New Pieri Type Formulas for Jack Polynomials and their Applications to Interpolation Jack Polynomials

We present new Pieri type formulas for Jack polynomials. As an application, we give a new derivation of higher order difference equations for interpolation Jack polynomials originally found by Knop and Sahi. We also propose Pieri formulas for interpolation Jack polynomials and intertwining relations for a kernel function for Jack polynomials.


Introduction
Given a positive integer r and a non-zero complex parameter d, we define a second-order differential operator in the variables z = (z 1 , . . . , z r ) by where ∂ z j := ∂ ∂z j for j = 1, . . . , r. The Jack polynomials are a family of homogeneous symmetric polynomials P m z; This paper is a contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory. The full collection is available at https://www.emis.de/journals/SIGMA/elliptic-integrablesystems.html and the monomial symmetric polynomials m m (z) := n∈Sr.m z n , where S r denotes the symmetric group of degree r. Since Jack polynomials P m z; d 2 are orthogonal polynomials associated with the root system A r−1 and eigenstates of the Calogero-Surtherland model, they play important roles in various fields of mathematics and physics. For their fundamental properties and applications, see [7,16,17].
The interpolation Jack polynomials (or shifted Jack polynomials) are also a family of symmetric polynomials indexed by partitions m, which we denote by P ip m z; d 2 according to the notation of Koornwinder [4]. They are uniquely defined by the following two conditions: where δ denotes the staircase partition (r−1, r−2, . . . , 2, 1, 0) and k ⊆ m stands for the inclusion partial order defined by These polynomials P ip m z; d 2 were introduced by Sahi [13], Knop-Sahi [3] and Okounkov-Olshanski [10] as a continuous deformation of shifted Schur polynomials P ip m (z; 1) [6,11]. The interpolation Jack polynomials appear as a multivariate analogue of the falling factorials in explicit formulas of binomial type for multivariate hypergeometric functions (see [4,8,9]). We review some fundamental results on Jack and interpolation Jack polynomials relevant to the main results of this paper.
Theorem 1.1 (twisted Pieri formulas for Jack polynomials). For l = 0, 1, . . . , r, we have where J c := [r] \ J, J := j∈J j and Since the first-order Sekiguchi operator H Therefore twisted Pieri formulas (1.6) and (1.7) are regarded as a higher order analogue of the above Pieri type formulas (1.2) and (1.3) respectively. See also Corollary 5.1. From the twisted Pieri formulas for Jack of Theorem 1.1, we obtain three important results as follows. The first one is an alternative proof of the following theorem on difference equations for interpolation Jack polynomials due to Knop-Sahi [3] (see also Corollary 3.1).

Theorem 1.2 (Knop-Sahi).
For any x ∈ C r and k ∈ P, we have where The second result is the Pieri formulas for interpolation Jack polynomials (see also Corollary 4.1). Theorem 1.3. For any x ∈ C r and k ∈ P, we have Finally, we obtain the following intertwining relation for a kernel function of Jack polynomials [5,17] which is a multivariate analogue of Theorem 1.4. For any l = 0, 1, . . . , r, we have It is a multivariate analogue of the relation ∂ z e zw = e zw w and a higher order analogue of the formula in [5] Section 14 The contents of this article are as follows. In Section 2, we prove the twisted Pieri formulas for Jack polynomials. From the twisted Pieri formulas for Jack polynomials, we give another proof of Theorem 1.2 in Section 3 and prove Theorem 1.3 in Section 4. We also prove the intertwining relation (1.10) for the kernel function 0 F 0 (d) (z, w) in Section 5. Finally, we mention some future works for twisted Pieri formulas and their applications in Section 6.

Twisted Pieri formulas for Jack polynomials
To prove Theorem 1.1, we need the following summation formula.

1)
where |I| is the cardinality of I and .
If r = 1, then (2.1) is equal to a trivial summation Proof . For convenience, we put By Pieri type formulas for the Jack polynomials (1.2) and (1.4), we have On the other hand, Then, we obtain (2.1).
Proof of Theorem 1.1. Since (1.6) and (1.7) can be similarly proved, we only prove (1.6). These formulas are proved by induction on l.
The case of l = 0 is This is (1.1) exactly. If l = 1, then Here, the first and second equalities follow from (1.1) and (1.2) respectively. Assume the n = l case holds. Hence, from the induction hypothesis and (1.2) we have x (z) where δ j,ν is the Kronecker's delta and From a simple calculation, we have Since the summation is our mysterious summation (2.1) exactly, we obtain the conclusion (1.6).

Another proof of difference equations for interpolation Jack polynomials
Proof of Theorem 1.2. Since the difference equation (1.8) is a relation for rational function of (x 1 , . . . , x r ), it is enough to prove (1.8) for any partition x ∈ P. To prove (1.8), we compute in two different ways. First, a simple calculation shows that x (z).
Since the highest derivative in H (d) r,p (z) has degree p, the sum terminates after (− ad |∂ z |) r . Then, we have x (z).
By applying the twisted Pieri (1.6) and the binomial (1.5), we have On the other hand, from the binomial formula (1.5) and (1.1), we have (3.2) By comparing coefficients for Ψ Comparing coefficients for u r−l in (1.8), we obtain higher order difference formulas for interpolation Jack polynomials.

Pieri formulas for interpolation Jack polynomials
Proof of Theorem 1.3. As with the proof of Theorem 1.2, it is enough to prove (1.9) for x ∈ P. For the purpose, we compute in two different ways. From the binomial (1.5) and the twisted Pieri (1.6) The third equality follows from I By comparing coefficients for u r−l in (1.9), we obtain the Pieri type formulas for the interpolation Jack polynomials, which are a higher order analogue of equation (5.3) in [10] or equation (14.2) in [5]. 5 Some intertwining relations for the kernel function 0 F 0 (d) (z, w) By comparing the coefficients for u r−l of the twisted Pieri formulas (1.6) and (1.7), we obtain the following twisted Pieri type formulas.