Trivariate Kampé de Fériet Function

• Trivariate Kampé de Fériet function is a multivariate hypergeometric series expressed as a triple power series in complex variables, characterized by rich parameter symmetries and reduction identities.
• It satisfies a system of partial differential equations and features analytic properties such as convergence conditions, recurrence relations, and Mellin–Barnes integral representations.
• The function finds practical applications in computing Feynman integrals, cosmological collider physics, and reductions to classical hypergeometric series in mathematical physics.

The trivariate Kampé de Fériet function is a prototypical multivariate hypergeometric function defined as a triple power series in three complex variables. It generalizes various classical hypergeometric series (such as ${}_pF_q$, Appell, and Lauricella functions) and serves as a central object in the paper of multivariable hypergeometric equations, reduction identities, and analytic computations in mathematical physics. The trivariate form arises naturally in the computation of certain Feynman integrals, cosmological correlators involving triple-exchange diagrams, and summation identities related to multiple series. It is characterized by rich parameter symmetries, reduction formulas, analytic continuation properties, and connections to Mellin–Barnes integral representations.

1. Formal Definitions and Series Expansions

The general trivariate Kampé de Fériet function can be written in several notational variants, each emphasizing different symmetries or parameterizations. In the notation used in “Massive Inflationary Amplitudes: New Representations and Degenerate Limits” (Xianyu et al., 11 Nov 2025), a typical trivariate KdF function is expressed as

${}^{p+q}\mathcal{F}_{\,r+s}\!\Bigl[ \begin{smallmatrix} a_1,\dots,a_p;\; c_1,\dots,c_r\[6pt] b_1,\dots,b_q;\;b'_1,\dots,b'_q;\;b''_1,\dots,b''_q\[3pt] d_1,\dots,d_s;\;d'_1,\dots,d'_s;\;d''_1,\dots,d''_s \end{smallmatrix} \Bigm|\,x,y,z\Bigr] = \sum_{m,n,k=0}^\infty \mathcal{A}_{m,n,k} \,\frac{x^m\,y^n\,z^k}{m!\,n!\,k!}$

where the triple-indexed coefficient

$$\mathcal{A}_{m,n,k} = \Gamma\!\Bigl[\! \begin{smallmatrix} a_1+m+n+k,\dots,a_p+m+n+k\ c_1+m+n+k,\dots,c_r+m+n+k \end{smallmatrix}\!\Bigr] \prod_{\text{cyclic}} \Gamma \text{ factors in } m, n, k.$$

Here, upper parameters $a_i$, $c_j$ are associated with the total index $m+n+k$, and lower parameters $b_i$, $d_j$ (and their primed variants) are distributed over $m$, $n$, $k$. The Pochhammer and Gamma notation follow the conventions detailed in the data. Alternative notations, as in (Verma, 2020, Rathie, 2020), arrange parameters into blocks and use generalized multi-index summation.

Convergence conditions for the triple series are described by inequalities on parameter sums, typically requiring $1 + m_1 + m_2 + m_3 - p - (q_1 + q_2 + q_3) \geq 0$ for the parameters defined in (Verma, 2020), with boundary cases requiring modulus restrictions on $x_i$.

2. Differential Equations, Analytic Properties, and Recurrences

Trivariate Kampé de Fériet functions are Horn-type hypergeometric series and satisfy a system of independent partial differential equations (PDEs), one for each variable. Using the Euler-type differential operators $\Theta_x = x\,\partial_x$, etc., a typical equation reads: $$\left(\Theta_{u_i} - \frac{3}{2}\right)^2 + \nu_i^2 - u_i (\Theta_{u_i} - 1)\left(\sum_{j} \Theta_{u_j} + p_0 + 1\right) = 0.$$ All three variables enter symmetrically, and the system is annihilated by the trivariate KdF. This PDE system underlies its application in cosmological amplitudes, permitting formal transformations to folded/degenerate limits (Xianyu et al., 11 Nov 2025).

The coefficients $\mathcal{A}_{m,n,k}$ factor into products of Pochhammer symbols, directly yielding linear recurrence (contiguous) relations in the parameters. Parameter-shifting or recurrence formulas typically relate functions with upper parameter incremented by one or more units to finite sums over KdF functions with shifted lower parameters. For example: $$d_{m,\ell,\{\dots\}} = (-1)^\ell\,(\ell+q_j+p_j+3)_m \cdots$$ encodes the effect of an additional index in one summation variable, corresponding to a linear relation among KdF functions with shifted indices (Xianyu et al., 11 Nov 2025). Finite summation (truncation) arises when upper parameters become negative integers, making the series a polynomial (Verma, 2020).

3. Reductions, Special Cases, and Transformation Properties

Significant structural simplifications occur under variable or parameter limits:

• Reductions: Setting a variable, e.g., $z=0$, reduces the trivariate KdF to a bivariate (Appell or Lauricella) function. Setting two variables to zero collapses further to a univariate $_pF_q$ (Xianyu et al., 11 Nov 2025, Verma, 2020).
• Colinear/Folded (Degenerate) Limit: The variable change $x \to \frac{2x}{1+x}$ implements the kinematic “folded limit” in physical applications, preserving symmetries while reducing the transcendental weight by one at each step.
• Parameter Collapse: Choices such as $d_i=1$ collapse Pochhammers to factorials, reducing the independent parameter count and sometimes recovering classical Lauricella or Appell functions.
• Specialization: With particular parameter choices, e.g., $p=1$, $\ell=1$, $q_2=q_3=0$, $m_2=m_3=0$, one recovers the Lauricella $F_D^{(3)}$ function. Its confluent limits provide Appell’s $F_A, F_B$ and related three-variable functions (Verma, 2020).

Reduction identities involving Beta and Gamma integral representations allow the expression of a trivariate KdF function as terminating single-variable $_pF_q$ series with shifted parameters (Rathie, 2020).

4. Finite Summation Formulas and Contiguous Relations

Systematic finite summation identities for trivariate KdF functions are derived using derivative operations or combinations of Gamma functions and binomial/Vandermonde identities. For example: $$\sum_{k=0}^r \frac{(a_i)_k [b^{(1)}]_k x_1^k}{k!} F^{p : q_1,q_2,q_3}_{\ell : m_1,m_2,m_3} \Bigl[\cdots|x_1, x_2, x_3\Bigr] = \frac{D^r}{dx_1^r}\{ x_1^{\alpha_1 + r - 1} F^{p : q_1, q_2, q_3}_{\ell : m_1, m_2, m_3} [\cdots]\}$$ yields the $r$th derivative formula in $x_1$ (Verma, 2020).

A Vandermonde-type formula relates finite sums over shifted upper parameters to single KdF functions with larger discrete shifts, leveraging $_2F_1$ summations within the multi-index sum. These identities generalize the contiguous relations in the univariate hypergeometric theory.

Closed-form simplifications arise in cases where upper parameters are negative integers—these instances truncate the otherwise infinite sums to polynomials.

5. Integral Representations, Reduction Formulas, and Special Values

Though not always required in physical applications, Mellin–Barnes (MB) representations for trivariate KdF functions can be written as triple complex integrals, with Gamma factors encoding the series coefficients: $${}^{p+q}\mathcal{F}_{r+s}(x, y, z) = \frac{1}{(2\pi i)^3} \int_{-i\infty}^{+i\infty} ds\, dt\, du\; [\text{product of } \Gamma \text{ factors}]\, x^{-s} y^{-t} z^{-u}$$ subject to appropriate contour closure conditions (Xianyu et al., 11 Nov 2025). In the literature (Rathie, 2020), Beta and Gamma integrals are utilized to perform decouplings in the sum indices, resulting in reduction formulas. For instance, in cases with $m, n \in \mathbb{N}_0$: $$F_{1:1;1}^{1:1;1} [d:a;B;\;e:2a+m;2B+n;\;x,x] = \sum_{j,k=0}^\infty \cdots x^{j+k}\; {}_4F_5[\text{shifted parameters}; -x^2]$$ with ${}_4F_5$ arguments depending on $j, k$.

Further reductions cover cases with symmetries or parameters chosen such that two or more variables or parameters coincide, and yield ${}_4F_5$, ${}_4F_3$, Legendre, or Gegenbauer polynomials as special cases.

6. Applications in Mathematical Physics and Model Building

The trivariate Kampé de Fériet function has found recent application in the analysis of cosmological collider physics, specifically in the analytic computation of inflationary amplitudes involving triple exchanges—see (Xianyu et al., 11 Nov 2025). In these computations, the fully folded (degenerate) limit of specific tree-level diagrams produces amplitudes described in terms of trivariate KdF functions.

An explicit example is the inflaton bispectrum with triple massive exchange, which is represented as

$$\langle\varphi\varphi\varphi\rangle' \propto \sum_{\iota_1,\iota_2,\iota_3=0,\pm1}\; \sum_{m_1,m_2,m_3=0}^\infty [\text{coefficients}] u_1^{\,m_1+\cdots}u_2^{\,m_2+\cdots}u_3^{\,m_3+\cdots}$$

Recognized as a linear combination (sum over eight terms) of trivariate KdF functions ${}^{1+2}\mathcal{F}_{0+2}(u_1, u_2, u_3)$.

In the “squeezed” kinematic limit, systematic reduction to bivariate and univariate KdF functions is effected by expanding in small $u_1$ and resumming the relevant terms, reflecting the hierarchical structure of the degeneration/reduction identities.

Other applications appear in mathematical physics (three-particle scattering, Feynman integral computations), statistics (array distributions), and multivariate orthogonal polynomials (Verma, 2020).

7. Convergence, Analytic Continuation, and Further Developments

The series convergence domain is dictated by inequalities involving the parameters and sums of indices: $1 + m_1 + m_2 + m_3 - p - (q_1 + q_2 + q_3) \ge 0,$ with boundary cases imposing $|x_1|^{1/p} + |x_2|^{1/p} + |x_3|^{1/p} < 1$ (for $p > q_1 + q_2 + q_3$) or $\max{|x_i|} < 1$ (for $p < q_1+q_2+q_3$) (Verma, 2020). Analytic continuation is achievable via parameter-shifting contiguous relations and, when needed, by Mellin–Barnes representations (Xianyu et al., 11 Nov 2025).

A plausible implication is that further reductions and summation formulas can be systematically developed for the trivariate KdF and its confluent or degenerate forms, as demonstrated by the extensive catalog in (Rathie, 2020). These developments highlight the connection between multivariate hypergeometric functions and the structure of Feynman/graph amplitudes, promoting further exploration in special function theory and mathematical physics.

Table: Trivariate KdF Properties and Reductions

Property/Operation	Effect/Outcome	Reference/Paper
Variable set to zero	Reduces to bivariate KdF/Appell/Lauricella	(Xianyu et al., 11 Nov 2025, Verma, 2020)
Negative integer upper param	Series truncates to a polynomial of total degree	(Verma, 2020)
Folded/colinear limit	Reduces transcendental weight, structure simplifies	(Xianyu et al., 11 Nov 2025)
Beta/Gamma integral	Enables reduction to $_pF_q$ series, closed-form identities	(Rathie, 2020)
Parameter-shift recurrence	Finite sum of shifted KdF functions (“contiguous relations”)	(Verma, 2020)

These structural features contribute to both the mathematical richness and the broad applicability of the trivariate Kampé de Fériet function across mathematical and physical domains.