Whittaker Integrals

• Whittaker integrals are a class of integral constructions built from classical and generalized Whittaker functions, providing explicit representations in harmonic analysis and automorphic theory.
• They employ diverse representations—such as beta-type, Mellin–Barnes, and Bessel-kernel forms—to analyze spectral transforms and L-function evaluations.
• Applications span integrable probability, quantum deformations, and wave physics, where these integrals yield exact formulas and underpin spectral decompositions.

A Whittaker integral, or Whittaker–type integral, refers to a large class of integral constructions involving Whittaker functions, their extensions, and their occurrence as kernels in harmonic analysis, representation theory, integrable probability, and wave physics. These integrals range from explicit formulas for special functions (including extensions of the classical Whittaker function), through generalized kernel representations in automorphic forms, to multidimensional transforms on groups and special models in quantum integrable systems. The following sections present a comprehensive overview of Whittaker integrals, their extensions, methods of construction, spectral theory, and representative applications in modern mathematics and mathematical physics.

1. Classical and Generalized Whittaker Functions

The classical Whittaker function arises as a specific solution to the confluent hypergeometric equation. For parameters $\kappa,\,\mu\in\mathbb{C}$ and $x>0$, the standard forms are $M_{\kappa,\mu}(x)$ and $W_{\kappa,\mu}(x)$, with multiple equivalent integral representations. For example, the beta-type representation for $M_{\kappa,\mu}$ is

$$M_{\kappa,\mu}(x) = \frac{x^{\mu+\frac12} e^{-x/2}}{B(\mu+\kappa+2,\,\mu-\kappa+1)} \int_0^1 t^{\mu+\kappa+1} (1-t)^{\mu-\kappa} e^{x t} dt$$

with convergence for $\Re(\mu\pm\kappa+1)>0$ (Apelblat et al., 2023). Whittaker functions have Laplace- and Mellin-Barnes representations, and, via the Jacquet integral, they are realized as matrix coefficients and test vectors in representations of real and $p$-adic reductive groups (Kurinczuk et al., 2017, Hirano et al., 31 Aug 2024, Kim, 15 Nov 2024).

Generalizations include $b$-Whittaker functions (quantum group deformations) (Schrader et al., 2018), $(p,q)$-Whittaker functions parameterized by multiple, possibly non-classical parameters (Rahman et al., 2017), and extensions utilizing generalized confluent hypergeometric functions as kernels (Rahman et al., 2018).

2. Integral Representations and Extensions

Whittaker functions admit a rich landscape of integral representations, serving as building blocks for spectral analysis, harmonic analysis, and special function theory. Five principal classes, illustrated by (Rahman et al., 2018, Rahman et al., 2017), are summarized below:

Representation	Main Formula	Kernel Structure
Beta-type	$\int_0^1 t^{\alpha}(1-t)^\beta e^{zt} dt$	Algebraic + exponential
Bessel-kernel	$\int_0^\infty t^\gamma e^{-t} I_\nu(2\sqrt{xt}) dt$	Bessel (or modified Bessel)
Mellin-Barnes	$$\int_{c-i\infty}^{c+i\infty} \Gamma(\cdots)x^{-s} ds$$	Gamma functions, contour integral
Transformed intervals	$\int_a^b \cdots du$	Changing integration domain
Extended kernel	Use of $K_{\nu}(f(t))$, or insertion of exponential deformation factors	Extended via extra parameters

For $(p,q)$-Whittaker and similar extensions, the kernel or exponent is modified, e.g., by $\exp(z t - p/t - q/(1-t))$ (Rahman et al., 2017) or by inclusion of $K_{v+1/2}(p\, t(1-t))$ (Rahman et al., 2018). These explicit forms systematically recover the classical representations as special cases when the extra parameters vanish.

Integral representations are also obtained for parameter derivatives and "integral Whittaker functions" (i.e., integrating $M_{\kappa,\mu}(t)/t$ or $W_{\kappa,\mu}(t)/t$ over various domains) (Apelblat et al., 2023, Apelblat et al., 2023, Apelblat et al., 2021).

3. Whittaker Integrals in Harmonic Analysis and Automorphic Theory

Whittaker integrals underpin the realization of Whittaker models for representations of general linear and reductive groups over local fields. The Jacquet integral, central in this theory, is given for $f$ in the induced model and $g$ in the group by

$J_{\psi}(f)(g) = \int_{N} \psi(n) f(gn)\, dn$

with convergence in a suitable right half-plane and meromorphic continuation in spectral parameters (Kim, 15 Nov 2024, Hirano et al., 31 Aug 2024).

Explicit formulas for test vectors and their utilization in Rankin–Selberg integrals, exterior and symmetric-square L-factors, and period integrals across various subgroups have been derived, particularly using the structure of essential or newform Whittaker vectors on $p$-adic groups (Jo, 2021, Kurinczuk et al., 2017). In the archimedean context, Stade's Mellin–Barnes integral for $GL(n,\mathbb{R})$ Whittaker functions provides a kernel for explicit L-factor evaluations (Hirano et al., 31 Aug 2024).

Whittaker integrals further appear in the spectral expansion of automorphic forms and in the explicit construction of local components of automorphic L-functions, via either integral transforms or as pieces in the functional equations of Eisenstein series.

4. Spectral Transforms and Whittaker Analysis on Reductive Groups

A central analytic feature is that Whittaker integrals realize generalized Fourier (spectral) transforms on $G/N$-type spaces. For $G$ a real reductive group and $N$ a maximal unipotent subgroup, the Whittaker Schwartz space admits a Fourier transform of the type

$$\mathcal{F}_P f(\nu) = \int_{G/N_0} f(g) W_P(\cdot, -\bar{\nu}; g)^*\, dg$$

mapping into a vector-valued Schwartz space on $i\mathfrak{a}_P^*$ (Ban, 24 Nov 2025, Ban, 2023). The inversion uses the normalized Whittaker integral constructed using Maass–Selberg relations to ensure holomorphicity and temperedness in spectral variables.

Maass–Selberg relations provide functional equations for intertwining operators and normalized Whittaker integrals, crucial for establishing regularity and control of the spectral decomposition (Ban, 24 Nov 2025). Uniform temperedness and analytic continuation are obtained by a combination of explicit integration-by-parts arguments (e.g., via explicit Bruhat cell decompositions (Kim, 15 Nov 2024)), functional equations, and shift operators, ultimately yielding rapid decay estimates for Whittaker transforms and spectral Plancherel formulas (Ban, 2023).

5. Whittaker Integrals in Integrable Probability and Mathematical Physics

Whittaker integrals appear in exact formulas for models in integrable probability, such as partition function laws in log-gamma polymer models and stochastic growth. Notably, the GL$(N,\mathbb{R})$–Whittaker function arises as the eigenfunction in the geometric RSK model and determines the Laplace transform of the polymer free energy (Corwin et al., 2011, Barraquand, 13 Sep 2024).

The Laplace transform of the partition function, under the "Whittaker measure," reduces to a Barnes-type multidimensional integral

$$\mathbb E[e^{-sZ_{N,1}(n)}] = \int_{(i\mathbb{R})^N} s^{\sum_{i}(\lambda_i - \theta_i)} \frac{\prod_{i,j} \Gamma(\lambda_i - \theta_j)}{\prod_{m=1}^n\Gamma(\lambda_i + \eta_m)} \mathfrak{s}_N(\lambda)\, d\lambda$$

where $\mathfrak{s}_N(\lambda)$ is the Sklyanin measure (Corwin et al., 2011).

Multilayer generalizations and refined joint distributions (two-layer Whittaker processes) have been developed using contour-integral formulas directly related to the Whittaker transform and the orthogonality relations of Whittaker functions (Barraquand, 13 Sep 2024).

In mathematical physics, the Whittaker integral also appears as the "transverse Whittaker integral," providing an exact representation for three-dimensional accelerating solutions of the Helmholtz equation with angular spectra built from special functions (Mathieu, Weber, Fresnel), showing semicircular trajectories and non-diffracting behaviors (Zhang et al., 2014).

6. Discrete Index and Quantum Generalizations

Discrete index analogs of the Whittaker transform have been constructed by considering expansions in $W_{\mu, i n}(x)$ for integer $n$ and Mellin–Barnes or Bessel-kernel type series and transforms, together with explicit inversion formulas. These transforms provide discrete orthogonality and completeness properties, with explicit integral kernels involving parabolic cylinder functions (Yakubovich, 2020).

Quantum deformations (the $b$-Whittaker functions) emerge as eigenfunctions in the $q$-deformed $\mathfrak{gl}_n$ Toda system, with integral representations paralleling the undeformed case but involving modular $q$-gamma and quantum dilogarithm functions. These $b$-Whittaker integrals satisfy deformed Cauchy–Littlewood (Plancherel) identities and possess unitary transform and inversion properties (Schrader et al., 2018).

7. Applications, Special Cases, and Further Developments

Whittaker integrals pervade harmonic analysis, automorphic forms, quantum integrable systems, stochastic processes, and applied wave theory. Applications encompass:

• Test vector and period integral computation for local and global L-functions (Jo, 2021, Kurinczuk et al., 2017, Hirano et al., 31 Aug 2024).
• Construction and regularization of spectral decompositions on $G/N$ (Ban, 24 Nov 2025, Ban, 2023).
• Exact formulas for partition functions and joint laws of integrable probabilistic models (Corwin et al., 2011, Barraquand, 13 Sep 2024).
• Special function theory, including closed-form and hypergeometric representations (Apelblat et al., 2021, Apelblat et al., 2023, Rahman et al., 2018).
• Generalizations via parameter extensions, quantum deformation, and non-classical integration kernels (Rahman et al., 2017, Schrader et al., 2018, Yakubovich, 2020).
• Construction of nonparaxial accelerating beams in physics (Zhang et al., 2014).

Whittaker integrals thus constitute a unifying analytic structure connecting representation theory, special functions, harmonic and Fourier analysis, as well as integrable and stochastic models across mathematics and physics.