TIGHT FRAMES OF JACOBI POLYNOMIALS ON A SIMPLEX

SHAYNE WALDRON AND YUAN XU

Abstract. The Jacobi polynomials {R_{α}^{κ}} of degree n on the d–simplex are the orthogonal projection of X^{α} := x_{1}^{α_{1}} … x_{d}^{α_{d}}(1 − x_{1} − ⋯ − x_{d})^{α_{d+1}}, |α| = n onto the orthogonal polynomials of degree n with respect to the weight function X^{κ}, κ_{i} > −1 on the simplex. Although the set {R_{α}^{κ} : |α| = n, α ∈ ℕ_{0}^{d+1}} is not a basis, we show that these polynomials enjoy a property akin to being an orthonormal basis, i.e., after a suitable scaling they form a tight frame. The proof is based on the fact that the inner product of R_{α}^{κ} and an element of an orthonormal basis can be expressed in terms of the Hahn polynomials in several variables.

1. Introduction

The Jacobi polynomials P_{n}^{(α,β)}(1 − 2x) are orthogonal polynomials with respect to the weight function x^{α}(1 − x)^{β}, α, β > −1 on [0, 1]. They can be expressed in terms of the _{2}F_{1} hypergeometric function as

(1.1)  \frac{P_{n}^{(α,β)}(1 − 2x)}{P_{n}^{(α,β)}(1)} = _{2}F_{1}( − n, n + α + β + 1; α + 1; x).

In this paper we consider the analogue of these polynomials on the simplex

T^{d} := {x ∈ ℝ^{d} : x_{1} ≥ 0, …, x_{d} ≥ 0, x_{d+1} ≥ 0}, x_{d+1} := 1 − x_{1} − ⋯ − x_{d}

in ℝ^{d} with respect to the weight function W_{κ}, κ_{i} > −1 defined by

W_{κ}(x) := w_{κ}x_{1}^{κ_{1}} ⋯ x_{d}^{κ_{d}}x_{d+1}^{κ_{d+1}}, w_{κ} := \frac{Γ(|κ| + d + 1)}{∏_{i=1}^{d+1} Γ(κ_{i} + 1)}.

Here κ_{i} > −1 ensures that W_{κ} is integrable over the simplex T^{d}, and the integral is 1 by the choice of w_{κ}. Denote the corresponding inner product by

⟨f, g⟩_{W_{κ}} := ∫_{T^{d}} f(x)g(x)W_{κ}(x)dx.

Several bases of orthogonal polynomials with respect to ⟨·, ·⟩_{W_{κ}} are known explicitly (see [2] and the following section). Here we introduce another family of orthogonal polynomials which is a generalization of (1.1) using a multivariate analogue of the 2F_{1} hypergeometric function.

Throughout this paper, we shall use standard multiindex notation. For example, for α, β ∈ ℕ_{0}^{d} and x ∈ ℝ^{d},

x^{α} := x_{1}^{α_{1}} … x_{d}^{α_{d}}, α! := α_{1}! ⋯ α_{d}!, (α)_{β} := (α_{1})_{β_{1}} ⋯ (α_{d})_{β_{d}},

Date: June 22, 2001, 2001.

1991 Mathematics Subject Classification. 33C50.

Key words and phrases. Jacobi polynomials, Hahn polynomials, simplex.

The work of the second author is supported by the National Science Foundation under Grant DMS-9802265.

where (x)_{n} := x(x + 1) ⋯ (x + n − 1) is the Pochhammer symbol, and

|α| := α_{1} + ⋯ + α_{d}.

When working with the simplex T^{d} it is natural to use the barycentric coordinates

X(x) := (x_{1}, …, x_{d}, 1 − x_{1} − ⋯ − x_{d}), x ∈ ℝ^{d},

since the symmetry group of T^{d} acts as a permutation group on these coordinates. In this notation, the weight function can be written W_{κ} = w_{κ} X^{κ}. For a ∈ ℝ, α ∈ ℤ^{d+1} and β ∈ ℕ_{0}^{d+1}, the Lauricella function of type A is defined by

(1.2)  F_{A}(a, α; β; X) := ∑_{γ∈ℕ_{0}^{d+1}} \frac{(a)_{|γ|}(α)_{γ}}{(β)_{γ}γ!} X^{γ},

which converges on T^{d} (see [4] or [3, vol. 1, p. 224]). This is one of the four generalizations of _{2}F_{1} to several variables given by Appell. Definition 1.1. For every α ∈ ℕ_{0}^{d+1} and κ = (κ_{1}, …, κ_{d+1}), κ_{i} > −1, define

R_{α}^{κ} := F_{A}(|α| + |κ| + d, −α; κ + 1, X), x ∈ ℝ^{d},

where |κ| := κ_{1} + ⋯ + κ_{d+1} and 1 := (1, 1, …, 1).

Since (−α)_{γ} = 0 for γ > α, R_{α}^{κ} is a linear combination of powers X^{γ}, γ ≤ α, and so defines a polynomial of degree n = |α| on ℝ^{d}. Furthermore, R_{α}^{κ} are orthogonal polynomials of exact degree n with respect to the weight W_{κ}, i.e., belong to the space

V_{n}^{d}(W_{κ}) := {P ∈ Π_{n}^{d} : ⟨P, Q⟩_{W_{κ}} = 0, ∀Q ∈ Π_{n−1}^{d}}.

Here Π_{n}^{d} denotes the space of polynomials of degree at most n in d variables. Proposition 1.2. For each α ∈ ℕ_{0}^{d+1} with |α| = n, R_{α}^{κ} ∈ V_{n}^{d}(W_{κ}). Furthermore,

(1.3)  R_{α}^{κ} = (−1)^{n} \frac{(n + |κ| + d)_{n}}{(κ + 1)_{α}} X^{α} + p_{α}^{κ},  p_{α}^{κ} ∈ Π_{n−1}^{d},

and so is a constant multiple of the orthogonal projection of X^{α} onto V_{n}^{d}(W_{κ}). It is known that

dim V_{n}^{d}(W_{κ}) = dim P_{n}^{d} = \binom{n + d − 1}{n},

where P_{n}^{d} denotes the space of homogeneous polynomials of degree n in d variables. On the other hand, the number of R_{α}^{κ} of degree n is

#\{R_{α}^{κ} : |α| = n, α ∈ ℕ_{0}^{d+1}\} = dim P_{n}^{d+1} = \binom{n + d}{n} > dim V_{n}^{d}(W_{κ}),

and so {R_{α}^{κ} : |α| = n} is not a basis of V_{n}^{d}(W_{κ}). For example, if d = 2, then {R_{α}^{κ} : |α| = n} contains (n + 2)(n + 1)/2 elements, whilst a basis of V_{n}^{2}(W_{κ}) has n+1 elements. The amount of redundancy in this set can be measured by the ratio

\frac{#\{R_{α}^{κ} : |α| = n\}}{dim V_{n}^{d}(W_{κ})} = 1 + \frac{n}{d}.

The set does contain a basis of V_{n}^{d}(W_{κ}), e.g., the subset {R_{α}^{κ} : |α| = n, α_{d+1} = 0} (or take α_{i} = 0 for any i = 1, 2, …, d). For many applications, such as computing an orthogonal expansion, this redundancy is a problem. The main result of this paper shows, however, that {R_{α}^{κ}} satisfy the following property akin to being an orthonormal basis.

Theorem 1.3. For any f ∈ V_{n}^{d}(W_{κ}),

(1.4)  f = C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ⟨f, R_{α}^{κ}⟩_{W_{κ}}R_{α}^{κ},

where C_{n,κ} > 0 is the constant defined by

(1.5)  C_{n,κ} := \frac{(|κ| + d + 1)_{2n}}{(n + |κ| + d)_{n}^{2}} .

To put this result in perspective, let us reformulate it in terms of the reproducing kernel P_{n}(W_{κ}; x, y) of V_{n}^{d}(W_{κ}), which is defined by the property

(1.6)  ⟨P_{n}(W_{κ}; x, ·), f⟩_{W_{κ}} = f(x),  ∀f ∈ V_{n}^{d}(W_{κ}).

If {P_{ν}} forms an orthogonal basis of V_{n}^{d}(W_{κ}), that is, ⟨P_{ν}, P_{μ}⟩_{W_{κ}} = 0 for ν ≠ μ, then it follows that

(1.7)  P_{n}(W_{κ}; x, y) = ∑_{ν} h_{ν}^{−1}P_{ν}(x)P_{ν}(y), h_{ν} := ⟨P_{ν}, P_{ν}⟩_{W_{κ}}.

The orthogonal projection of x^{ν}, ν ∈ ℕ_{0}^{d}, |ν| = n onto V_{n}^{d}(W_{κ}), denoted by V_{ν}(x), gives the so–called monomial basis of V_{n}^{d}(W_{κ}) (cf. [2, p. 47]). The polynomial V_{ν} is a constant multiple of R_{α}^{κ} for α = (ν, 0). There is another basis U_{ν} of V_{n}^{d}(W_{κ}) which is biorthogonal to V_{ν} in the sense that ⟨U_{ν}, V_{μ}⟩_{W_{κ}} = 0 for ν ≠ μ. The reproducing kernel can be written in terms of these as

(1.8)  P_{n}(W_{κ}; x, y) = ∑_{|ν|=n} b_{ν}^{−1}V_{ν}(x)U_{ν}(y), b_{ν} := ⟨V_{ν}, U_{ν}⟩_{W_{κ}}.

There is also a compact formula of this kernel as an integral of a Gegenbauer polynomial, see [11]. An equivalent form of Theorem 1.3 is the following result: Theorem 1.4. Let C_{n,κ} be the constant defined as above. For n ∈ ℕ_{0},

(1.9)  P_{n}(W_{κ}; x, y) = C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} R_{α}^{κ}(x)R_{α}^{κ}(y).

In view of formula (1.8) and the fact V_{ν} is a constant multiple of R_{α}^{κ}, α = (ν, 0), the formula (1.9) is highly nontrivial. The full set {R_{α}^{κ}} appropriately reflects the symmetries of the simplex, which allows (1.9). Moreover, the formulae for R_{α}^{κ} are more symmetric than those for U_{ν} or any known orthonormal P_{ν}.

A formula of the type (1.4) is suggested in the case d = 2 by a result of [7], but this does not give the constants (κ + 1)_{α}/α! (which are computed only in some specific cases). Several corollaries of our main result are discussed in Section 3, including

⟨f, f⟩_{W_{κ}} = C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ⟨f, R_{α}^{κ}⟩_{W_{κ}}^{2}, ∀f ∈ V_{n}^{d}(W_{κ}).

In the language of frames (see, e.g., Young [12]), this means that the family

\{\sqrt{\frac{(κ + 1)_{α}}{α!}} R_{α}^{κ} : |α| = n, α ∈ ℕ_{0}^{d+1}\}

forms a tight frame of V_{n}^{d}(W_{κ}) with frame bound 1/C_{n,κ}.

The proof of the main result depends on Hahn polynomials of several variables. For one variable, the Hahn polynomial Q(x; a, b, N) is defined using the _{3}F_{2} series by (see, for example, [1, p. 345])

(1.10) Q_{n}(x; a, b, N) := _{3}F̃_{2} (\begin{matrix}−n, n + a + b + 1, −x \\ a + 1, −N \end{matrix}; 1) ,  n = 0, 1, …, N,

where _{3}F̃_{2} is defined as the usual _{3}F_{2} with the summation terminating at N. These polynomials are the discrete orthogonal polynomials defined on the set {0, 1, …, N}, which are orthogonal with respect to the binomial distribution, i.e.,

∑_{x=0}^{N} \frac{(a + 1)_{x}(b + 1)_{N−x}}{x!(N − x)!} Q_{n}(x; a, b, N)Q_{m}(x; a, b, N)

= \frac{(−1)^{n}n!(b + 1)_{n}(n + a + b + 1)_{N+1}}{N!(2n + a + b + 1)(−N)_{n}(a + 1)_{n}} δ_{n,m},  n, m ≤ N.

Hahn polynomials of several variables were studied by Karlin and McGregor in [6], and they also appear in the recent work of Rosengren [8] (the biorthogonal ones are studied in [10]). Their definition and basic properties will be given in the following section. For the proof of the main result, we express the polynomials R_{α}^{κ} in terms of an orthogonal basis {P_{ν}} of V_{n}^{d}(W_{κ}). It turns out that the inner product ⟨R_{α}^{κ}, P_{ν}⟩_{W_{κ}} is a constant multiple of a Hahn polynomial in the variable α, and so the orthogonality of the Hahn polynomials can be used to prove (1.4).

The rest of this paper is organized as follows. In Section 2, we introduce the orthogonal basis {P_{ν}} and the Hahn polynomials in several variables, and prove Proposition 1.2. In Section 3, we prove the main results and discuss some further consequences.

2. Jacobi polynomials and Hahn polynomials on a simplex

For x = (x_{1}, …, x_{d}) ∈ ℝ^{d}, we introduce the following notation,

(2.1)  x_{0} := 0,  x_{j} := (x_{1}, …, x_{j}),  1 ≤ j ≤ d.

Note x_{d} = x. The following formula is useful in dealing with integrals on T^{d},

(2.2)  ∫_{T^{d}} f(x)dx = ∫_{T^{d−1}} ∫_{0}^{1} f(x_{d−1}, (1 − |x_{d−1}|)y)dy(1 − |x_{d−1}|)dx_{d−1}.

This follows from the change of variables x_{d} = y(1 − |x_{d−1}|). We start by proving the orthogonality of R_{α}^{κ} given in Proposition 1.2, which we restate as follows.

Proposition 2.1. For each α ∈ ℕ_{0}^{d+1} with |α| = n, R_{α}^{κ} ∈ V_{n}^{d}(W_{κ}). Furthermore, the set {R_{α}^{κ} : α ∈ ℕ_{0}^{d+1}, |α| = n, α_{d+1} = 0} is a basis of V_{n}^{d}(W_{κ}).

Proof. The polynomials {X^{γ} : |γ| = n, γ ∈ ℕ_{0}^{d+1}} form a (Bernstein) basis of Π_{n}^{d}. Thus it suffices to show that R_{α}^{κ} is orthogonal to X^{γ} where |γ| = n − 1. The multivariate beta integral shows that

∫_{T^{d}} X^{τ} dx = \frac{∏_{j=1}^{d+1} Γ(τ_{j} + 1)}{Γ(|τ| + d + 1)}, τ_{i} > −1, 1 ≤ i ≤ d + 1.

Using (2.2) and the formula for R_{α}^{κ} it follows for |γ| = n − 1 and |α| = n that

∫_{T^{d}} R_{α}^{κ}(x)X^{γ}W_{κ}(x)dx = w_{κ} ∑_{β∈ℕ_{0}^{d+1}} \frac{(n + |κ| + d)_{|β|}(−α)_{β}}{(κ + 1)_{β}β!} ∫_{T^{d}} X^{β+γ+κ}dx

= ∑_{β∈ℕ_{0}^{d+1}} \frac{(n + |κ| + d)_{|β|}(−α)_{β}}{(κ + 1)_{β}β!} \frac{(κ + 1)_{β+γ}}{(|κ| + d + 1)_{|β|+|γ|}}

= \frac{(κ + 1)_{γ}}{(|κ| + d + 1)_{|γ|}} ∑_{β∈ℕ_{0}^{d+1}} \frac{(γ + κ + 1)_{β}(−α)_{β}}{(κ + 1)_{β}β!}

= \frac{(κ + 1)_{γ}}{(|κ| + d + 1)_{|γ|}} ∏_{i=1}^{d+1} ∑_{0≤β_{i}≤α_{i}} \frac{(−α_{i})_{β_{i}}(γ_{i} + κ_{i} + 1)_{β_{i}}}{(κ_{i} + 1)_{β_{i}}β_{i}!} .

A Chu–Vandermonde sum shows that the sum over β_{i} is equal to (−γ_{i})_{α_{i}}/(κ_{i}+1)_{α_{i}}, which is zero if γ_{i} < α_{i}. Since |γ| = n − 1 < n = |α|, there is at least one i such that γ_{i} < α_{i}. Hence we conclude that R_{α}^{κ} is orthogonal to X^{γ}, i.e., R_{α}^{κ} ∈ V_{n}^{d}(W_{κ}). The fact that {R_{α}^{κ} : |α| = n, α_{d+1} = 0} is a basis of V_{n}^{d}(W_{κ}) follows by considering their leading terms (which are the monomials of degree n). □

The polynomials R_{α}^{κ} are generalizations of the monomial polynomials V_{ν}, and the proof above is similar to the proof that the V_{ν} ∈ V_{n}^{d}(W_{κ}) (see, [2, p. 48]).

Amongst the various known bases of V_{n}^{d}(W_{κ}), the most important to us is an orthonormal basis that dates back to Proriol for d = 2 and κ = 0 (see [5] and [2]). To define this basis, we need to introduce further notations.

Throughout this paper, we use ν, μ to denote multiindicies in ℕ_{0}^{d} and use α, β, γ for multiindices in ℕ_{0}^{d+1}, unless stated otherwise. For ν = (ν_{1}, …, ν_{d}) ∈ ℕ_{0}^{d} and κ = (κ_{1}, …, κ_{d+1}) ∈ 𝕂^{d+1}, where 𝕂 = ℝ or 𝕂 = ℕ_{0}, we introduce the notation

(2.3) ν^{j} := (ν_{j}, …, ν_{d}) and κ^{j} := (κ_{j}, …, κ_{d+1}),  j = 1, 2, …, d + 1,

Note |ν^{d+1}| = 0. These we treat as elements in ℕ_{0}^{d−j+1} and 𝕂^{d−j+2}, respectively, so that the quantities |ν^{j}| and |κ^{j}| are defined as before (e.g., |κ^{j}| = κ_{j+1} +…+κ_{d+1}). The orthogonal polynomials are defined as follows (see, e.g., [2, p. 47]).

Definition 2.2. For ν ∈ ℕ_{0}^{d} and x ∈ ℝ^{d},

P_{ν}(x) := ∏_{j=1}^{d} (\frac{1 − |x_{j}|}{1 − |x_{j−1}|})^{|ν^{j+1}|} P_{ν_{j}}^{(a_{j},κ_{j})} (\frac{2x_{j}}{1 − |x_{j−1}|} − 1) ,

where |x_{j}| = x_{1} + … + x_{j} and a_{j} is defined by

(2.4)  a_{j} := a_{j}(κ, ν) := |κ^{j+1}| + 2|ν^{j+1}| + d − j,  1 ≤ j ≤ d + 1.

The notation a_{j} is reserved for the value in (2.4) throughout the paper. The polynomials {P_{ν} : |ν| = n} form an orthogonal basis of V_{n}^{d}(W_{κ}), with

h_{ν} := ⟨P_{ν}, P_{ν}⟩_{W_{κ}} = \frac{1}{(|κ| + d + 1)_{2|ν|}} ∏_{j=1}^{d}(κ_{j} + a_{j} + 2)_{2ν_{j}}h_{ν_{j}}^{(a_{j},κ_{j})}, (2.5)

where h_{n}^{(a,b)} denotes the L_{2}–norm of the Jacobi polynomials, i.e.,

(2.6)  h_{n}^{(a,b)} := \frac{Γ(a + b + 2)}{2^{a+b+1}Γ(a + 1)Γ(b + 1)} ∫_{−1}^{1}[P_{n}^{(a,b)}(x)]^{2}(1 − x)^{a}(1 + x)^{b}dx

= \frac{(a + 1)_{n}(b + 1)_{n}(a + b + n + 1)}{n!(a + b + 2)_{n}(a + b + 2n + 1)}.

In the following section, we will express the polynomials R_{α}^{κ} in terms of P_{ν}. The inner product of R_{α}^{κ} and P_{ν} turns out to be a constant multiple of a Hahn polynomial in the variable α. We now define these Hahn polynomials.

The Hahn polynomials of several variables are studied in [6]. They can be defined by a generating function. A generating function for the Hahn polynomials of one variable is

(2.7)  (1 + t)^{N} \frac{P_{j}^{(a,b)}(\frac{1−t}{1+t})}{P_{j}^{(a,b)}(1)} = ∑_{n=0}^{N} \binom{N}{n}Q_{j}(n; a, b, N)t^{n}.

For several variables we denote the Hahn polynomials by ϕ_{ν}(α; κ, N). These are discrete orthogonal polynomials indexed by ν ∈ ℕ_{0}^{d}, |ν| ≤ N which are defined on the set {α ∈ ℕ_{0}^{d+1} : |α| = N} and are orthogonal with respect to the binomial distribution given by the parameter κ = (κ_{1}, …, κ_{d+1}). They are defined by the following generating function, where x^{j} with x ∈ ℝ^{d+1} is defined as in (2.3). Definition 2.3. Suppose κ ∈ ℝ^{d+1} with κ_{i} > −1, and N ∈ ℕ. For ν ∈ ℕ_{0}^{d}, |ν| ≤ N define the Hahn polynomials ϕ_{ν}(α; κ, N) at α ∈ ℕ_{0}^{d+1}, |α| = N by (2.8)

|x|^{N−|ν|} ∏_{j=1}^{d} |x^{j}|^{ν_{j}} \frac{P_{ν_{j}}^{(a_{j},κ_{j})}(2x_{j}/|x^{j}| − 1)}{P_{ν_{j}}^{(a_{j},κ_{j})}(1)} = ∑_{|α|=N} \frac{N!}{α!}ϕ_{ν}(α; κ, N)x^{α}, x ∈ ℝ^{d+1},

where a_{j} = a_{j}(κ, ν) is defined in (2.4).

Setting d = 1, x_{1} = t, x_{2} = 1 in the above, the left hand side becomes

(1 + t)^{N−ν_{1}}(1 + t)^{ν_{1}} \frac{P_{ν_{j}}^{(κ_{2},κ_{1})}(2t/(t + 1) − 1)}{P_{ν_{j}}^{(κ_{2},κ_{1})}(1)} = (1 + t)^{N} \frac{(−1)^{ν_{j}}P_{ν_{j}}^{(κ_{2},κ_{1})}(\frac{1−t}{1+t})}{P_{ν_{j}}^{(κ_{2},κ_{1})}(1)}

and the right hand side becomes

∑_{α_{1}=0}^{N} \frac{N!}{α_{1}!(N − α_{1})!}ϕ_{ν}(α_{1}, N − α_{1}; κ_{1}, κ_{2}, N)x_{1}^{α_{1}}x_{2}^{N−α_{1}}

= ∑_{α_{1}=0}^{N} \binom{N}{α_{1}}ϕ_{ν}(α_{1}, N − α_{1}; κ_{1}, κ_{2}, N)t^{α_{1}}.

Hence

ϕ_{ν_{1}}(α_{1}, N − α_{1}; κ_{1}, κ_{2}, N) = (−1)^{ν_{1}}Q_{ν_{1}}(α_{1}; κ_{2}, κ_{1}, N).

Let us indicate how our definition agrees with that given in [6]. There the

generating function is denoted by G_{r,N}(w̄ ; ᾱ|ν̄), which is defined by an inductive

formula (see [6, (5.7), p. 278] and the first equation on page 279). We make the following substitutions: r = d + 1, ᾱ = (κ_{d+1}, κ_{d}, …, κ_{1}), w̄ = (x_{d+1}, x_{d}, …, x_{1}), ν̄ = (ν_{d}, ν_{d−1}, …, ν_{1}), and work out the generating function explicitly to obtain the form presented in Definition 2.3. We now give an explicit formula, and the orthogonality relation for ϕ_{ν}(α; κ, N). Both are stated in [6] by inductive formulae, from which the explicit formulae can be worked out using the aforementioned substitutions. The formula for ϕ_{ν}(α; κ, N) presented below was obtained by further simplifications, and we give a complete proof.

Proposition 2.4. For α ∈ ℕ_{0}^{d+1}, |α| = N and ν ∈ ℕ_{0}^{d}, |ν| ≤ N,

ϕ_{ν}(α; κ, N) = \frac{(−1)^{|ν|}}{(−|α|)_{|ν|}} ∏_{j=1}^{d} \frac{(κ_{j} + 1)_{ν_{j}}}{(a_{j} + 1)_{ν_{j}}} (−|α^{j}|+|ν^{j+1}|)_{ν_{j}}Q_{ν_{j}}(α_{j}; κ_{j}, a_{j}, |α^{j}|−|ν^{j+1}|).

Proof. Recall the definitions of α^{j}, ν^{j} and κ^{j} from (2.3). First we show that ϕ_{ν}(α; κ, N) satisfies the inductive formula

(2.9) ϕ_{ν}(α;κ, N)
= \frac{(−|α^{2}|)_{|ν^{2}|}}{(−|α|)_{|ν^{2}|}} Q_{ν_{1}}(|α^{2}| − |ν^{2}|; a_{1}, κ_{1}, |α| − |ν^{2}|)ϕ_{ν^{2}}(α^{2}; κ^{2}, N − α_{1}).

Denote the right hand side of the above by ψ_{ν}(α; κ, N). Then splitting the sum over |α| = N, α ∈ ℕ_{0}^{d+1} into two sums using α = (α_{1}, α^{2}), |α| = α_{1} + |α^{2}|, we have

∑_{|α|=N} \frac{N!}{α!}ψ_{ν}(α; κ, N)x^{α} = ∑_{α_{1}=0}^{N} \frac{N!}{α_{1}!(N − α_{1})!} \frac{(−N + α_{1})_{|ν^{2}|}}{(−N)_{|ν^{2}|}}
× Q_{ν_{1}}(N − α_{1} − |ν^{2}|; a_{1}, κ_{1}, N − |ν^{2}|)x_{1}^{α_{1}}

× ∑_{|α^{2}|=N−α_{1}} \frac{(N − α_{1})!}{(α^{2})!}ϕ_{ν^{2}}(α^{2}; κ^{2}, N − α_{1})x_{2}^{α_{2}} ⋯ x_{d+1}^{α_{d+1}}.

The second sum on the right hand side can be evaluated using the generating function of Definition 2.3, the fact that a_{j}(κ^{2}, ν^{2}) = a_{j+1}(κ, ν) = a_{j+1}, and the change of index j + 1 → j, to obtain

|x^{2}|^{N−α_{1}−|ν^{2}|} ∏_{j=2}^{d} |x^{j}|^{ν_{j}} \frac{P_{ν_{j}}^{(a_{j},κ_{j})}(2x_{j}/|x^{j}| − 1)}{P_{ν_{j}}^{(a_{j},κ_{j})}(1)}.

Here only the factor |x^{2}|^{−α_{1}} depends on the index in the first summation.Since (−N + α_{1})_{|ν^{2}|} = 0 if N − α_{1} < |ν^{2}|, the first sum times |x^{2}|^{−α_{1}} simplifies to

∑_{α_{1}=0}^{N−|ν^{2}|} \frac{(N − |ν^{2}|)!}{α_{1}!(N − |ν^{2}| − α_{1})!}Q_{ν_{1}}(N − α_{1} − |ν^{2}|; a_{1}, κ_{1}, N − |ν^{2}|)(\frac{x_{1}}{|x^{2}|})^{α_{1}}

= (\frac{x_{1}}{|x^{2}|})^{N−|ν^{2}|} ∑_{n=0}^{N−|ν^{2}|} \binom{N − |ν^{2}|}{n}Q_{ν_{1}}(n; a_{1}, κ_{1}, N − |ν^{2}|)(\frac{|x^{2}|}{x_{1}})^{n}.

Taking t = |x^{2}|/x_{1} in (2.7) and using |x^{1}| = |x| = x_{1} + |x^{2}|, this can be summed to

(\frac{x_{1}}{|x^{2}|})^{N−|ν^{2}|}(\frac{|x^{1}|}{x_{1}})^{N−|ν^{2}|} \frac{P_{ν_{1}}^{(a_{1},κ_{1})}(2x_{1}/|x^{1}| − 1)}{P_{ν_{1}}^{(a_{1},κ_{1})}(1)}.

Combining these, we see that ψ_{ν}(α; κ, N) has the same generating function as ϕ_{ν}(α; κ, N), and so (2.9) is proved. We may apply (2.9) repeatedly, giving

ϕ_{ν}(α; κ, N) = ∏_{j=1}^{d} \frac{(−|α^{j+1}|)_{|ν^{j+1}|}}{(−|α^{j}|)_{|ν^{j+1}|}} Q_{ν_{j}}(|α^{j+1}| − |ν^{j+1}|; a_{j}, κ_{j}, |α^{j}| − |ν^{j+1}|).

To obtain the desired formula for ϕ_{ν}(α; κ, N) from the above, use the transformation of the Hahn polynomials (see, for example, [1, p. 346])

Q_{n}(x; a, b, N) = (−1)^{n} \frac{(b + 1)_{n}}{(a + 1)_{n}} Q_{n}(N − x; b, a, N)

and simplify the constants using

∏_{j=1}^{d} \frac{(−|α^{j+1}|)_{|ν^{j+1}|}}{(−|α^{j}|)_{|ν^{j+1}|}} = ∏_{j=1}^{d}(−|α^{j}|+|ν^{j+1}|)_{ν_{j}} \frac{(−|α^{j+1}|)_{|ν^{j+1}|}}{(−|α^{j}|)_{|ν^{j}|}} = ∏_{j=1}^{d} \frac{(−|α^{j}| + |ν^{j+1}|)_{ν_{j}}}{(−|α|)_{|ν|}} ,

and the fact that ν^{d+1} = 0 and |α^{1}| = |α|. □

Since (−|α^{j}|+|ν^{j+1}|)_{ν_{j}}/(−|α^{j}|+|ν^{j+1}|)_{i} is a polynomial in α if i ≤ ν_{j}, it follows that the formula in Proposition 2.4 is a polynomial in α of degree |ν|. The proof that the ϕ_{ν}(α; κ, N) are orthogonal with respect to the binomial distribution is given in [6] and the constant A_{ν} below is given by inductive formulae (5.13), (5.14), (5.18) in [6]. The verification (using the substitution that we mentioned earlier) is straightforward, we omit the details. Proposition 2.5. For ν, μ ∈ ℕ_{0}^{d} with |ν|, |μ| ≤ N,

∑_{|α|=N} \frac{(κ + 1)_{α}}{α!} ϕ_{ν}(α; κ, N)ϕ_{μ}(α; κ, N) = A_{ν}δ_{ν,μ},

where A_{ν} is given by

(2.10) A_{ν} := \frac{(−1)^{|ν|}(|κ| + d + 1)_{N+|ν|}Γ(|κ| + d + 1)}{(−N)_{|ν|} N! Γ(κ_{d+1} + 1)}

× ∏_{j=1}^{d} \frac{Γ(a_{j} + 1)ν_{j}! (κ_{j} + 1)_{ν_{j}}}{Γ(κ_{j} + a_{j} + ν_{j} + 1)(a_{j} + 1)_{ν_{j}}(a_{j} + κ_{j} + 2ν_{j} + 1)}.

For other properties of these polynomials, such as recurrence relations, see [6].

3. Proof of the main result and some consequences

Since {P_{ν} : |ν| = n} is an orthogonal basis of V_{n}^{d}(W_{κ}), each R_{α}^{κ} can be written

(3.1)  R_{α}^{κ} = ∑_{|ν|=n} \frac{⟨R_{α}^{κ}, P_{ν}⟩_{W_{κ}}}{⟨P_{ν}, P_{ν}⟩_{W_{κ}}} P_{ν},  |α| = n.

In order to prove the main result, we need an explicit formula for ⟨R_{α}^{κ}, P_{ν}⟩_{W_{κ}}. We start with the following lemma. Lemma 3.1. For κ_{1}, κ_{2} > −1 and λ_{1}, λ_{2} ∈ ℕ_{0},

∫_{0}^{1} P_{j}^{(κ_{2},κ_{1})}(2t − 1)t^{λ_{1}+κ_{1}}(1 − t)^{λ_{2}+κ_{2}}dt = \frac{Γ(κ_{1} + 1)Γ(κ_{2} + 1)(κ_{2} + 1)_{j}}{Γ(κ_{1} + κ_{2} + 1) j!}

× \frac{(κ_{1} + 1)_{λ_{1}}(κ_{2} + 1)_{λ_{2}}(−λ_{1} − λ_{2})_{j}}{(κ_{1} + κ_{2} + 2)_{λ_{1}+λ_{2}+j}} Q_{j}(λ_{1}; κ_{1}, κ_{2}, λ_{1} + λ_{2}).

Proof. We start with a formula for the hypergeometric functions (see [1, p. 67]):

∫_{0}^{1} x^{p}(1 − x)^{q}_{2}F_{1}(a, b, c; x)dx = \frac{Γ(p + 1)Γ(q + 1)}{Γ(p + q + 2)}_{3}F_{2}(\begin{matrix}a, b, p + 1 \\ c, p + q + 2 \end{matrix}; 1).

Using (1.1) and the fact that P_{n}^{(a,b)}(1) = (a + 1)_{n}/n!, we conclude that

∫_{0}^{1} P_{j}^{(κ_{2},κ_{1})}(2t − 1)t^{λ_{1}+κ_{1}}(1 − t)^{λ_{2}+κ_{2}}dt

= \frac{(κ_{2} + 1)_{j}Γ(λ_{1} + κ_{1} + 1)Γ(λ_{2} + κ_{2} + 1)}{j! Γ(λ_{1} + λ_{2} + κ_{1} + κ_{2} + 2)}_{3}F_{2}(\begin{matrix}−j, j + κ_{1} + κ_{2} + 1, λ_{1} + κ_{1} + 1 \\ κ_{2} + 1, λ_{1} + λ_{2} + κ_{1} + κ_{2} + 2 \end{matrix}; 1).

Using the transformation of the _{3}F_{2} ([1, Corollary 3.3.4, p. 141])

3F_{2}(\begin{matrix}−j, a, b \\ d, e \end{matrix}; 1) = \frac{(d − a)_{n}(e − a)_{n}}{(d)_{n}(e)_{n}}_{3}F_{2}(\begin{matrix}−j, a, a + b − n − d − e + 1 \\ a − n − d + 1, a − n − e + 1 \end{matrix}; 1)

and the fact that (−j − A)_{j} = (−1)^{j}(A + 1)_{j}, the stated result follows from the definition of the Hahn polynomials given in (1.10). □

For the proof of the main result, the essential ingredient is the following result, which is of interest in itself. Proposition 3.2. If α ∈ ℕ_{0}^{d+1}, ν ∈ ℕ_{0}^{d} and |α| = |ν| = n, then

⟨P_{ν}, R_{α}^{κ}⟩_{W_{κ}} = (−1)^{n} \frac{(n + |κ| + d)_{n} n!}{(|κ| + d + 1)_{2n}} ∏_{j=1}^{d} \frac{(a_{j} + 1)_{ν_{j}}}{ν_{j}!} ϕ_{ν}(α; κ, n).

Proof. Since both R_{α}^{κ} and P_{ν} are elements of V_{n}^{d}(W_{κ}), it follows from (1.3) that

(3.2)  ⟨P_{ν}, R_{α}^{κ}⟩_{W_{κ}} = (−1)^{n} \frac{(n + |κ| + d)_{n}}{(κ + 1)_{α}} ⟨P_{ν}, X^{α}⟩_{W_{κ}}.

Using the definition of P_{ν}, the formula (2.2) and Lemma 3.1, the last integral can be evaluated

⟨P_{ν}, X^{α}⟩_{W_{κ}} = w_{κ} ∏_{i=1}^{d} ∫_{0}^{1} P_{ν_{j}}^{(a_{j},κ_{j})}(2t − 1)t^{α_{j}}(1 − t)^{|α^{j+1}|−|ν^{j+1}|}t^{κ_{j}}(1 − t)^{a_{j}}dt

= w_{κ} ∏_{j=1}^{d} \frac{(κ_{j} + 1)_{ν_{j}}(κ_{j} + 1)_{α_{j}}(a_{j} + 1)_{|α^{j+1}|−|ν^{j+1}|}(−|α^{j}| + |ν^{j+1}|)_{ν_{j}}}{ν_{j}! (a_{j} + κ_{j} + 2)_{|α^{j}|−|ν^{j+1}|+ν_{j}}}

× ∏_{j=1}^{d} \frac{Γ(κ_{j} + 1)Γ(a_{j} + 1)}{Γ(κ_{j} + a_{j} + 2)}Q_{ν_{j}}(α_{j}; κ_{j}, a_{j}, |α^{j}| − |ν^{j+1}|).

Recalling the definition of a_{j} = a_{j}(κ, ν) and the notation κ^{j+1}, we have

∏_{j=1}^{d} \frac{(a_{j} + 1)_{|α^{j+1}|−|ν^{j+1}|}}{(a_{j} + κ_{j} + 2)_{|α^{j}|−|ν^{j+1}|+ν_{j}}}

= ∏_{j=1}^{d} \frac{Γ(a_{j} + κ_{j} + 2)}{Γ(a_{j} + 1)} \frac{Γ(|κ^{j+1}| + |ν^{j+1}| + |α^{j+1}| + d − j + 1)}{Γ(|κ^{j}| + |ν^{j}| + |α^{j}| + d − j + 2)}

= ∏_{j=1}^{d} \frac{Γ(a_{j} + κ_{j} + 2)}{Γ(a_{j} + 1)} · \frac{Γ(κ_{d+1} + α_{d+1} + 1)}{Γ(|κ| + |ν| + n + d + 1)},

where the second equality follows from telescoping the product and using the fact that ν^{d+1} = 0. Substituting this into the formula for ⟨X^{α}, P_{ν}⟩_{W_{κ}} and using the definition of w_{κ}, it follows that

⟨P_{ν}, X^{α}⟩_{W_{κ}} = \frac{Γ(|κ| + d + 1)}{Γ(|κ| + |ν| + |α| + d + 1)} ∏_{j=1}^{d} \frac{(κ_{j} + 1)_{ν_{j}}(κ_{j} + 1)_{α_{j}}}{ν_{j}!}

× ∏_{j=1}^{d}(−|α^{j}| + |ν^{j+1}|)_{ν_{j}}Q_{ν_{j}}(α_{j}; κ_{j}, a_{j}, |α^{j}| − |ν^{j+1}|)

= \frac{(−1)^{|ν|}(−n)_{|ν|}(κ + 1)_{α}}{(|κ| + d + 1)_{|ν|+n}} ∏_{j=1}^{d} \frac{(a_{j} + 1)_{ν_{j}}}{ν_{j}!} ϕ_{ν}(α; κ, n).

Here the last equality follows from the formula for ϕ_{ν}(α; κ, n) in Proposition 2.4. Since |ν| = |α| = n and (−1)^{n}(−n)_{n} = n!, substituting the above into (3.2) finishes the proof. □

Clearly the representation of Theorem 1.3, i.e.,

(3.3)  f = C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ⟨f, R_{α}^{κ}⟩_{W_{κ}}R_{α}^{κ},  ∀f ∈ V_{n}^{d}(W_{κ})

implies the Parseval type equality (take the inner product with g)

(3.4) ⟨f, g⟩_{W_{κ}} = C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ⟨f, R_{α}^{κ}⟩_{W_{κ}}⟨R_{α}^{κ}, g⟩_{W_{κ}}  ∀f, g ∈ V_{n}^{d}(W_{κ}).

On the other hand, taking g = P_{n}(W_{κ}; x, ·) shows that (3.4) implies (3.3), so they are equivalent. Since the condition f, g ∈ V_{n}^{d}(W_{κ}) can be replaced by f, g belong to some basis of V_{n}^{d}(W_{κ}), e.g., take {P_{ν}}, it suffices to prove the following equivalent form of the main result. Theorem 3.3. If ν, μ ∈ ℕ_{0}^{d} with |ν| = |μ| = n, then

(3.5)  C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ⟨P_{ν}, R_{α}^{κ}⟩_{W_{κ}}⟨R_{α}^{κ}, P_{μ}⟩_{W_{κ}} = ⟨P_{ν}, P_{ν}⟩_{W_{κ}}δ_{ν,μ},

where the constant C_{n,κ} is given by

C_{n,κ} = \frac{(|κ| + d + 1)_{2n}}{(n + |κ| + d)_{n}^{2}} .

Proof. By Proposition 3.2

E_{ν,μ} := ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ⟨P_{ν}, R_{α}^{κ}⟩_{W_{κ}}⟨R_{α}^{κ}, P_{μ}⟩_{W_{κ}}

= ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} (\frac{(n + |κ| + d)_{n} n!}{(|κ| + d + 1)_{2n}})^{2} ∏_{j=1}^{d} \frac{(a_{j} + 1)_{ν_{j}}}{ν_{j}!} \frac{(a_{j} + 1)_{μ_{j}}}{μ_{j}!} ϕ_{ν}(α; κ, n)ϕ_{μ}(α; κ, n)

= (\frac{(n + |κ| + d)_{n} n!}{(|κ| + d + 1)_{2n}})^{2} ∏_{j=1}^{d} \frac{(a_{j} + 1)_{ν_{j}}}{ν_{j}!} \frac{(a_{j} + 1)_{μ_{j}}}{μ_{j}!} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ϕ_{ν}(α; κ, n)ϕ_{μ}(α; κ, n).

By the orthogonality of the ϕ_{ν}(α; κ, n) given in Proposition 2.5, this becomes, after some simplification,

E_{ν,μ} = (\frac{(n + |κ| + d)_{n} n!}{(|κ| + d + 1)_{2n}})^{2} ∏_{j=1}^{d} \frac{(a_{j} + 1)_{ν_{j}}^{2}}{(ν_{j}!)^{2}} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} [ϕ_{ν}(α; κ, n)]^{2}δ_{μ,ν}

= (\frac{(n + |κ| + d)_{n}}{(|κ| + d + 1)_{2n}} )^{2}(|κ| + d + 1)_{2n} \frac{Γ(|κ| + d + 1)}{Γ(κ_{d+1} + 1)}

× ∏_{j=1}^{d} \frac{Γ(a_{j} + 1)(a_{j} + 1)_{ν_{j}}(κ_{j} + 1)_{ν_{j}}}{ν_{j}! Γ(κ_{j} + a_{j} + 1)(a_{j} + κ_{j} + 1)_{ν_{j}}(a_{j} + κ_{j} + 2ν_{j} + 1)}δ_{μ,ν}.

¿From the definition of a_{j} = a_{j}(κ, ν), a_{d} = κ_{d+1} and a_{0} = |κ| + 2|ν| + d, it follows

∏_{j=1}^{d}(a_{j} + κ_{j} + 2)_{2ν_{j}} = ∏_{j=1}^{d} Γ(|κ^{j}| + 2|ν^{j}| + d − j + 2)Γ(|κ^{j}| + 2|ν^{j+1}| + d − j + 2)

= ∏_{j=1}^{d} \frac{Γ(a_{j−1} + 1)}{Γ(κ_{j} + a_{j} + 2)} = \frac{Γ(|κ| + 2n + d + 1)}{Γ(κ_{d+1} + 1)} ∏_{j=1}^{d} \frac{Γ(a_{j} + 1)}{Γ(κ_{j} + a_{j} + 2)},

that is,

∏_{j=1}^{d} \frac{Γ(a_{j} + 1)}{Γ(κ_{j} + a_{j} + 2)} = \frac{Γ(κ_{d+1} + 1)}{Γ(|κ| + 2n + d + 1)} ∏_{j=1}^{d}(a_{j} + κ_{j} + 2)_{2ν_{j}}.

Substituting this equation into the formula of E_{ν,μ} and using (2.6) gives

E_{ν,μ} = \frac{(n + |κ| + d)_{n}^{2}}{(|κ| + d + 1)_{2n}} \frac{∏_{j=1}^{d}(κ_{j} + a_{j} + 2)_{2ν_{j}}}{(|κ| + d + 1)_{2|ν|}} ∏_{j=1}^{d} h_{ν_{j}}^{(a_{j},κ_{j})}.

The desired result now follows from (2.5). □

The equivalence of Theorems 1.3 and 1.4 follows by checking that (1.9) satisfies the equation (1.6) which defines the reproducing kernel. Corollary 3.4. (Tight frame) For any f ∈ V_{n}^{d}(W_{κ}),

⟨f, f⟩_{W_{κ}} = C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ⟨f, R_{α}^{κ}⟩_{W_{κ}}^{2}.

The constant C_{n,κ} satisfies

C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ⟨R_{α}^{κ}, R_{α}^{κ}⟩_{W_{κ}}^{2} = dim(V_{n}^{d}(W_{κ})) = \binom{n + d − 1}{d}.

Proof. The first part is (3.4) with f = g. The second part is proved as follows. Let {e_{ν}} be any orthonormal basis of V_{n}^{d}(W_{κ}), then

dim V_{n}^{d}(W_{κ}) = ∑_{ν}⟨e_{ν}, e_{ν}⟩ = ∑_{ν} C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ⟨e_{ν}, R_{α}^{κ}⟩_{W_{κ}}^{2}

= C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ∑_{ν}⟨e_{ν}, R_{α}^{κ}⟩_{W_{κ}}^{2}.

By Parseval’s equality the sum over ν above equals ⟨R_{α}^{κ}, R_{α}^{κ}⟩_{W_{κ}} which completes the proof. □

Corollary 3.5. (Tight frame for L_{2}(W_{κ})) For any f ∈ L_{2}(W_{κ}),

f = ∑_{n=0}^{∞} C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ⟨f, R_{α}^{κ}⟩_{W_{κ}}R_{α}^{κ},

and, in particular,

⟨f, f⟩_{W_{κ}} = ∑_{n=0}^{∞} C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ⟨f, R_{α}^{κ}⟩_{W_{κ}}^{2}.

Proof. Since ∪_{n=0}^{∞}{P_{ν} : |ν| = n} forms a complete orthogonal system for L_{2}(W_{κ}),

f = ∑_{n=0}^{∞} ∑_{|ν|=n} \frac{⟨f, P_{ν}⟩_{W_{κ}}}{⟨P_{ν}, P_{ν}⟩_{W_{κ}}} P_{ν} = ∑_{n=0}^{∞}⟨f, P_{n}(W_{κ}; x, ·)⟩_{W_{κ}},

where P_{n}(W_{κ}; x, y) is the reproducing kernel of V_{n}^{d}(W_{κ}). Hence, Theorem 1.4 implies the stated result. □

Equating the compact formula of [11] for the kernel P_{n}(W_{κ}; x, y) as an integral of a Gegenbauer polynomial with that of Theorem 1.4 gives

C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} R_{α}^{κ}(x)R_{α}^{κ}(y) = \frac{2n + |κ| + 1}{|κ| + 1}

× ∫_{[−1,1]^{d+1}} C_{2n}^{|κ|+1}(\sqrt{x_{1}y_{1}} t_{1} + ⋯ + \sqrt{x_{d+1}y_{d+1}} t_{d+1}) ∏_{i=1}^{d+1} c_{κ_{i}}(1 + t_{i})(1 − t_{i}^{2})^{κ_{i}−1}dt,

where

x_{d+1} := 1 − x_{1} − ⋯ − x_{d},  y_{d+1} := 1 − y_{1} − ⋯ − y_{d},  c_{μ} := \frac{Γ(μ + 1/2)}{\sqrt{π} Γ(μ)}.

Denote the multivariate Hahn polynomials of exact degree n for N = n by

U_{n}^{d}(κ) := span{ϕ_{ν}(·; κ, n) : |ν| = n, ν ∈ ℕ_{0}^{d}}

and the inner product by

⟨ϕ, ψ⟩_{κ,n} := ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ϕ(α)ψ(α).

Proposition 3.2 shows that the map L_{κ,n} : V_{n}^{d}(W_{κ}) → U_{n}^{d}(κ) defined by

(L_{n,κ}p)(α) := \sqrt{C_{n,κ}}⟨p, R_{α}^{κ}⟩_{W_{κ}}, ∀α ∈ ℕ_{0}^{d+1}, |α| = n,

is well-defined. Corollary 3.6. The map L_{n,κ} : V_{n}^{d}(W_{κ}) → U_{n}^{d}(κ) is an isometry.

Proof. For p, q ∈ V_{n}^{d}(W_{κ}),

⟨L_{n,κ}p, L_{n,κ}q⟩_{κ,n} = C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ⟨p, R_{α}^{κ}⟩_{W_{κ}}⟨R_{α}^{κ}, q⟩_{W_{κ}} = ⟨p, q⟩_{κ,n}

by the formula (3.4). □

Setting f = R_{β}^{κ}, g = R_{γ}^{κ} in (3.4) gives

(3.6)  ⟨R_{β}^{κ}, R_{γ}^{κ}⟩_{W_{κ}} = C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} ⟨R_{β}^{κ}, R_{α}^{κ}⟩_{W_{κ}}⟨R_{α}^{κ}, R_{γ}^{κ}⟩_{W_{κ}}.

Expressing this in terms of the Gram matrix for {R_{α}^{κ} : |α| = n} gives: Corollary 3.7. Normalize the R_{α}^{κ} via

R̃_{α}^{κ} := \sqrt{C_{n,κ}(κ + 1)_{α}/α!} R_{α}^{κ}, |α| = n.

Then the Gram matrix G_{n,κ} := [⟨R̃_{α}^{κ}, R̃_{β}^{κ}⟩_{W_{κ}}]_{|α|,|β|=n} has rank \binom{n+d−1}{d} and satisfies

G_{n,κ}^{2} = G_{n,κ}.

Moreover, its Hadamard (matrix pointwise) product satisfies

(G_{n,κ} ∘ G_{n,κ})1 = (⟨R̃_{α}^{κ}, R̃_{α}^{κ}⟩_{W_{κ}})_{|α|=n}.

Proof. The first part is (3.6). Since {R_{α}^{κ} : |α| = n} spans V_{n}^{d}(W_{κ}) the rank of the Gram matrix is dim V_{n}^{d}(W_{κ}) = \binom{n+d−1}{n}. The last part is a restatement of

⟨f, f⟩_{W_{κ}} = ∑_{|β|=n}⟨f, R̃_{β}^{κ}⟩_{W_{κ}}^{2}, f = R̃_{α}^{κ}, |α| = n,

the first formula of Corollary 3.4. □

Finally, as an illustration of our main result, we consider the case d = 1. Here α = (α_{1}, α_{2}) and R_{α}^{κ}, |α| = n is (a scalar multiple of) the orthogonal projection of x^{α_{1}}(1 − x)^{α_{2}} onto the 1–dimensional space V_{n}^{1}(W_{κ}). Since this is spanned by P_{n}^{(κ_{1},κ_{2})}(1 − 2x) we compare leading coefficients, to obtain

R_{α}^{κ}(x) = \frac{n!(−1)^{α_{2}}}{(κ + 1)_{α}} P_{n}^{(κ_{1},κ_{2})}(1 − 2x).

Hence

∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} R_{α}^{κ}(x)R_{α}^{κ}(y)

= (n!)^{2}P_{n}^{(κ_{1},κ_{2})}(1 − 2x)P_{n}^{(κ_{1},κ_{2})}(1 − 2y) ∑_{|α|=n} \frac{1}{α!(κ + 1)_{α}} .

Writing α = (j, n − j) and using a Chu–Vandermonde sum shows that

∑_{|α|=n} \frac{1}{α!(κ + 1)_{α}} = ∑_{j=0}^{n} \frac{1}{j!(n − j)!(κ_{1} + 1)_{j}(κ_{2} + 1)_{n−j}}

= \frac{1}{n!(κ_{2} + 1)_{n}}_{2}F_{1}(−n, −n − κ_{2}; κ_{1} + 1; 1) = \frac{(|κ| + n + 1)_{n}}{n!(κ_{2} + 1)_{n}(κ_{1} + 1)_{n}} .

Consequently, using the constant h_{n}^{(κ_{1},κ_{2})} of (2.6), it follows that

C_{n,κ} ∑_{|α|=n} \frac{(κ + 1)_{α}}{α!} R_{α}^{κ}(x)R_{α}^{κ}(y) = \frac{1}{h_{n}^{(κ_{1},κ_{2})}} P_{n}^{(κ_{1},κ_{2})}(1 − 2x)P_{n}^{(κ_{1},κ_{2})}(1 − 2y),

which agrees with the formula (1.7).

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Department of Mathematics, University of Auckland, Auckland, New Zealand.

E-mail address: waldron@math.auckland.ac.nz

Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222.

E-mail address: yuan@math.uoregon.edu