Before we study the theory of orthogonal polynomials in several variables, we give various examples of families of orthogonal polynomials in several variables in this chapter. More detailed study on some of the families will be given in later chapters, here we deal with the basic properties.

Notation and Preliminary

Throughout this book, we will use the standard multi-index notation. We denote by ℤ0 the set of nonnegative integers. A multi-index is usually denoted by α, α = (α1, …, αd) ∈ ℤd0. Whenever α appears with subscript, it means a multi-index. In this spirit we define, for example, α = α1… αd and |a| = α1 + …+ αd; and if α, β ∈ ℤd0, then we define For∈ ℤd0 and x = (x1, …, Xd), a monomial in variables x1, …Xd is a product

The number |a| is called the total degree of xα. A polynomial P in d variables is a linear combination of monomials,

where the coefficients cα are in a field κ, usually the rational numbers ℚ, the real numbers ℝ or the complex numbers ℂ. The degree of a polynomial is defined as the highest total degree of its monomials. The collection of all polynomials in x with coefficients in κ is denoted by κ[x1, …, xd], which has the structure of a commutative ring. We shall use the abbreviation ∏d to denote κ[xi, …,xd]. We also denote the space of polynomials of degree at most n by. When d = 1, we drop the superscript and write ∏ and ∏n instead.