Radon Hypergeometric Function (Radon HGF)

• Radon HGF is a multivariable hypergeometric function defined on Grassmannians using Radon transforms and matrix models.
• It unifies various special functions, including Gauss, Kummer, and Bessel analogues, by framing them as determinant-power integrals with covariance under GL(m) and Hλ.
• Contiguity relations derived from the Capelli identity and Cayley’s formula govern parameter shifts, yielding recurrences for Hermitian matrix beta and gamma functions.

Searching arXiv for papers on Radon hypergeometric functions and related Grassmannian constructions. The Radon hypergeometric function (Radon HGF) is a multi-variable hypergeometric function on the Grassmannian $\operatorname{Gr}(m,N)$, with $N=rn$, defined as a Radon transform of a character of the universal covering group of a subgroup $H_\lambda\subset GL(N)$ determined by a partition $\lambda$ of $n$. In Kimura’s formulation it includes confluent and non-confluent types, reduces to the Gelfand hypergeometric function when $r=1$, and provides a unified realization of Hermitian matrix integral analogues of the Gauss hypergeometric function and its confluent family (Kimura, 25 Jul 2025). Subsequent work on contiguity relations shows that the Capelli identity and Cayley’s formula control the parameter-shift structure of Radon HGFs and yield recurrences for beta and gamma functions defined by Hermitian matrix integrals (Kimura, 30 Sep 2025).

1. Geometric framework and Radon-transform construction

The ambient geometry is a Grassmannian double-fibration setting. With $V=\mathbb{C}^N$, $N=nr$, one considers

$$M_1=\operatorname{Gr}(r,V),\qquad M_2=\operatorname{Gr}(m,V),$$

together with the flag manifold

$$\operatorname{Flag}(r,m,V)=\{(v_1,v_2)\mid v_1\subset v_2\subset V,\ \dim v_1=r,\ \dim v_2=m\}.$$

The Radon transform is defined by pulling a function on $N=rn$0 to the flag manifold, restricting it to a fiber $N=rn$1, and integrating along a suitable top-degree chain $N=rn$2 in that fiber. In homogeneous coordinates this becomes

$N=rn$3

where $N=rn$4 is a full-rank $N=rn$5 matrix representing a point of $N=rn$6, $N=rn$7, and $N=rn$8 is a canonical top-degree form on $N=rn$9 (Kimura, 25 Jul 2025).

A useful matrix model identifies

$H_\lambda\subset GL(N)$0

This model is the basis for the explicit formulas for the Radon HGF, for its differential operators, and for its covariance properties. It also makes clear that the external variables of the function are Grassmannian variables represented by full-rank matrices rather than local affine coordinates alone (Kimura, 25 Jul 2025).

The terminology is not fully uniform across the 2025 literature. One paper works formally with Gelfand hypergeometric functions on $H_\lambda\subset GL(N)$1, but defines them as Radon transforms of characters of a maximal abelian subgroup $H_\lambda\subset GL(N)$2; in that setting the Radon-transform viewpoint is primary, and the case $H_\lambda\subset GL(N)$3 is precisely the scalar limit of the later Radon HGF formalism (Kimura, 13 Jun 2025).

2. The subgroup $H_\lambda\subset GL(N)$4, Jordan blocks, and characters

Let $H_\lambda\subset GL(N)$5 be a partition of $H_\lambda\subset GL(N)$6. For each positive integer $H_\lambda\subset GL(N)$7, the generalized Jordan group $H_\lambda\subset GL(N)$8 is formed from block upper-triangular Toeplitz matrices, equivalently from the unit group of the truncated algebra $H_\lambda\subset GL(N)$9 with $\lambda$0. The subgroup attached to $\lambda$1 is

$\lambda$2

so $\lambda$3. In the non-confluent case $\lambda$4, one has

$\lambda$5

and for $\lambda$6 this becomes the diagonal Cartan subgroup of $\lambda$7 (Kimura, 30 Sep 2025).

Characters of $\lambda$8 are expressed using the logarithm of the unipotent part. Writing

$\lambda$9

a character has the form

$n$0

with $n$1. For $n$2, the full character is a product over the blocks: $n$3 In the non-confluent case this simplifies to

$n$4

The parameters therefore record determinant exponents for non-confluent blocks and additional exponential-trace data for confluent blocks (Kimura, 30 Sep 2025).

The standard assumptions are that $n$5 for all $n$6, that $n$7 when $n$8, and that

$n$9

The last condition ensures the correct $r=1$0-weight for the integrand and is what allows the character factor to combine with the canonical form $r=1$1 into a well-defined top form on $r=1$2 (Kimura, 30 Sep 2025).

3. Definition of the Radon HGF and its analytic structure

Write $r=1$3 in blocks adapted to $r=1$4,

$r=1$5

and define the Zariski-open set

$r=1$6

The integration space is

$r=1$7

If $r=1$8 on the standard affine chart with $r=1$9 and $V=\mathbb{C}^N$0, the canonical top-degree form is

$V=\mathbb{C}^N$1

and it satisfies

$V=\mathbb{C}^N$2

This compensates exactly for the character weight imposed by $V=\mathbb{C}^N$3 (Kimura, 25 Jul 2025).

For fixed $V=\mathbb{C}^N$4, the branch divisors are

$V=\mathbb{C}^N$5

The integrand may be written as

$V=\mathbb{C}^N$6

where

$V=\mathbb{C}^N$7

determines a rank-one local system $V=\mathbb{C}^N$8, and the exponential factor $V=\mathbb{C}^N$9 controls the irregular behavior in confluent cases through a family of supports $N=nr$0 in the sense of Pham. The Radon HGF is then

$N=nr$1

In affine coordinates this becomes

$N=nr$2

with $N=nr$3. In the non-confluent case all exponential terms disappear and one recovers the determinant-power integral familiar from Gelfand-type constructions (Kimura, 25 Jul 2025).

Analytically, for a fixed sector of cycles the Radon HGF is multivalued holomorphic in $N=nr$4 on a Zariski-open subset and meromorphic in the parameters $N=nr$5. The non-confluent branch behavior comes from determinantal divisors, whereas the confluent case introduces exponential growth and decay governed by the $N=nr$6-terms. This separation between algebraic monodromy and irregular behavior is one of the structural features distinguishing confluent from non-confluent Radon HGFs (Kimura, 25 Jul 2025).

4. Differential equations, covariance, and the $N=nr$7 Gelfand limit

The Radon HGF satisfies an overdetermined linear PDE system whose highest-order part expresses the image constraints of the Radon transform. For $N=nr$8, let

$N=nr$9

and for subsets $$M_1=\operatorname{Gr}(r,V),\qquad M_2=\operatorname{Gr}(m,V),$$0, $$M_1=\operatorname{Gr}(r,V),\qquad M_2=\operatorname{Gr}(m,V),$$1 with $$M_1=\operatorname{Gr}(r,V),\qquad M_2=\operatorname{Gr}(m,V),$$2, define

$$M_1=\operatorname{Gr}(r,V),\qquad M_2=\operatorname{Gr}(m,V),$$3

Then

$$M_1=\operatorname{Gr}(r,V),\qquad M_2=\operatorname{Gr}(m,V),$$4

for all such $$M_1=\operatorname{Gr}(r,V),\qquad M_2=\operatorname{Gr}(m,V),$$5. These equations are the main equations characterizing the image of the Radon transform. They are complemented by first-order equations obtained by differentiating the covariance relations under $$M_1=\operatorname{Gr}(r,V),\qquad M_2=\operatorname{Gr}(m,V),$$6 and $$M_1=\operatorname{Gr}(r,V),\qquad M_2=\operatorname{Gr}(m,V),$$7, so the full Radon hypergeometric system consists of these $$M_1=\operatorname{Gr}(r,V),\qquad M_2=\operatorname{Gr}(m,V),$$8-st order determinantal equations together with Lie-algebraic covariance equations (Kimura, 25 Jul 2025).

The covariance itself is explicit. For $$M_1=\operatorname{Gr}(r,V),\qquad M_2=\operatorname{Gr}(m,V),$$9 and $$\operatorname{Flag}(r,m,V)=\{(v_1,v_2)\mid v_1\subset v_2\subset V,\ \dim v_1=r,\ \dim v_2=m\}.$$0,

$$\operatorname{Flag}(r,m,V)=\{(v_1,v_2)\mid v_1\subset v_2\subset V,\ \dim v_1=r,\ \dim v_2=m\}.$$1

This describes $$\operatorname{Flag}(r,m,V)=\{(v_1,v_2)\mid v_1\subset v_2\subset V,\ \dim v_1=r,\ \dim v_2=m\}.$$2 as a relative invariant under the $$\operatorname{Flag}(r,m,V)=\{(v_1,v_2)\mid v_1\subset v_2\subset V,\ \dim v_1=r,\ \dim v_2=m\}.$$3-action and is the basic source of the first-order differential relations (Kimura, 25 Jul 2025).

When $$\operatorname{Flag}(r,m,V)=\{(v_1,v_2)\mid v_1\subset v_2\subset V,\ \dim v_1=r,\ \dim v_2=m\}.$$4, the system reduces to the Gelfand hypergeometric system on a Grassmannian. In the $$\operatorname{Flag}(r,m,V)=\{(v_1,v_2)\mid v_1\subset v_2\subset V,\ \dim v_1=r,\ \dim v_2=m\}.$$5 setting, the second-order operators

$$\operatorname{Flag}(r,m,V)=\{(v_1,v_2)\mid v_1\subset v_2\subset V,\ \dim v_1=r,\ \dim v_2=m\}.$$6

are presented as the part of the system that characterizes the image of the Radon transform, while additional first-order equations encode covariance under $$\operatorname{Flag}(r,m,V)=\{(v_1,v_2)\mid v_1\subset v_2\subset V,\ \dim v_1=r,\ \dim v_2=m\}.$$7 and $$\operatorname{Flag}(r,m,V)=\{(v_1,v_2)\mid v_1\subset v_2\subset V,\ \dim v_1=r,\ \dim v_2=m\}.$$8. Restricting to a distinguished slice yields a hyperbolic subsystem, and the corresponding Gelfand HGF furnishes solutions of the 2-dimensional Toda-Hirota equation; the paper states that the contiguity relations play an important role in constructing the Laplace sequence and Bäcklund transformations (Kimura, 13 Jun 2025).

A common misconception is to treat the Radon HGF merely as a reformulation of classical hypergeometric integrals. The differential-system viewpoint shows that its defining content is not only the integral representation but also the Grassmannian Radon-image condition encoded by the determinantal PDEs. In the scalar case this specializes to the classical Gelfand framework; for $$\operatorname{Flag}(r,m,V)=\{(v_1,v_2)\mid v_1\subset v_2\subset V,\ \dim v_1=r,\ \dim v_2=m\}.$$9 it produces a higher-rank system rather than a direct tensor product of one-variable equations (Kimura, 25 Jul 2025).

5. Contiguity relations, Capelli identity, and Cayley’s formula

The 2025 contiguity theory identifies exact parameter-shift operators for the Radon HGF. In the non-confluent case $N=rn$00, writing

$N=rn$01

one introduces first-order operators

$N=rn$02

and the order-$N=rn$03 operator

$N=rn$04

The contiguity relation is

$N=rn$05

Thus a differential operator of order $N=rn$06 increments one parameter by $N=rn$07 and decrements another by $N=rn$08, with scalar factor given by the $N=rn$09-function (Kimura, 30 Sep 2025).

For general $N=rn$10, the operator becomes

$N=rn$11

If $N=rn$12, the same $N=rn$13-function appears: $N=rn$14 If $N=rn$15, the confluent formula is

$N=rn$16

The shift always acts on the leading parameters by $N=rn$17 and $N=rn$18, while the scalar factor depends on whether the target block is non-confluent or confluent (Kimura, 30 Sep 2025).

These formulas are derived from the Capelli identity

$N=rn$19

and Cayley’s formula

$N=rn$20

The role of the Capelli identity is to convert invariant differential operators into determinant-lowering operators, while Cayley’s formula supplies the scalar factor that appears in the shifted integral (Kimura, 30 Sep 2025).

The same machinery yields contiguity relations for Hermitian matrix beta and gamma functions: $N=rn$21

$N=rn$22

$N=rn$23

These are obtained by realizing $N=rn$24 and $N=rn$25 as special Radon HGFs on $N=rn$26 (Kimura, 30 Sep 2025).

6. Weyl-group analogue and symmetry structure

A further structural layer is the Weyl-group analogue

$N=rn$27

which governs symmetries of the Radon HGF. If

$N=rn$28

then

$N=rn$29

where $N=rn$30 is the permutation group of equal-size Jordan blocks, and $N=rn$31 is the continuous factor identified with automorphisms of the truncated polynomial algebra $N=rn$32. In the non-confluent case $N=rn$33, one recovers

$N=rn$34

(Kimura, 27 Oct 2025).

The symmetry acts simultaneously on the Grassmannian variables and on the parameter vector. Writing the induced linear action on parameters as $N=rn$35, the main symmetry statement is

$N=rn$36

and for the finite permutation part $N=rn$37,

$N=rn$38

The continuous part therefore generates nontrivial linear transformations of the parameters, whereas the finite part produces genuine symmetry identities of the function (Kimura, 27 Oct 2025).

One consequence is parameter reduction. The continuous subgroup can normalize higher confluent parameters so that, with the leading exponents fixed, each block may be reduced to a form

$N=rn$39

This explains why several matrix analogues in the confluent family carry fewer essential external parameters than the raw character formula might suggest. In the $N=rn$40 applications, the same symmetry yields transformation formulas analogous to part of Kummer’s 24 solutions for the Gauss hypergeometric function and to Kummer’s first transformation for the confluent case (Kimura, 27 Oct 2025).

7. Specializations, classical analogues, and related developments

A notable feature of the theory is that the Radon HGF organizes several scalar and matrix-valued special functions into a single Grassmannian framework. The following specializations are stated explicitly.

Partition $N=rn$41	Grassmannian	Specialization
$N=rn$42	$N=rn$43	Hermitian matrix beta function
$N=rn$44	$N=rn$45	Hermitian matrix gamma function
$N=rn$46	$N=rn$47	Gaussian matrix integral
$N=rn$48	$N=rn$49	Gauss matrix HGF
$N=rn$50	$N=rn$51	Kummer analogue
$N=rn$52	$N=rn$53	Bessel analogue
$N=rn$54	$N=rn$55	Hermite–Weber analogue
$N=rn$56	$N=rn$57	Airy analogue

For $N=rn$58, the paper states that the Hermitian matrix integral analogues of the Gauss HGF and its confluent family are obtained in a unified manner from the partitions $N=rn$59, $N=rn$60, $N=rn$61, $N=rn$62, and $N=rn$63. This is one of the main points of the definition paper and places matrix hypergeometric functions of Faraut–Korányi and Muirhead inside the same Radon-transform construction (Kimura, 25 Jul 2025).

The symmetry paper goes further by deriving matrix analogues of classical transformation formulas. For the Gauss analogue it obtains

$N=rn$64

which is identified as the matrix analogue of a classical Kummer transformation. For the Kummer analogue it derives

$N=rn$65

These formulas arise from the finite part of the Weyl-group analogue acting on normal forms in $N=rn$66 (Kimura, 27 Oct 2025).

At $N=rn$67, the same general framework specializes to Gelfand hypergeometric functions on Grassmannians, including confluent and non-confluent types. On $N=rn$68, the partitions $N=rn$69, $N=rn$70, $N=rn$71, $N=rn$72, and $N=rn$73 correspond respectively to Gauss, Kummer, Bessel, Hermite–Weber, and Airy functions. The integrable-systems development shows that, on the slice $N=rn$74, the Gelfand HGF enters the 2-dimensional Toda-Hirota equation through its contiguity operators and the Laplace sequence of the associated hyperbolic operators. This suggests a broader role for Radon-type Grassmannian hypergeometric functions in the interface between integral geometry, holonomic PDEs, and integrable systems (Kimura, 13 Jun 2025).