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On Borodin-Gorin presentation of Heckman-Opdam $\mathfrak{gl}_n$ hypergeometric function

This paper establishes the equivalence between the Borodin-Gorin interlacing integral presentation of the Heckman-Opdam $\mathfrak{gl}_n$ hypergeometric function and two other integral representations, while also examining related Baxter operators and the limiting procedure from the Ruijsenaars hyperbolic model.

Original authors: L. Cherepanov, S. Khoroshkin

Published 2026-10-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: L. Cherepanov, S. Khoroshkin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematical physics, there exists a class of problems concerned with how collections of particles interact when they are confined to a line and repel one another. These systems, often modeled by equations that describe waves moving through a medium, are not just abstract puzzles; they appear in the study of quantum mechanics, random matrix theory, and even the statistical behavior of complex networks. For decades, mathematicians have sought to understand the specific "wave functions" that describe the state of these systems. A wave function, in this context, is a mathematical object that encodes all the possible information about the system's configuration. Over time, researchers discovered that these wave functions could be described in different ways, much like a complex landscape can be mapped using different coordinate systems. One particularly elegant approach involves building the solution step-by-step, layer by layer, using a process of integration that links variables in a specific, interlocking order. This method, known as an interlacing integral, offers a unique perspective on the underlying structure of the physical model.

The paper at hand tackles a specific question regarding these interlacing integrals and their relationship to other known descriptions of the same physical system. The authors, working within the field of mathematical physics, focus on a particular type of wave function associated with a model called the Heckman–Opdam system. This system is a sophisticated generalization of the well-known Calogero–Sutherland model, which describes particles on a line with specific interaction strengths. The researchers set out to prove that a relatively new way of writing down these wave functions, proposed by mathematicians Alexander Borodin and Vadim Gorin, is mathematically identical to two older, established formulas. While the Borodin–Gorin approach uses a chain of integrals where the variables are strictly ordered—like a set of nested boxes where each inner box must fit perfectly within the outer one—the older formulas use different integral structures involving complex numbers and special functions. The core achievement of this work is the rigorous demonstration that these seemingly different mathematical expressions are, in fact, the same thing.

To reach this conclusion, the authors did not simply compare the final formulas side-by-side. Instead, they constructed a bridge between the different descriptions using a set of mathematical tools called operators. Think of these operators as machines that take a function describing a system with a certain number of particles and transform it into a function for a system with one more particle. The paper introduces a new family of these machines, specifically designed to work with the interlacing structure of the Borodin–Gorin formula. By studying how these new machines interact with each other and with the older, known machines, the authors were able to show that they produce the exact same results. They proved that if you start with a simple, basic wave and apply the new interlacing machines in a specific sequence, you arrive at the same final wave function as if you had used the older, more traditional methods. This equivalence is not just a matter of numerical coincidence; it is a structural identity that holds true for a wide range of interaction parameters.

The researchers also explored the properties of these new machines, specifically looking at how they behave when applied to the wave functions they help create. They discovered that these machines act as "diagonalizers," meaning they simplify the complex wave functions into a form where their essential properties become clear and easy to calculate. This finding is significant because it confirms that the new interlacing approach is not just an alternative notation, but a robust framework that reveals the same deep symmetries as the established theories. Furthermore, the paper investigates how these new results connect to a broader, more complex model known as the hyperbolic Ruijsenaars model. The authors show that their new interlacing machines can be derived as a limiting case of the machines used in that more complex model. However, they also found that this limiting process is subtle; while the machines that add particles (the raising operators) emerge cleanly from the limit, the machines that analyze the system's energy (the Baxter operators) do not appear to arise in the same straightforward way. This distinction suggests that while the interlacing structure is a natural feature of the simpler model, its connection to the more complex hyperbolic world is not a simple matter of scaling down, but rather a more intricate relationship that requires careful handling.

Ultimately, the work provides a unified view of these mathematical objects. By proving the equivalence of the Borodin–Gorin presentation with the classical integral formulas, the authors have confirmed that the interlacing integral is a valid and powerful tool for understanding these quantum systems. They have also mapped out the precise rules governing the new operators, showing how they commute and interact, which is essential for using them in future calculations. The paper does not claim to have solved the entire mystery of these systems, but it has firmly established a new, rigorous foundation for one of their most elegant descriptions. The results are presented as mathematical proofs, relying on the careful analysis of integrals and the properties of special functions, ensuring that the connection between the different views is absolute. This clarity allows other researchers to use the interlacing approach with confidence, knowing it leads to the same physical truths as the methods that came before it.

Technical Summary: On Borodin–Gorin Presentation of Heckman–Opdam $\mathfrak{gl}_n$ Hypergeometric Function

Problem Statement
The paper addresses the relationship between different integral representations of the Heckman–Opdam $\mathfrak{gl}_n$ hypergeometric function, $F_{\lambda_n}(x_n)$ . Specifically, it investigates the equivalence between the Borodin–Gorin presentation, defined via interlacing integrals over a specific domain ( $x_1 < y_1 < \dots < y_{n-1} < x_n$ ), and two other integral presentations previously established in the literature (specifically those by Hallnäs–Ruijsenaars and Khoroshkin–Khoroshkin). The authors aim to prove that the Borodin–Gorin form is equivalent to the standard form up to a known normalization constant. Additionally, the paper seeks to characterize the spectral properties of this function with respect to a new family of "interlacing" Baxter operators and raising operators, and to understand the limiting procedure from the hyperbolic Ruijsenaars model to this rational/trigonometric degeneration.

Methodology
The authors employ a combination of operator theory and integral identities to establish their results:

Operator Construction: The paper defines a new set of "interlacing" raising operators ( $\Lambda^*_n$ ) and Baxter operators ( $Q^*_{n,\pm}$ ). Unlike the standard operators which integrate over $\mathbb{R}^n$ , these operators integrate over interlacing domains defined by the partial order $y_m \succ y_{m-1}$ (i.e., $y_{m,1} < y_{m-1,1} < y_{m,2} < \dots$ ).
Exchange Relations: The core of the methodology involves proving exchange relations (commutation relations) between these new operators and the standard raising/Baxter operators. These relations are derived using the rational degeneration of E. Rains' integral identities. Specifically, the authors utilize a limiting procedure on Rains' identities involving the double sine function $S_2$ to transition from the hyperbolic Ruijsenaars model to the trigonometric Sutherland model.
Convergence Analysis: A significant portion of the technical work is dedicated to proving the absolute convergence of the multiple integrals defining these operators, particularly in the presence of singularities along the diagonals ( $y_i = x_j$ ) and at infinity. The authors use specific bounds (Lemma 3.1) and asymptotic analysis of the wave functions to justify the application of Fubini's theorem and the dominated convergence theorem.
Inductive Proofs: The equivalence of the integral representations and the diagonalization properties of the new operators are proven by induction on the rank $n$ , utilizing the established exchange relations.

Key Contributions and Results

Equivalence of Representations (Theorem 3.1): The authors prove that the Borodin–Gorin interlacing integral representation, denoted $\Psi^*_{\lambda_n}(x_n)$ , is equivalent to the standard wave function $\Psi_{\lambda_n}(x_n)$ (which generates the Heckman–Opdam hypergeometric function $F_{\lambda_n}$ ) up to a normalization factor $\alpha_n^{-1} = \prod_{k=1}^n \frac{\Gamma(g)}{\Gamma(kg)}$ .
$\Psi^*_{\lambda_n}(x_n) = \alpha_n^{-1} F_{\lambda_n}(x_n)$
This is achieved by showing that the Borodin–Gorin wave function can be generated by successive application of the new interlacing raising operators $\Lambda^*_n$ , and that these operators commute with the standard raising operators up to Gamma function factors.
New Baxter Operators and Spectral Properties (Theorem 3.2): The paper introduces a commuting family of symmetric operators $Q^*_n(\mu) = Q^*_{n,+}(\mu) + Q^*_{n,-}(\mu)$ . It is proven that the wave functions $\Psi_{\lambda_n}(x_n)$ are eigenfunctions of these operators. The eigenvalues are explicitly calculated as a sum of products of Gamma functions:
$\prod_{j=1}^n \frac{\Gamma(\frac{1-g}{2} - i(\lambda_j - \mu))}{\Gamma(\frac{1+g}{2} - i(\lambda_j - \mu))} + \prod_{j=1}^n \frac{\Gamma(\frac{1-g}{2} + i(\lambda_j - \mu))}{\Gamma(\frac{1+g}{2} + i(\lambda_j - \mu))}$
This establishes that the Borodin–Gorin presentation diagonalizes a specific family of Baxter operators distinct from those in the standard presentation.
Limiting Procedure from Hyperbolic Model (Section 4): The authors demonstrate that the interlacing raising operator $\Lambda^*_n$ can be obtained as a strict limit of the reflected raising operator $\Lambda^{R*}_n$ from the hyperbolic Ruijsenaars model as the parameter $\omega_2 \to 0^+$ . This is non-trivial because the limit introduces additional integrable singularities along the diagonals. The authors provide rigorous bounds to justify this limit using the dominated convergence theorem.
Limitations on Baxter Operator Limits: The paper notes a distinction in the limiting behavior: while the raising operators arise naturally as limits, the corresponding Baxter operators $Q^*_n$ do not arise as direct limits from the hyperbolic system (the limit diverges). The authors suggest that these operators likely arise from more sophisticated limits involving compositions of operators, though this is only rigorously established for a restricted range of the coupling constant $g \in (1, 3/2)$ in the discussion.

Significance and Claims
The paper claims to provide a rigorous proof of the equivalence between the Borodin–Gorin interlacing integral presentation and the established Heckman–Opdam hypergeometric function, resolving a question regarding the coincidence of these dual integral presentations. By introducing and analyzing the new interlacing operators, the authors extend the algebraic structure (commuting families and exchange relations) known for the standard presentation to this interlacing context.

The work also clarifies the relationship between the hyperbolic Ruijsenaars model and its trigonometric degeneration (the Sutherland model), specifically highlighting how the "interlacing" structure emerges in the rational limit. The authors modestly note that while the raising operators are well-understood as limits, the origin of the interlacing Baxter operators from the hyperbolic system remains an open technical challenge requiring further investigation into compositions of operators. The results are presented as a step toward a deeper understanding of the duality and spectral properties of these integrable systems.

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