Vertical Recurrence Techniques

• Vertical recurrence is a technique that builds hierarchical structures along a defined axis, such as incrementing angular momentum in quantum chemistry integrals.
• It employs recursive schemes in combinatorial matrices and Riordan arrays, enabling efficient matrix decomposition and systematic construction of operator hierarchies.
• Applications span evolution equations, dynamical systems, and nested neural architectures, offering practical methods for modeling complex, multilevel phenomena.

Vertical recurrence encompasses a range of specialized mathematical and computational techniques, all rooted in the notion of building up structure or dependencies in the "vertical" direction—either within recursion relations for multi-index integrals, combinatorial constructions, operator hierarchies, matrix decompositions, or time-series phenomena. The term finds precise pertinence and utility in quantum chemistry, combinatorial matrix theory, evolution equations, nonlinear time series analysis, and dynamical systems. Across these fields, vertical recurrence deploys recursive schemes focusing on a particular axis: for instance, incrementing angular momentum on one center in electron integrals, stacking decompositions in combinatorial arrays, elevating differential order in operator semigroups, or extracting slow time-scale trapping via recurrence quantification analysis.

1. Vertical Recurrence Relations in Quantum Chemistry

Vertical recurrence relations (VRRs) are foundational in the computation of multi-electron integrals over Gaussian basis functions, particularly in the context of explicitly correlated F12 methods. VRRs enable the systematic "build-up" of angular momentum on one center, starting from a momentumless (s-type) fundamental integral. For example, in three-electron integrals, the algorithm derived in "Many-electron integrals over gaussian basis functions. I. Recurrence relations for three-electron integrals" (Barca et al., 2016) constructs higher-momentum integrals via formulas such as

$$[a_3^+]^m = (Z_3 - A_3)[a_3]^m\{0\} + \zeta_1\zeta_2 Y_{13}[a_3]^m\{2\} + \ldots,$$

where the auxiliary index m and increment/decrement annotations encode the combinatorial structure arising from Laplace integrals and angular momentum operators. For four-electron integrals, a related approach yields 24-term VRRs for each center ("Recurrence relations for four-electron integrals over Gaussian basis functions" (Barca et al., 2017)). The method allows direct computation without recourse to the resolution-of-the-identity (RI) approximation, thereby preserving both numerical exactness and computational efficiency for high-accuracy molecular orbital calculations.

VRRs are often contrasted with transfer recurrence relations (TRRs), which transfer momentum among centers but generally require a larger number of intermediate terms, and horizontal recurrence relations (HRRs), which shift angular momentum from "bra" to "ket" centers. The hierarchy VRR–TRR–HRR constitutes an exact and algebraically tractable scheme for generating complex integrals with controlled computational scaling.

2. Vertical Recurrence in Combinatorial Matrix Theory

Vertically-recurrent matrices, as defined in "On Vertically-Recurrent Matrices and Their Algebraic Properties" (Faal, 2022), generalize classical triangular arrays via a multidimensional recurrence relation

$$a_{n, k} = \sum_{l=k-1}^{n-1} \lambda_{n-1-l} a_{l, k-1},\quad (n \geq k \geq 1),$$

where $\Lambda = \{\lambda_n\}$ is the associated sequence. The structure admits an explicit lower-triangular Toeplitz-block decomposition:

$$V_n[\Lambda] = \overline{T}_n[\Lambda] \cdot \overline{T}_{n-1}[\Lambda] \cdots \overline{T}_1[\Lambda],$$

with $\overline{T}_k[\Lambda]$ as block Toeplitz matrices derived from $\Lambda$. This formalism supports matrix powers (e.g., if $\lambda_n$ is constant, the $m$th power has an associated sequence $\lambda^m((\lambda^m-1)/(\lambda-1))^n$), applications to the decomposition of admissible matrices, and ladder network transfer matrices.

Open problems include determining the associated sequence for general powers $(V_n[\Lambda])^m$, characterizing minimal polynomials over finite fields, and understanding recursively defined combinatorial arrays (such as the Catalan triangle) within this vertically-recurrent framework.

3. Vertical Recursive Relations in Riordan Arrays

Riordan arrays admit a vertical recursive relation that expresses each entry as a linear combination of previous row entries, parameterized by a generating sequence. In "The Vertical Recursive Relation of Riordan Arrays and Their Matrix Representation" (He, 2022), the key recursion is:

$$d_{n,k} = \sum_{j=1}^{n-k+1} f_j d_{n-j, k-1},\quad (n, k \geq 1),$$

where $f_j$ are coefficients derived from the generating function $f$. The structure is recast as a matrix representation, with the quasi-Riordan group capturing ensembles of such matrices under matrix multiplication. This allows construction and transformation of finite and weighted Riordan arrays, underpinning combinatorial identities and connecting to weighted (c)- or (C)-Riordan arrays for objects such as the rook and Laguerre triangles.

4. Vertical Recurrence in Higher Order Evolution Equations

An operator-theoretic manifestation of vertical recurrence appears in the theory of evolution equations, where the n-th order infinitesimal generator is recursively defined:

$$A_n(t) = (\partial_t + A_1(t)) A_{n-1}(t),\quad n \geq 2,$$

with $A_1(t)$ from the first-order equation $\partial_t \psi(t) = A_1(t) \psi(t)$ ("Recurrence formula for some higher order evolution equations" (Iwata, 2022)). The logarithmic representation of operators encodes this hierarchy,

$$A_n(t) = \prod_{k=1}^n \left( (\kappa \mathcal{U}_k(s, t) + I) \partial_t \log (\mathcal{U}_k(t, s) + \kappa I) \right),$$

mapping to Riccati-type and Miura transforms. This recurrence formalism translates operator evolution in Banach spaces, providing systematic progression to higher-order dynamics.

5. Vertical Recurrence in Dynamical Systems and Recurrence Quantification

In nonlinear time-series analysis, vertical recurrence captures the system's tendency to linger in similar dynamical states, operationalized as vertical line structures in recurrence plots. In "Recurrence based quantification of dynamical complexity in the Earth's magnetosphere at geospace storm timescales" (Donner et al., 2018), trapping time (TT) quantifies vertical recurrence:

$$TT = \frac{ \sum_{v=v_{\min}}^{v_{\max}} v p(v)}{ \sum_{v=v_{\min}}^{v_{\max}} p(v) },$$

where $p(v)$ is the probability density for vertical line lengths. High TT in the vertical component of the interplanetary magnetic field ($B_z$) corresponds to persistent southward conditions and correlates with storm-time magnetospheric reconfiguration. Vertical recurrence thus functions as a physically meaningful classifier and diagnostic for distinguishing between externally forced and internally evolving processes.

6. Applications in Random Walks, Lattice Sums, and Reflection Equations

Vertical recurrence is also found in probabilistic models such as the horizontal-vertical walk on $\mathbb{Z}^2$ ("Recurrence of horizontal-vertical walks" (Chan, 2020)), where it refers to the process's propensity to revisit sites under stochastic label/resampling dynamics. Recurrence is achieved for renewal parameters $q \in (1/3, 1]$, with martingale methods ensuring every vertex is visited infinitely often.

In combinatorics, vertical sum recurrences underpin counting modular and distributive lattices ("Cartesian lattice counting by the vertical 2-sum" (Kohonen, 2020)), with intricate symmetry factors controlling the number of nonisomorphic structures.

In algebraic recurrence relations with reflection ("Green's Functions of Recurrence Relations with Reflection" (Tojo, 2019)), vertical recurrence extends to nonlocal difference equations involving shift and reflection operators, leading to explicit Green's function representations for boundary value problems.

7. Recursion in Recursion and Hierarchical Deep Models

In machine learning, vertical recurrence is central to two-level nested recursion in recursive neural architectures. "Recursion in Recursion: Two-Level Nested Recursion for Length Generalization with Scalability" (Chowdhury et al., 2023) introduces a framework (RIR) wherein outer k-ary balanced-tree recursion is combined with inner Beam Tree RvNNs. The total non-linear recursion depth is bounded by $k \log_k n$, delivering efficient, scalable sequence composition with high length-generalization accuracy on tasks such as ListOps and competitive performance on the Long Range Arena (LRA). The approach balances computational efficiency with hierarchical expressiveness by nesting expressive, beam-search-based compositional recursion inside a scalable outer structure.

Summary Table: Representative Forms of Vertical Recurrence

Domain	Recurrence Structure	Key Functional Role
Quantum Chemistry	Angular momentum VRRs	Fast computation of multi-electron integrals
Matrix Theory	Multidimensional array recursion	Structured decomposition, combinatorial identities
Evolution Eqns.	Operator hierarchy (Aₙ(t))	Systematic elevation of differential order
Time Series	Vertical line trapping time (TT)	Quantification of dynamical persistence
Machine Learning	Nested RvNN recursion (RIR)	Hierarchical sequence composition, efficient deep computation
Lattice Counting	Vertical 2-sum recurrence	Asymptotic enumeration, modular/distributive lattices
Reflection Eqns.	Shift-reflection algebraic recursion	Green's function solutions to nonlocal difference equations

Vertical recurrence, as formalized across these areas, enables robust, algebraically precise solutions to constructing, counting, and analyzing complex, hierarchically dependent entities—be they molecular integrals, combinatorial matrices, operator hierarchies, or structured neural computations.