/2l−jljBjlu1−w=1−wmwn−m−1∑j=0lηj,lλBjlu1−w.
Note that span ofHl,mλu,v,w comprises the set of Bernstein polynomials Bjmu/1−w1−wmwn−m−1=Bjmu,vwn−m−1, j=0,…,m; m=0,…,n−1 spanning Πn−1. Accordingly, it is sufficient to show that(18)I=∬Tℭn,r,d∗λ,K0,K1u,v,wHl,mλu,v,wWλ,γ,K0,K1u,v,wdA=0,l=0,…,m;m=0,…,n−1.
Now, by substitutingℭn,r,d∗λ,K0,K1u,v,w, Hl,mλu,v,w, and Wλ,γ,K0,K1u,v,w into the integral(19)I=Δ∫01∫01−wℭn,r,d∗λ,K0,K1u,v,wHl,mλu,v,wWλ,γ,K0,K1u,v,wdudw,and by collecting, reordering the terms, and applying t=u/1−w, we get(20)IΔ=Γ2λ+1Γλ+1/22∑j=0λ−1/2−1jλ−1/2j∫01Cr∗λtCl∗λttλ−1/21−tjdt∫01wn−m−11−wj+γ+r+m+λ−1/2qn,rwdw+K0∫01Cr∗λtCl∗λtδ0tdt∫01wn−m−11−wr+m+1qn,rwdw+K1∫01Cr∗λtCl∗λtδ1tdt∫01wn−m−11−wr+m+1qn,rwdw+Γ2λ+1Γλ+1/22∑k=0nΛk∑j=0λ−1/2−1jλ−1/2j∫01Cl∗λtCd∗λttλ−1/21−tjdt∫01wn−m−11−wj+γ+d+m+λ−1/2qk,dwdw+K0∑k=0nΛk∫01Cl∗λtCd∗λtδ0tdt∫011−wd+m+1wn−m−1qk,dwdw+K1∑k=0nΛk∫01Cl∗λtCd∗λtδ1tdt∫011−wd+m+1wn−m−1qk,dwdw.
Note thatl<d<r, when m<d<r, and so the orthogonality property of the shifted Gegenbauer polynomials can be used, and when r≤d≤m≤n−1, use the orthogonality property given in equation (12). Hence, I=0, which completes the proof.
Note that using the generalized weight functionΓ2λ+1/Γλ+1/22uλ−1/21−vλ−1/21−wγ−1Iu∈0,1−wIw∈0,1dudw+K0δ0u+K1δw−1u enables us to separate the integrand, and taking the constrained 2λ+γ=1 facilitates the usage of the orthogonality property given in equation (12).
Now we introduce Theorem4 which will be proved to finalize the construction of the BGS Gegenbauer polynomials orthogonal system over T, ℭn,r,d∗λ,K0,K1u,v,wn=0,r=0,d=0∞,n,n.Theorem 4.
LetK0,K1≥0, λ>−1/2, γ>−1, and Wλ,γ,K0,K1u,v,w be the generalized weight function defined as Γ2λ+1/Γλ+1/22uλ−1/21−vλ−1/21−wγ−1Iu∈0,1−wIw∈0,1dudw+K0δ0u+K1δw−1u. Then, for r≠s, d≠m, we have ℭn,r,d∗λ,K0,K1u,v,w⊥ℭn,s,m∗λ,K0,K1u,v,w.Proof.
LetT=u,v,w:u,v,w≥0;u+v+w=1 be a triangular domain, and for r,s=0,…,n; m=0,…,s; and d=0,…,r, where r≠s and d≠m, let(21)I=∬Tℭn,r,d∗λ,K0,K1u,v,wℭn,s,m∗λ,K0,K1u,v,wWλ,γ,K0,K1u,v,wdA,which can be written as(22)I=∫01∫01−wℭn,r,d∗λ,K0,K1u,v,wℭn,s,m∗λ,K0,K1u,v,wWλ,γ,K0,K1u,v,wΔdudw.
Note that forn∈0∪ℕ; r,k=0,…,n; and d=0,1,…,k, the BGS GP defined in equation (14) can be expressed in terms of qn,rw as(23)ℭn,r,d∗λ,K0,K1u,v,w=1+ΛnCr∗λu1−w1−wrqn,rw+Cd∗λu1−w1−wd∑k=0n−1Λkqk,dw.
By substituting the terms into the integralI, making the change of variables u=t1−w, separating the integrand, and reordering the integrals, the value of I/Δ equals(24)Γ2λ+1Γλ+1/221+Λn2∑i=0λ−1/2−1iλ−1/2i∫01Cr∗λtCs∗λt1−titλ−1/2dt∫01qn,rwqn,sw1−wi+r+s+γ+λ−1/2dw+Γ2λ+1Γλ+1/221+Λn∑j=0n−1Λj∑i=0λ−1/2−1iλ−1/2i∫01Cr∗λtCm∗λt1−titλ−1/2dt∫01qn,rwqj,mw1−wi+r+m+γ+λ−1/2dw+Γ2λ+1Γλ+1/221+Λn∑k=0n−1Λk∑i=0λ−1/2−1iλ−1/2i∫01Cs∗λtCm∗λt1−titλ−1/2dt∫01qn,sqn,rwqj,mw1−wi+r+m+γ+λ−1/2dw+Γ2λ+1Γλ+1/22∑k=0n−1Λk∑j=0n−1Λj∑i=0λ−1/2−1iλ−1/2i∫01Cd∗λtCm∗λt1−titλ−1/2dt∫01qk,dwqj,mw1−wi+d+m+γ+λ−1/2dw+K01+Λn2∫01Cr∗λtCs∗λtδ0tdt∫01qn,rwqn,sw1−wr+s+1dw+K01+Λn∑j=0n−1Λj∫01Cr∗λtCm∗λtδ0tdt∫01qn,rwqj,mw1−wr+m+1dw+K01+Λn∑k=0n−1Λk∫01Cs∗λ∫01qn,swqk,dw1−ws+d+1dw+K0∑k=0n−1Λk∑j=0n−1Λj∫01Cd∗λtCm∗λtδ0tdt∫01qk,dwqj,mw1−wd+m+1dw+K11+Λn2∫01Cr∗λtCs∗λtδ1tdt∫01qn,rwqn,sw1−wr+s+1dw+K11+Λn∑j=0n−1Λj∫01Cr∗λtCm∗λtδ1tdt∫01qn,rwqj,mw1−wr+m+1dw+K11+Λn∑k=0n−1Λk∫01Cs∗λtCd∗λtδ1tdt∫01qn,swqk,dw1−ws+d+1dw+K1∑k=0n−1Λk∑j=0n−1Λj∫01Cd∗λtCm∗λtδ1tdt∫01qk,dwqj,mw1−wd+m+1dw.
Sinced≠r and m≠s, then r≠s and d≠m, and so the orthogonality property of the shifted Gegenbauer polynomials is utilized. Then, I=0, and thus ℭn,r,d∗λ,K0,K1u,v,w⊥ℭn,s,m∗λ,K0,K1u,v,w.
Hence, forλ>−1/2, γ>−1, ℭn,r,d∗λ,K0,K1u,v,wn=0,r=0,d=0∞,n,n form an orthogonal system over the triangular domain T, with respect to the generalized weight Wλ,γ,K0,K1u,v,w.


## 2.1. The Triangular DomainT

Letp1,p2,p3 be noncollinear vertices for a triangle T. Then, any point p inside the triangle can be written uniquely in terms of the vertices pi,i=1,2,3 utilizing a triple of non-negative coordinates, called “barycentric coordinates,” defined as u=areap,p2,p3/areap1,p2,p3, v=areap1,p,p3/areap1,p2,p3, w=areap1,p2,p/areap1,p2,p3, such that u+v=1−w and p=up1+vp2+wp3. We only have two degrees of freedom, despite the fact that we have three “barycentric coordinates.”Thus, any point is exclusively defined by any two of the barycentric coordinates, and thus the triangular domain is defined asT=u,v,w:u,v,w≥0;w=1−u+v.For linearly independent and mutually orthogonal polynomials of some degreen, 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+22.It is worthwhile to mention that generalized Bernstein polynomials,Bijknu,v,w, are non-negative and form a partition of unity, ∑0≤i,j,k≤ni+j+k=nn!/i!j!k!uivjwk=1, and are equally weighted as basis functions with equal integrals over the domain T.Lemma 1 (see [12]).
The generalized Bernstein polynomialsBijknu,v,w such that i+j+k=n on the triangular domain T satisfy ∬TBijknu,v,wdA=Δ/n+22,Δ=2∗areaT.


## 2.2. Construction

Forγ>−1, n∈0∪ℕ; k=0,…,n; r=0,1,…,n; d=0,1,…,k; Λk defined by equation (3), ηi,rλ defined by equation (6), where 2λ+γ=1; and analogues to the construction in [13] using barycentric coordinates u,v,w, the explicit representation of degree-ordered system of ℭn,r,d∗λ,K0,K1u,v,w, BGS Gegenbauer polynomials in Bernstein form over the triangular domain T is written as(9)ℭn,r,d∗λ,K0,K1u,v,w=∑i=0rηi,rλBiru,v∑j=0n−r−1jn+r+1jBjn−rw,u+v+∑k=0nΛk∑i=0dηi,dλBidu,v∑j=0k−d−1jk+d+1jBjk−dw,u+v,these polynomials create an orthogonal system with respect to the weight function Wλ,γ,K0,K1u,v,w given as(10)Γ2λ+1Γλ+1/22uλ−1/21−vλ−1/21−wγ−1Iu∈0,1−wIw∈0,1dudw+K0δ0u+K1δw−1u.The idea of the construction of this system is to let the bivariateℭn,r,d∗λ,K0,K1u,v,w concur with the univariate shifted Gegenbauer polynomial Cn∗λx, along one side of T, and to make its variation along each chord parallel to that edge a scaled version of ℭn,r,d∗λ,K0,K1u,v,w.The variation ofℭn,r,d∗λ,K0,K1u,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 ℭn,r,d∗λ,K0,K1u,v,w for r≠s.Fori=0,…,k, k∈0∪ℕ, employ the polynomial qk,iw introduced by Farouki et al. [12] defined as(11)qk,iw=∑j=0k−i−1jk+i+1jBjk−iw.Then, any polynomialpw of degree ≤k−i−1, qk,iw is orthogonal to 1−w2i+l+1 on 0,1 for l=0,…,k−i−1. Thus,(12)∫01qk,iw1−w2i+1pwdw=0.By choosingℭ0,0,0∗λ,K0,K1=1, for 0≤r≤n and n=0,1,2,…, the bivariate polynomials ℭn,r,d∗λ,K0,K1u,v,w form a degree-ordered orthogonal system over the triangular domain T.Using the fact thatu+vrBiru/1−w=Biru,v, we rephrase polynomials (9) as(13)ℭn,r,d∗λ,K0,K1u,v,w=∑i=0r−1r−ir+λ−1/2ir+λ−1/2r−iriBiru1−w1−wrqn,rw+∑k=0nΛk∑i=0d−1d−id+λ−1/2id+λ−1/2d−idiBidu1−w1−wdqk,dw.Using Theorem1, for d=0,…,k, r=0,…,n, we get(14)ℭn,r,d∗λ,K0,K1u,v,w=Cr∗λu1−w1−wrqn,rw+Cd∗λu1−w1−wd∑k=0nΛkqk,dw,where qk,dw is defined in equation (11) and Cr∗λt denotes the univariate shifted Gegenbauer polynomial of degree r.Two steps are needed to prove that the polynomialsℭn,r,d∗λ,K0,K1u,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 thatℭn,r,d∗λ,K0,K1u,v,w∈ℒn, n≥1, r=0,1,…,n, where for n≥1, the space ℒn=p∈Πn:p⊥Πn−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 Theorem 3.(2)
Step 2: forr≠s and d≠m, we need to show that ℭn,r,d∗λ,K0,K1⊥ℭn,s,m∗λ,K0,K1.Letr=0,…,n and T be a triangular domain; we show that the system of polynomials ℭn,r,d∗λ,K0,K1u,v,w is orthogonal to all polynomials of degree less than n as in the following theorem.Theorem 3.
Forr=0,1,…,n; n∈ℕ; K0,K1≥0; and λ>−1/2, γ>−1, such that 2λ+γ=1, we have ℭn,r,d∗λ,K0,K1u,v,w with respect to the weight function Wλ,γ,K0,K1u,v,w written as(15)Γ2λ+1Γλ+1/22uλ−1/21−vλ−1/21−wγ−1Iu∈0,1−wIw∈0,1dudw+K0δ0u+K1δw−1u.Proof.
Forl=0,…,m; m=0,…,n−1, construct a set of bivariate polynomials:(16)Hl,mλu,v,w=Cl∗λu1−w1−wmwn−m−1,where Cl∗λt denotes the univariate shifted Gegenbauer polynomial of degree l. Employ equation (6); then, Hl,mλu,v,w can be expressed as(17)Hl,mλu,v,w=1−wmwn−m−1∑j=0l−1l−jl+λ−1/2jl+λ−1/2l−jljBjlu1−w=1−wmwn−m−1∑j=0lηj,lλBjlu1−w.
Note that span ofHl,mλu,v,w comprises the set of Bernstein polynomials Bjmu/1−w1−wmwn−m−1=Bjmu,vwn−m−1, j=0,…,m; m=0,…,n−1 spanning Πn−1. Accordingly, it is sufficient to show that(18)I=∬Tℭn,r,d∗λ,K0,K1u,v,wHl,mλu,v,wWλ,γ,K0,K1u,v,wdA=0,l=0,…,m;m=0,…,n−1.
Now, by substitutingℭn,r,d∗λ,K0,K1u,v,w, Hl,mλu,v,w, and Wλ,γ,K0,K1u,v,w into the integral(19)I=Δ∫01∫01−wℭn,r,d∗λ,K0,K1u,v,wHl,mλu,v,wWλ,γ,K0,K1u,v,wdudw,and by collecting, reordering the terms, and applying t=u/1−w, we get(20)IΔ=Γ2λ+1Γλ+1/22∑j=0λ−1/2−1jλ−1/2j∫01Cr∗λtCl∗λttλ−1/21−tjdt∫01wn−m−11−wj+γ+r+m+λ−1/2qn,rwdw+K0∫01Cr∗λtCl∗λtδ0tdt∫01wn−m−11−wr+m+1qn,rwdw+K1∫01Cr∗λtCl∗λtδ1tdt∫01wn−m−11−wr+m+1qn,rwdw+Γ2λ+1Γλ+1/22∑k=0nΛk∑j=0λ−1/2−1jλ−1/2j∫01Cl∗λtCd∗λttλ−1/21−tjdt∫01wn−m−11−wj+γ+d+m+λ−1/2qk,dwdw+K0∑k=0nΛk∫01Cl∗λtCd∗λtδ0tdt∫011−wd+m+1wn−m−1qk,dwdw+K1∑k=0nΛk∫01Cl∗λtCd∗λtδ1tdt∫011−wd+m+1wn−m−1qk,dwdw.
Note thatl<d<r, when m<d<r, and so the orthogonality property of the shifted Gegenbauer polynomials can be used, and when r≤d≤m≤n−1, use the orthogonality property given in equation (12). Hence, I=0, which completes the proof.
Note that using the generalized weight functionΓ2λ+1/Γλ+1/22uλ−1/21−vλ−1/21−wγ−1Iu∈0,1−wIw∈0,1dudw+K0δ0u+K1δw−1u enables us to separate the integrand, and taking the constrained 2λ+γ=1 facilitates the usage of the orthogonality property given in equation (12).
Now we introduce Theorem4 which will be proved to finalize the construction of the BGS Gegenbauer polynomials orthogonal system over T, ℭn,r,d∗λ,K0,K1u,v,wn=0,r=0,d=0∞,n,n.Theorem 4.
LetK0,K1≥0, λ>−1/2, γ>−1, and Wλ,γ,K0,K1u,v,w be the generalized weight function defined as Γ2λ+1/Γλ+1/22uλ−1/21−vλ−1/21−wγ−1Iu∈0,1−wIw∈0,1dudw+K0δ0u+K1δw−1u. Then, for r≠s, d≠m, we have ℭn,r,d∗λ,K0,K1u,v,w⊥ℭn,s,m∗λ,K0,K1u,v,w.Proof.
LetT=u,v,w:u,v,w≥0;u+v+w=1 be a triangular domain, and for r,s=0,…,n; m=0,…,s; and d=0,…,r, where r≠s and d≠m, let(21)I=∬Tℭn,r,d∗λ,K0,K1u,v,wℭn,s,m∗λ,K0,K1u,v,wWλ,γ,K0,K1u,v,wdA,which can be written as(22)I=∫01∫01−wℭn,r,d∗λ,K0,K1u,v,wℭn,s,m∗λ,K0,K1u,v,wWλ,γ,K0,K1u,v,wΔdudw.
Note that forn∈0∪ℕ; r,k=0,…,n; and d=0,1,…,k, the BGS GP defined in equation (14) can be expressed in terms of qn,rw as(23)ℭn,r,d∗λ,K0,K1u,v,w=1+ΛnCr∗λu1−w1−wrqn,rw+Cd∗λu1−w1−wd∑k=0n−1Λkqk,dw.
By substituting the terms into the integralI, making the change of variables u=t1−w, separating the integrand, and reordering the integrals, the value of I/Δ equals(24)Γ2λ+1Γλ+1/221+Λn2∑i=0λ−1/2−1iλ−1/2i∫01Cr∗λtCs∗λt1−titλ−1/2dt∫01qn,rwqn,sw1−wi+r+s+γ+λ−1/2dw+Γ2λ+1Γλ+1/221+Λn∑j=0n−1Λj∑i=0λ−1/2−1iλ−1/2i∫01Cr∗λtCm∗λt1−titλ−1/2dt∫01qn,rwqj,mw1−wi+r+m+γ+λ−1/2dw+Γ2λ+1Γλ+1/221+Λn∑k=0n−1Λk∑i=0λ−1/2−1iλ−1/2i∫01Cs∗λtCm∗λt1−titλ−1/2dt∫01qn,sqn,rwqj,mw1−wi+r+m+γ+λ−1/2dw+Γ2λ+1Γλ+1/22∑k=0n−1Λk∑j=0n−1Λj∑i=0λ−1/2−1iλ−1/2i∫01Cd∗λtCm∗λt1−titλ−1/2dt∫01qk,dwqj,mw1−wi+d+m+γ+λ−1/2dw+K01+Λn2∫01Cr∗λtCs∗λtδ0tdt∫01qn,rwqn,sw1−wr+s+1dw+K01+Λn∑j=0n−1Λj∫01Cr∗λtCm∗λtδ0tdt∫01qn,rwqj,mw1−wr+m+1dw+K01+Λn∑k=0n−1Λk∫01Cs∗λ∫01qn,swqk,dw1−ws+d+1dw+K0∑k=0n−1Λk∑j=0n−1Λj∫01Cd∗λtCm∗λtδ0tdt∫01qk,dwqj,mw1−wd+m+1dw+K11+Λn2∫01Cr∗λtCs∗λtδ1tdt∫01qn,rwqn,sw1−wr+s+1dw+K11+Λn∑j=0n−1Λj∫01Cr∗λtCm∗λtδ1tdt∫01qn,rwqj,mw1−wr+m+1dw+K11+Λn∑k=0n−1Λk∫01Cs∗λtCd∗λtδ1tdt∫01qn,swqk,dw1−ws+d+1dw+K1∑k=0n−1Λk∑j=0n−1Λj∫01Cd∗λtCm∗λtδ1tdt∫01qk,dwqj,mw1−wd+m+1dw.
Sinced≠r and m≠s, then r≠s and d≠m, and so the orthogonality property of the shifted Gegenbauer polynomials is utilized. Then, I=0, and thus ℭn,r,d∗λ,K0,K1u,v,w⊥ℭn,s,m∗λ,K0,K1u,v,w.
Hence, forλ>−1/2, γ>−1, ℭn,r,d∗λ,K0,K1u,v,wn=0,r=0,d=0∞,n,n form an orthogonal system over the triangular domain T, with respect to the generalized weight Wλ,γ,K0,K1u,v,w.


## 3. 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 ℭn,r,d∗λ,K0,K1u,v,w, r=0,1,…,n, d=0,…,k be written in terms of Bernstein–Bézier form as follows:(25)ℭn,r,d∗λ,K0,K1u,v,w=∑i+j+k=nai,j,kn,r,dBi,j,knu,v,w.Our goal is to obtain a closed formula for coefficientsai,j,kn,r,d and produce a recurrence relation equation to calculate coefficients efficiently.Theorem 5.
Forn∈0∪ℕ, r=0,1,…,n, and d=0,…,k and non-negative integers i,j,k such that i+j+k=n, Bernstein–Bézier coefficients ai,j,kn,r,d of BGS Gegenbauer polynomials ℭn,r,d∗λ,K0,K1u,v,w=∑i+j+k=nai,j,kn,r,dBi,j,knu,v,w are expressed by(26)aijkn,r,d=−1kn+r+1kn−rkAi,rn−k+Λk−1jk+d+1jk−djAi,dk−j,0≤k≤n−r,0,k>n−r.where Ai,rn−k are given in equation (7).Proof.
To find the coefficientsai,j,kn,r,d, we have the following cases:Case 1. k>n−r. From equation (9), it is obvious that the polynomial ℭn,r,d∗λ,K0,K1u,v,w has degree less than or equal to n−r in the variable w. Therefore, aijkn,r=0.Case 2. 0≤k≤n−r. Equate equation (25) with equation (9), and then we obtain the following:(27)∑i+j=n−kaijkn,rBijknu,v,w=−1kn+r+1kBkn−rw,u+v∑i=0rηi,rλBiru,v+Λk∑j=0k−d−1jk+d+1jBjk−dw,u+v∑i=0dηi,dλBidu,v,where Λk is defined in equation (3) and ηi,rλ is defined in equation (6). In the last equation, we compare similar powers of w to obtain(28)∑i=0n−kaijkn,rnkBin−ku,v=∑i=0n−kaijkn,rn!i!j!k!uivj=−1kn+r+1kn−rku+vn−r−k∑i=0rηi,rλBidu,v+Λk∑j=0k−d−1jk+d+1jk−dju+vk−d−j∑i=0dηi,dλBidu,v.
Then, we utilize Corollary1 and some combinatorial identities to write ∑i=0n−kaijkn,rnkBin−ku,v in above equation as(29)−1kn+r+1kn−rk∑i=0rAi,rn−kBin−ku,v+Λk∑j=0k−d−1jk+d+1jk−dj∑i=0dAi,dk−jBik−ju,v.
Therefore, we obtain a closed form for Bernstein–Bézier coefficients.
We will incorporate a result of Farouki et al. [12] to find the recursive formula for aijkn,r, which will be used in the process of obtaining the recurrence equation.Lemma 2.
Letgx=∑i+j+k=mdi,j,kBi,j,kmx∈Πm. Then, the following holds:(30)gx∈ℒm⇔∑i+j+k=mdi,j,kci,j,k=0,∀hx=∑i+j+k=mci,j,kBi,j,kmx∈Πm−1.Now, useBijkn−1u,v,w=u+v+wBijkn−1u,v,w to express the generalized n−1th degree Bernstein polynomial Bijkn−1u,v,w as(31)1ni+1Bi+1,j,knu,v,w+j+1Bi,j+1,knu,v,w+k+1Bi,j,k+1nu,v,w.Fori+j+k=n−1, using the construction of the BGS Gegenbauer polynomials ℭn,r,d∗λ,K0,K1u,v,w, we get Bijkn−1u,v,w,ℭn,r,d∗λ,K0,K1u,v,w=0. Finally, use Lemma 2 to attain the desired recurrence relation to generate the coefficients ai,j,kn,r recursively on k, ai,j,k+1n,r=−1/k+1i+1ai+1,j,kn,r+j+1ai,j+1,kn,r, such that ai,n−i,0n,r=Ai,rn for i∈0∪ℕ where Ai,rn is given in equation (7).


## 4. Applications and Conclusion

Orthogonal polynomials provide a natural approach to the extension, implementation, and study of solutions to multiple differential equations. They are vital to stability and efficiency of numerical methods, the characteristics of which are useful for solving engineering and physical problems. So, discovering novel types of orthogonal polynomials serve in the advancements of applications in diverse disciplines.Therefore, orthogonal polynomials have gained notable attention. They are key to numerous problems in mathematical analysis and approximation theory. They play a significant role in splines, Fredholm theory, fractional calculus, integrodifferential equations, stochastic differential equations (SDEs), spectral element method (SEM), and data integration.It is therefore more practical to construct orthogonal polynomial systems recursively in lieu of absolute forms. The recursive relation makes it possible to establish any polynomial of a given degree directly.In this paper, we investigated the GS Gegenbauer orthogonal polynomialsCr∗λ,K0,K1x using Bernstein representation on the interval 0,1 with regard to the measure [2] 2λΓ2λ/Γλ+1/22x−x2λ−1/2Ix∈0,1dx+K0δ0+K1δ1.Then, forλ>−1/2, we constructed ℭn,r,d∗λ,K0,K1u,v,w=∑i+j+k=nai,j,kn,r,dBi,j,knu,v,w, the BGS Gegenbauer orthogonal polynomials, on the triangular domain T in terms of generalized Bernstein coefficients ai,j,kn,r,d, with respect to the weight Γ2λ+1/Γλ+1/22uλ−1/21−vλ−1/21−wγ−1Iu∈0,1−wIw∈0,1dudw+K0δ0u+K1δw−1u, for γ>−1, 2λ+γ=1.We have also shown thatℭn,r,d∗λ,K0,K1u,v,w∈ℒn,n≥1,r=0,…,n, and shown that for r≠s and d≠m, ℭn,r,d∗λ,K0,K1u,v,w⊥ℭn,s,m∗λ,K0,K1 to conclude that these bivariate polynomials form an orthogonal system over T. The coefficients ai,j,kn,r,d were given in an closed and explicit form, and then a recurrence relationship was defined to allow us to calculate the polynomials effectively and efficiently.The construction in this paper can be extended to ofi-variate polynomials over a i-dimensional simplex where(32)Ti=u=u0,…,ui:uj≥0,∑j=0iuj=1denotes the i-dimensional unit simplex in barycentric coordinates. Moreover,(33)Bζnu=nζuζ=n!ζ0!,…,ζk!u0ζ0…uiζi,ζ=n,where ζ=ζ0,…,ζd∈0∪ℕi+1 and ζ=ζ0+⋯+ζk=n denotes the Bernstein basis for nth degree polynomial over the simplex Ti.



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