A formula for the nonsymmetric Opdam's hypergeometric function of type $A_2$

TL;DR

This paper derives an explicit formula for the nonsymmetric Heckman-Opdam hypergeometric function of type A2 by differentiating the symmetric version, advancing understanding of special functions related to root systems.

Contribution

It provides the first explicit formula for the nonsymmetric hypergeometric function of type A2, obtained through differentiation of the symmetric function.

Findings

Explicit formula for nonsymmetric hypergeometric function of type A2

Method based on differentiation of symmetric hypergeometric function

Enhances analytical tools for special functions in root system theory

Abstract

The aim of this paper is to give an explicit formula for the nonsymmetric Heckman-Opdam's hypergeometric function of type $A_2$. This is obtained by differentiating the corresponding symmetric hypergeometric function.

Equations

$$R=\{e_i-e_j,1\le i\ne j\le n\}$$

$$R_{+}=\{e_i-e_j,1\le i<j\le n\}.$$

$$C=\{x\in \mathbb{V},x_1>x_2>...>x_n\}.$$

$${\pi }_n(x)=x-\frac{1}{n}\left(\sum _{j=1}^n{x}_j\right)e,x\in {\mathbb{R}}^n.$$

$$P_{+}=\sum _{j=1}^{n-1}{\mathbb{Z}}_{+}{\beta }_j,{\beta }_j={\pi }_n(e_1+e_2+...+e_j).$$

$$T_{\xi }^k={\partial }_{\xi }+k\sum _{i<j}({\xi }_i-{\xi }_j)\frac{1-s_{i,j}}{1-e^{{}^{x_j-x_i}}}-⟨{\rho }_k,\xi ⟩$$

$$p(T_{\pi (e_1)}^k,...T_{\pi (e_n)}^k)F_k(\lambda ,.)=p(\lambda )F_k(\lambda ,.);F(\lambda ,0)=1,$$

$${\Delta }_k{F}_k(\lambda ,.)=\Vert \lambda {\Vert }^2{F}_k(\lambda ,.)$$

$$L_k=\Delta +k\sum _{\alpha \in R^{+}}\coth \frac{⟨\alpha ,x⟩}{2}{\partial }_{\alpha }+⟨{\rho }_k,{\rho }_k⟩$$

$$T_{\xi }(k)G_k(\lambda ,.)=⟨\lambda ,\xi ⟩G_k(\lambda ,.),\forall \xi \in {\mathbb{R}}^N,G_k(\lambda ,0)=1.$$

$$|G_k(\lambda ,x)|\le \sqrt{n!}{e}^{{\max }_{w\in W}⟨Re(\lambda ),w(x)⟩}.$$

$$F_k(\lambda ,x)=\frac{1}{n!}\sum _{w\in W}{G}_k(\lambda ,w.x).$$

$$G_k(\lambda ,x)=D_q{F}_k(\lambda ,x),x\in \mathbb{V}$$

$$D_q=\prod _{1\le i<j\le n}{\left(1-\frac{k}{{\lambda }_i-{\lambda }_j}\right)}^{-1}\prod _{w\in W,w\ne id}\left(\frac{T_{\xi }-⟨w(\lambda ),\xi ⟩}{⟨\lambda ,\xi ⟩-⟨w(\lambda ),\xi ⟩}\right)$$

$$F_{k,n}(\lambda +{\rho }_{k,n},x)=$$

$$\displaystyle\frac{c_{n}(\lambda+\rho_{k,n})}{c_{n-1}(\overline{\lambda}+\rho_{k,n-1})U_{n}(\widetilde{\lambda})V_{n}(x)^{2k-1}}\int_{x_{2}}^{x_{1}}...\int_{x_{n}}^{x_{n-1}}F_{k,n-1}\Big{(}\pi_{n-1}(\overline{\lambda})+\rho_{k,n-1}^{k},\pi_{n-1}(\nu)\Big{)}\;$$

$$e^{|\nu |\left(1+\frac{|\overline{\lambda }|}{n-1}\right)}{V}_{n-1}(\nu )W_k(x,\nu )d\nu$$

$$c_n(\lambda )=\prod _{\alpha \in R+}\frac{\Gamma (⟨\lambda ,\alpha ⟩)\Gamma (⟨{\rho }_{k,n},\alpha ⟩+k)}{\Gamma (⟨\lambda ,\alpha ⟩+k)\Gamma (⟨{\rho }_{k,n},\alpha ⟩)},U_n(\lambda )=\prod _{j=1}^{n-1}\beta ({\lambda }_j+(n-j)k,k)$$

$$\tilde{\lambda }=({\lambda }_1-{\lambda }_n,{\lambda }_2-{\lambda }_n,...,{\lambda }_{n-1}-{\lambda }_n,0),\overline{\lambda }=({\lambda }_1-{\lambda }_n,{\lambda }_2-{\lambda }_n,...,{\lambda }_{n-1}-{\lambda }_n),$$

$$|\nu |={\nu }_1+{\nu }_2+...+{\nu }_{n-1};V_n(x)=\prod _{1\le i<j\le n}(e^{x_i}-e^{x_j}),$$

$$W_k(x,\nu )=\prod _{1\le i\le n;1\le j\le n-1}|e^{x_i}-e^{{\nu }_j}{|}^{k-1}.$$

$$F_{k,n}(\lambda +{\rho }_{k,n},x)$$

$$H_1(\lambda )=e^{-⟨\lambda ,x⟩}{F}_{k,n}(\lambda +{\rho }_{k,n},x)$$

$$H_2(\lambda )=\frac{\Gamma (nk)}{\Gamma (k{)}^n{V}_n(x{)}^{2k-1}}{\int }_{x_2}^{x_1}...{\int }_{x_n}^{x_{n-1}}{e}^{-⟨\lambda ,x⟩+|\nu ||\overline{\lambda }|/(N-1)}$$

$$\displaystyle\qquad\qquad\quad\qquad F_{k,n-1}^{k}\Big{(}\pi_{n-1}(\overline{\lambda})+\rho_{k,n-1},\pi_{n-1}(\nu)\Big{)}e^{|\nu|}V_{n-1}(\nu)\;W_{k}(x,e^{\nu})\;d\nu$$

$$|H_1(\lambda )|\le \sqrt{n!}{e}^{⟨{\rho }_{k,n},x⟩}$$

$$\displaystyle\left|e^{-\langle\lambda,\;x\rangle+|\nu||\overline{\lambda}|/(n-1))}F_{k,n-1}\Big{(}\pi_{n-1}(\overline{\lambda})+\rho_{k,n-1},\pi_{n-1}(\nu)\Big{)}\right|$$

$$\le \sqrt{(n-1)!}{e}^{-⟨Re(\lambda ),x-{\nu }^{\ast }⟩+⟨{\rho }_{k,n-1},{\pi }_{n-1}(\nu )⟩}$$

$$\le \sqrt{(n-1)!}{e}^{⟨{\rho }_{k,n-1},{\pi }_{n-1}(\nu )⟩}$$

$$F_{k,n}(\lambda ,x)=$$

$$\displaystyle\frac{\Gamma(nk)}{\Gamma(k)^{n}V_{n}(x)^{2k-1}}\int_{x_{2}}^{x_{1}}...\int_{x_{n}}^{x_{n-1}}F_{k,n-1}\Big{(}\pi_{n-1}(\overline{\lambda}),\pi_{n-1}(\nu)\Big{)}\;e^{|\nu|\left(1-nk/2+|\overline{\lambda}|/(n-1)\right)}$$

$$V_{n-1}(\nu )W_k(x,\nu )d\nu$$

$$F_{k,2}(\lambda ,x)$$

$${\phi }_{i\eta }^{k-1/2,-1/2}(2t)={\phi }_{2i\eta }^{k-1/2,k-1/2}(t)$$

$${\left({\phi }_{i\eta }^{k-1/2,-1/2}\right)}^{\prime }(t)=\frac{1}{(2k+1)}({\eta }^2-k^2)\sinh (t){\phi }_{i\eta }^{k+1/2,-1/2}(t)$$

$${\left({\phi }_{i\eta }^{k-1/2,-1/2}\right)}^{\prime \prime }(t)+2k\coth (t){\left({\phi }_{i\eta }^{k-1/2,-1/2}\right)}^{\prime }(t)=({\eta }^2-k^2){\phi }_{i\eta }^{k+1/2,-1/2}(t)$$

$$F_k(\lambda ,x)=$$

Full text

** Keywords**. Root systems, Cherednik operators, Hypergeometric functions.

Mathematics Subject Classification. Primary 33C67;17B22. Secondary 33D52 .

Université Tunis El Manar, Faculté des sciences de Tunis,

Laboratoire d’Analyse Mathématique et Applications,

LR11ES11, 2092 El Manar I, Tunisie.

e-mail: bechir.amri@ipeit.rnu.tn, bedhiafi.mounir@yahoo.fr

1 Introduction

The theory of the hypergeometric functions associated to root systems started in the 1980s with Heckman and Opdam via a generalization of the spherical functions on Riemannian symmetric spaces of noncompact type. Several important aspects are studied by them in a series of publications [4, 9, 10, 11, 12]. One of the impressive developments came in 1995s with the work of Opdam [11], where he introduced a remarkable family of orthogonal polynomials ( the so called Opdam’s nonsymmetric polynomials ) as simultaneous eigenfunctions of Cherednik operators. It contains in particular the presentation of the non-symmetric hypergeometric functions where their investigations become an interesting topics in the theory of special functions and in the harmonic analysis. In this paper, we focus on the non-symmetric hypergeometric function associated to root systems of type $A$, for the purpose in finding an explicit formula for it, as it is done in the symmetric case [1, 2, 13, 3]. The setting is that non-symmetric hypergeometric function can be derived from the symmeric ones via application of a suitable polynomial of Cherednik operators ( [12], cor. 7.6 ). This paper deals with the case where the root system is of type $A_{2}$, using Opdam’s shifted operators and Cherednik operators, so the problem semble to be more robust for others $A_{n}$.

In order to describe our approach let us be more specific about $A$-type hypergeometric function. We assume that the reader is familiar with root systems and their basic properties. As a general reference, we mention Opdam [11, 12].

Let $(e_{1},e_{2},...,e_{n})$ be the standard basis of $\mathbb{R}^{n}$ and $\langle.,.\rangle$ be the usual inner product for which this basis is orthonormal. We denote by $\|\,.\,\|$ its Euclidean norm. Let $\mathbb{V}$ be the hyperplane orthogonal to the vector $e=e_{1}+...+e_{n}$. In $\mathbb{V}$ we consider the root system of type $A_{n-1}$

[TABLE]

with the subsystem of positives roots

[TABLE]

The associated Weyl group $W$ is isomorphic to symmetric group $S_{n}$, permuting the $n$ coordinates. We define the positive Weyl chamber

[TABLE]

Denote $\pi_{n}$ the orthogonal projection onto $\mathbb{V}$, which is given by

[TABLE]

The cone of dominant weights is the set

[TABLE]

For fixed $k>0$, the Dunkl-Cherednik operators $T_{\xi}$, $\xi\in\mathbb{R}^{n}$, is defined by

[TABLE]

where $\displaystyle{\rho_{k}=\frac{k}{2}\sum_{j=1}^{n}(n-2j+1)}e_{j}$ and $s_{i,j}$ acts on functions of variables $(x_{1},x_{2},...,x_{n})$ by interchanging the variables $x_{i}$ and $x_{j}$. For each $\lambda\in\mathbb{V}_{\mathbb{C}}$ ( the complexification of $\mathbb{V}$ ) there exists a unique holomorphic W-invariant function $F_{k}(\lambda,.)$ in a W-invariant tubular neighborhood of $\mathbb{V}$ in $\mathbb{V}_{\mathbb{C}}$ such that

[TABLE]

for all symmetric polynomial $p\in\mathbb{C}[X_{1},..,X_{n}]$. In particular

[TABLE]

where $\Delta_{k}=\sum_{i=1}^{n}\left(T_{\pi(e_{i})}\right)^{2}$ is the Heckman-Opdam Laplacian. Note that the restriction of $\Delta_{k}$ to the set of W-invariant functions is the differential operator

[TABLE]

where $\Delta$ is the ordinary Laplace operator.

There exists a unique solution $G_{k}(\lambda,x)$ of the eigenvalue problem

[TABLE]

holomorphic for all $\lambda$ and for $x$ in $\mathbb{V}+iU$ for a neighbourhood $U\subset\mathbb{V}$ of zero. The function $G_{k}$ is the so-called nonsymmetric Opdam’s hypergeometric function.

If $x\in\mathbb{V}$ then

[TABLE]

Moreover, the Heckman-Opdam hypergeometric function $F_{k}$ can be written as

[TABLE]

In other words, for $\lambda$ satisfying $\lambda_{i}-\lambda_{j}\neq 0;\pm k$ we have

[TABLE]

where

[TABLE]

and $\xi$ is any element in $\mathbb{V}$ satisfying $\langle\lambda,\xi\rangle-\langle w(\lambda),\xi\rangle\neq 0$ for all $w\neq id$.

However, (1.6) is far from being applied to find an expansion of $G_{k}$ when an explicit formula of $F_{k}$ is given, so the polynomial $q$ has degree $n!-1$. It would therefore be desirable to get another polynomial that is of suitable low degree, this will be described in section 3 when the root system is of type $A_{2}$.

2 An integral formula for Heckman-Opdam hypergeometric function of type A

In [1] an explicit and recursive formula on the dimension $n$ for the A-type Heckman-Opdam’s hypergeometric function is obtained as a consequence of similar formula for Jack polynomials. We have for $\lambda\in P_{+}$ and $x\in C$

[TABLE]

with the following notations

[TABLE]

It can be simplified to

[TABLE]

By analytic continuation, according to Carlson’s theorem ( see [14], p. 186 ), this formula is still valid for all $\lambda\in\mathbb{V}_{\mathbb{C}}$. Indeed, for $x\in C$ define the functions of vaiable $\lambda\in\mathbb{V}_{\mathbb{C}}$

[TABLE]

We have from (1.4)

[TABLE]

for all $\lambda\in\mathbb{V}_{\mathbb{C}}$, $Re(\lambda)\in C$ and $x\in C$. In order to estimate $H_{2}$ we note beforehand the following fact, for $\nu=(\nu_{1},\nu_{2},...,\nu_{n-1})$ and $x\in C$ with $x_{i+1}\leq\nu_{i}\leq x_{i}$, if we consider $\nu^{*}=(\nu_{1},\nu_{2},...,\nu_{n-1},\nu_{n})\in\mathbb{V}$ then we have $\nu^{*}\preceq x$ where $\preceq$ denotes the partial order on $\mathbb{V}$ associated to the dual cone $\sum_{i=1}^{n-1}\mathbb{R}_{+}(e_{i}-e_{i+1})$, which implies that $\langle Re(\lambda),\nu^{*}\rangle\leq\langle Re(\lambda),x\rangle$, for $\lambda\in\mathbb{V}_{\mathbb{C}}$ with $Re(\lambda)\in C$. Applying this fact and (1.4) it follows that

[TABLE]

Hence $H_{1}$ and $H_{2}$ are Holomorphic functions, bounded for $Re(\lambda)\in C$ and coincide on $P_{+}\simeq\mathbb{Z}_{+}^{n-1}$, the Carleson’s Theorem yields $H_{1}(\lambda)=H_{1}(\lambda)$ for all $\lambda\in\mathbb{V}_{\mathbb{C}}$, $Re(\lambda)\in C$ and thus for all $\lambda\in\mathbb{V}_{\mathbb{C}}$ by analytic continuation.

Now, using the fact that $\pi_{n-1}(\overline{\rho_{k,n}})=\rho_{k,n-1}^{k}$ and $|\overline{\rho_{k,n}}|=kn(n-1)/2$, we state the following final form of our recursive formula.

Theorem 2.1**.**

For all $\lambda\in\mathbb{V}_{\mathbb{C}}$ and $x\in C$.

[TABLE]

In the rank-one case, which corresponds to take $n=2$ and $\mathbb{V}=\mathbb{R}(e_{1}-e_{2})$, the formula (2.1) becomes

[TABLE]

where $\varphi^{k-1/2,-1/2}_{2i\lambda_{1}}$ is a Jacobi function see (1.4), (3.4) and (3.5) of [7]. We recall here various facts about the Jacobi function $\varphi^{k-1/2,-1/2}_{i\eta}$ that we shall need later, we refer to [7, 8].

[TABLE]

In the rank-two case, where $n=3$ which is our subject in the next section, Heckman-Opdam’s hypergeometric function has the following integral representation ( we omit here the dependance on $n$ )

[TABLE]

where

[TABLE]

and

[TABLE]

In order to find an expression for $F_{k}$ of Laplace type, we write

[TABLE]

With the change of variables

[TABLE]

we have that

[TABLE]

Now inserting

[TABLE]

with the use of Fubini’s Theorem and the fact that

[TABLE]

if follows that

[TABLE]

where

[TABLE]

if $\max(|y_{2}|,|y_{1}-x_{2}|)\leq\min(y_{1}-x_{3},x_{1}-y_{1})$ and $R_{k}(x,y_{1},y_{2})=0$, otherwise. We should note here that the condition

[TABLE]

is equivalent to

[TABLE]

an thus equivalents to $y=(y_{1}+y_{2},y_{1}-y_{2},-2y_{1})\in co(x)$, the convex hull of the orbit $W.x$, see proposition 3.6 of [1]. Also, we have $y=y_{1}\sqrt{6}\,\varepsilon_{1}+y_{2}\sqrt{2}\,\varepsilon_{2}$ in the orthonormal basis of $\mathbb{V}$

[TABLE]

Making the change of variables $z_{1}=\sqrt{6}\,y_{1},\;z_{2}=\sqrt{2}\,y_{2}$ in the formula (2.7) and identify $\mathbb{R}^{2}$ with $\mathbb{V}$ via the basis $(\varepsilon_{1},\varepsilon_{2})$ we finally write

[TABLE]

where $\mathcal{N}_{k}(x,z)=R_{k}\left(x,z_{1}/\sqrt{6},z_{2}/\sqrt{2}\right)/\sqrt{12}$.

Next, for $f\in C^{\infty}(\mathbb{V})$, $W$- invariant, we define $V_{k}(f)$ the $W$- invariant function on $\mathbb{V}=\mathbb{R}\varepsilon_{1}\oplus\mathbb{R}\varepsilon_{2}$ by

[TABLE]

Proposition 2.2**.**

For $f\in C_{c}^{\infty}(\mathbb{V})$ and $x\in C$, we have the following intertwining property

[TABLE]

Proof.

By inversion formula for Fourier transform and Fubini’s Theorem

[TABLE]

Here we define the Fourier transform of $f$ by

[TABLE]

From a general estimates of Heckman-Opdam’s hypergeometric function ( see for example corollary 6.2 in [12]), the last integral of (2.9) is a $C^{\infty}$ as a function of $x$ and then by (1.2) one has

[TABLE]

which proves the desired fact. ∎

3 Nonsymmetric Opdam’s hypergeometric function of type $A_{2}$

We begin with some backgrounds from Heckman-Opdam theory of hypergeometric functions associated to a root system $R$ with Weyl group $W$ of a finite-dimensional vector space $\mathfrak{a}$ , we refer to [11, 12] for a more detailed treatment. For a regular weight $\lambda\in P_{+}$ (the set of dominant weights) we denote by $P_{k}(\lambda,.)$ the Hekman-Opdam Jacobi polynomial and by $E_{k}(\lambda,.)$ the non symmetric Opdam polynomial. In the following we collect some properties and relationships.

• (i)

$\displaystyle{P_{k}(\lambda,x)=\sum_{w\in W}E_{k}(\lambda,wx)}$

• (ii)

$\displaystyle{\sum_{w\in W}det(w)E_{k}(\lambda+\delta,wx)=V(x)P_{k+1}(\lambda,x)}$, where $\displaystyle{\delta=\frac{1}{2}\sum_{\alpha\in R^{+}}\alpha}$ and

$\displaystyle{V(x)=\prod_{\alpha\in R^{+}}(e^{\alpha/2}-e^{-\alpha/2})}$

• (iii)

$\displaystyle{P_{k}(\lambda,0)=\frac{c_{k}(\rho_{k})}{c_{k}(\lambda+\rho_{k})}}$, where $\displaystyle{\rho_{k}(\lambda)=\frac{1}{2}\sum_{\alpha\in R+}k_{\alpha}}\alpha$, $\quad\displaystyle{c_{k}(\lambda)=\prod_{\alpha\in R+}\frac{\Gamma(\langle\lambda,\check{\alpha}\rangle)}{\Gamma(\langle\lambda,\check{\alpha}\rangle+k_{\alpha})}}$ and $\check{\alpha}=2\alpha/|\alpha|^{2}$.

• (iv)

$P_{k}(\lambda,x)=P_{k}(\lambda,0)F_{k}(\lambda+\rho_{k},x)$

• (v)

$\displaystyle{E_{k}(\lambda,x)=\frac{P_{k}(\lambda,0)}{|W|}\;G_{k}(\lambda+\rho_{k},x)}$.

From these facts we give the following consequence.

Proposition 3.1**.**

For all $\lambda,x\in\mathfrak{a}$,

[TABLE]

where,

[TABLE]

Proof.

This is closely related to $(ii)$, where by using the above relations one can write for regular $\lambda\in P_{+}$,

[TABLE]

since we have $\rho_{k}+\delta=\rho_{k+1}$. This identity can be extended for all $\lambda\in\mathfrak{a}$ via analytic continuation by means of Carlson’s theorem.

∎

Let us return now to the space $\mathbb{V}$, for $n=3$ and investigate (3.1). In this case we have

[TABLE]

We introduce the antisymmetric function

[TABLE]

and the following notations:

[TABLE]

So we can write

[TABLE]

The integral representation for the functions $F_{k}^{*}$ is given in the following.

Proposition 3.2**.**

We have for $\lambda\in\mathbb{V_{\mathbb{C}}}$ and $x\in C$,

[TABLE]

where

[TABLE]

and with $a=e^{x_{1}}+e^{x_{2}}+e^{x_{3}}$ and $b=e^{-x_{1}}+e^{-x_{2}}+e^{-x_{3}}$.

Proof.

We first write

[TABLE]

The formula (2.6) for parameter $k+1$, together with (2.4) yield

[TABLE]

So using integration by parts,

[TABLE]

and

[TABLE]

In the same way with the use of (2.5),

[TABLE]

Hence it follows that

[TABLE]

Now, it is straightforward computation to verify that

[TABLE]

∎

Next, we can provide an integral expansion for $G_{k}$ by differentiating (3.3) and (3.4). Let us introduce the operators

[TABLE]

where $\tau(\lambda)=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{1}\lambda_{2}$. Recall that $\pi(e_{1})=(\frac{2}{3},-\frac{1}{3},-\frac{1}{3})$ and $\pi(e_{2})=(-\frac{1}{3},\frac{2}{3},-\frac{1}{3})$. Our main result from this section is the following.

Theorem 3.3**.**

For $\lambda,\,x\in\mathbb{V}$ we have

[TABLE]

Proof.

The proof is purely computational. Using the following fact, see (5.1 ) of [11],

[TABLE]

we obtain

[TABLE]

and

[TABLE]

where $\sigma=s_{1,3}s_{1,2}$. Thus we have

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

So, formula (3.5) can be checked by a straightforward calculations. ∎