General Formula Of Three-variable Chebyshev Polynomial Of The Second Kind

In recent years,The research of fourier transform and orthogonal polyno-mials on irregular area has a further development. It is widely used in many fields. It provides the fast transform algorithm and core processes to scien-tific computing,and it provides a strong theoretical foundation to our further research.The author's main task is about the research of orthogonal polynomials on irregular area. That is the general formula problem of Chebyshev polynomials of the second kind in curved tetrahedral domain. This article is divided into two parts. In the first part,we introduce the orthogonal polynomials in gener-alized curved triangle domain briefly,it mainly about Chebyshev polynomials of the first kind and the second kind in Steiner hypocycloid domain;The second part is the main content of this article, in this part we describe the orthogonal polynomials in curved tetrahedral domain,and we obtain and prove some new results about the general formula of three-variables Chebyshev polynomials of the second kind.The Steiner hypocycloid domain is proposed by Steiner (Steiner,1796-1863) who is Swiss mathematician, Following definition: Definition 1 Steiner Hypocycloid domain is the area being enclosed by inequality. whereBy proof,we known that the curvature of Steiner is negative everywhere, Besides,the polynomials in Steiner hypocycloid domain have three-four recur-sive formula.that,We mainly consider two important examples in Steiner hypocycloid do-main,which are the so-called Chebyshev polynomials of the first kind and the second kind,they all have (3) recursive formula.Only the initial conditions and boundary conditions of them are different.According to two kinds of Chebyshev polynomial's recursive formula and initial conditions, We have obtained the general formula of two kinds of Cheby-shev polynomials.This is an important result in the field in recent years.Theorem 1 for n2≤n1,the following general formula sets upwhereω=3z,the the coefficient of general formula B_j,k^[n1,n2] is (n2>0) Note:n1=n2,j=n2,κ=0时,B_j,k^[n1,n2]the formula degenerates to (?) Theorem 2 for n2≤n1,the following general formula sets up whereω=3z,We will do hypocycloid domain(Steiner)extension to three-dimensional,that is curved tetrahedral domain,as follows:Three mixed variables(two conjugate variables z,z and a real variable r) records three-dimensional regional where J(z,z,r)is defined coefficient of determinant thatThe shape of the three-dimensional region J(z, z, r)> 0 likes curved tetra-hedral domain.The orthogonal polynomials in curved tetrahedral domain have three-tiered recursive formula also. As a special case,we still only discuss Chebyshev orthogonal polynomials of the first kind and the second kind, mainly about the definition of PDE of them,three-tiered recursive formula and some properties. Its focus is on the general formula of Chebyshev orthogonal polynomials of the second kind. The corresponding lemma and important conclusions as follows:Lemma 1 if n2=n3=0,for n1>1 then whereLemma 2 if n3=1,then Theorem 3 if n3=0,then...