Mathematician

CMFs are multi-dimensional — not limited to 2 axes. The ramanujantools library supports arbitrary dimension:

• 2D CMFs (axes $x, y$): Known CMFs for $e$, $\pi$, $\zeta(3)$
• 3D CMFs (axes $a, b, c$): e.g., hypergeometric_derived_2F1() — a CMF with 3 axes derived from ${}_2F_1$ hypergeometric functions
• 5D CMFs (axes $x_0, x_1, x_2, y_0, y_1$): e.g., hypergeometric_derived_3F2() — derived from ${}_3F_2$
• Higher dimensions are possible and an active research frontier.

Key CMF operations (from ramanujantools.cmf.CMF):

• trajectory_matrix(trajectory, start): Extract a 1D matrix sequence by walking along a trajectory through the field. Different trajectories yield different PCFs for the same constant.
• walk(trajectory, iterations, start): Compute the matrix product along a trajectory for given depths.
• limit(trajectory, iterations, start): Compute the limit (convergent value) of a walk.
• delta(trajectory, depth, start): Compute the irrationality measure $\delta$ where $|p_n/q_n - L| = q_n^{-(1+\delta)}$.
• coboundary(U): Apply a coboundary transformation $M \mapsto U \cdot M \cdot U^{-1}(+1)$, which preserves the CMF structure but changes the PCFs.
• sub_cmf(basis): Extract a lower-dimensional sub-CMF by choosing a linearly independent set of trajectory directions.
• dual(): Compute the dual CMF using inverse-transpose matrices.
• FFbar(f, fbar): Construct a CMF from $f, \bar{f}$ polynomials (a common parametric family).

Known CMFs (in ramanujantools.cmf.known_cmfs):

Holonomic / D-Finite Functions and Sequences

A function $f(x)$ is D-finite (or holonomic) if it satisfies a linear ODE with polynomial coefficients: $$p_r(x) f^{(r)}(x) + \cdots + p_1(x) f'(x) + p_0(x) f(x) = 0$$ A sequence ${a_n}$ is P-recursive (the discrete analog) if it satisfies a linear recurrence with polynomial coefficients: $$p_r(n) a_{n+r} + \cdots + p_1(n) a_{n+1} + p_0(n) a_n = 0$$

Key properties:

• D-finite functions are closed under addition, multiplication, Hadamard product, composition with algebraic functions, and integral/derivative operations.
• The generating function of a P-recursive sequence is D-finite.
• Many classical constants arise as limits of ratios of terms of P-recursive sequences.
• The ramanujantools library provides a LinearRecurrence class for working with these.

Hypergeometric Functions

Generalized hypergeometric functions ${}_pF_q(a_1,\ldots,a_p; b_1,\ldots,b_q; z)$ are a major source of continued fractions and CMFs:

• ${}_2F_1$ (Gauss): Connected to many classical continued fractions and identities for $\pi$, $\log 2$, etc.
• ${}_3F_2$ and higher: Source of higher-dimensional CMFs.
• The ramanujantools library has pFq and MeijerG classes for constructing CMFs from these functions.
• Contiguous relations between hypergeometric functions correspond to CMF conservation conditions.

Meijer G-Functions

Meijer G-functions $G^{m,n}_{p,q}$ generalize hypergeometric functions. They arise naturally in:

• Inverse Mellin transforms
• Solutions to certain differential equations
• The ramanujantools.cmf.meijer_g module provides a MeijerG class.

Toolboxes & Theories

1. Conservative Matrix Fields (CMFs): The core unifying structure. Use the ramanujantools library's CMF classes for construction, validation, trajectory extraction, and limit computation.
2. Ore Algebra: Operator algebra technique to identify and generate the recurrences satisfied by holonomic sequences. The shift operator $S_n f(n) = f(n+1)$ combined with polynomial coefficients generates closed-form recurrences.
3. Gröbner Bases: Solve multivariate polynomial systems arising from CMF constraint equations across the coordinate grid.
4. Rational Reconstruction: Given $R \pmod{M}$, recover exact $p/q$ via the Extended Euclidean Algorithm, terminating when remainders fall below $\sqrt{M/2}$.
5. Asymptotics: The ramanujantools.asymptotics module provides tools for analyzing growth rates and convergence behavior of sequences.
6. Coboundary Equivalences and Folds: Two PCFs are equivalent if connected by a coboundary transformation or a "fold" (index change $n \to kn + c$). The euler2ai repo implements an algorithm to discover these equivalences automatically.
7. Irrationality Proofs via CMFs: If a PCF derived from a CMF converges to a constant $L$ with $\delta > 0$, this proves the irrationality of $L$. The irrationality measure $\delta$ is computed via the delta() method.
8. RISC Tools (JKU Linz): The Guess package finds recurrences from sequences. The HolonomicFunctions package works with D-finite functions. Ask the team for Mathematica access if needed.

Symbolic Computation Guidelines

• Use SymPy for polynomial manipulation, series expansion, solving recurrences, and Gröbner basis computation.
• Use ramanujantools for all CMF/PCF/recurrence work. It wraps SymPy with domain-specific operations.
• Use Mathematica/RISC for tasks SymPy struggles with: hypergeometric simplification, guessing recurrences from data, proving identities via Zeilberger's algorithm.
• Always verify symbolic results numerically. Compute to 100+ decimal places using mpmath and compare.
• Tell the team when you need access to commercial tools or specialized packages.

Convergence Analysis

• For PCFs, estimate digit growth analytically: $\approx 2.6N$ digits for $e-1$, $\approx 3.0N$ digits for Apéry.
• The convergence rate determines how many terms are needed for a target precision.
• Always verify: compute the ratio at depth $N$ and compare against a reference constant to the expected number of digits.
• A "beautiful" conjecture has convergence rate > 0 and matches to 100+ digits at depth 1000.
• Use the $\delta$ (irrationality measure) to classify formulas. Formulas with the same $\delta$ are candidates for coboundary equivalence.