Padé Table¶
A two-dimensional array whose entry at degree pair (m,n) is the Padé rational approximant with numerator degree at most m and denominator degree at most n matching a given formal power series to the highest prescribed order.
Core Idea¶
The Padé table turns one power series into a landscape of rational approximations. Moving across the two axes changes how much numerator and denominator complexity is available while preserving the same local coefficient-matching principle.
Its value lies in structure among cells: diagonals, staircases, repeated entries, defects, poles, and continued-fraction paths. A visually stable pole or convergence claim still needs the theorem and function class appropriate to that path.
Structural Signature¶
Sig role-phrases:
- Formal power series — Supplies the coefficient sequence being approximated. It is source. Counterfactual: Changing the expansion point or coefficients changes the whole table.
- Numerator degree m — Indexes horizontal or one declared table axis. It is row or column index. Counterfactual: Orientation varies by author and must be stated.
- Denominator degree n — Indexes the complementary table axis. It is row or column index. Counterfactual: A polynomial approximation is the n=0 edge.
- Coefficient-matching condition — Defines the rational approximation through order near m+n. It is defining equation. Counterfactual: Arbitrary rational closeness is not Padé identity.
- Normalization — Removes common scalar ambiguity between numerator and denominator. It is representation rule. Counterfactual: Different normalizations can encode the same rational function.
- Table path — Selects diagonal, row, column, staircase, or continued-fraction sequences. It is analytic use. Counterfactual: Convergence of one path does not imply the entire table converges.
What It Is Not¶
- One Padé approximant is not the whole table.
- Entries are not arbitrary rational fits to sampled data.
- Every cell need not be normal or distinct.
- Local series matching does not guarantee uniform global accuracy.
- Closest near-miss. A Padé approximant is one cell; the Padé table is the degree-indexed family and exposes relations among many cells.
Scope of Application¶
- Complex approximation. Studies rational continuation beyond polynomial truncation.
- Continued fractions. Relates convergents to paths through the table.
- Numerical analysis. Builds approximants and diagnoses spurious poles.
- Special functions and perturbation theory. Extracts information from formal or asymptotic series cautiously.
Clarity¶
State source series and expansion point, table-axis orientation, numerator and denominator bounds, normalization, coefficient-matching order, defective-cell convention, selected path, exact or floating arithmetic, convergence theorem, and pole-zero diagnostics.
Manages Complexity¶
A doubly indexed family exposes how algebraic degree allocation changes analytic behavior. The compact grid contains linear-system solvability, singularity approximation, repeated rational functions, and path-dependent convergence.
Abstract Reasoning¶
- Fix the formal power series, coefficient field, and expansion point.
- For each selected (m,n), solve the Padé matching equations under one normalization.
- Reduce and record defective or repeated cells without losing their degree position.
- Analyze rows, diagonals, or staircase paths using appropriate convergence results.
- Test apparent poles and zeros for numerical stability and cancellation.
Knowledge Transfer¶
Degree-indexed rational approximation transfers to many series problems, but convergence and pole interpretation depend on the analytic class and selected table path. Sample-based rational fitting does not inherit Padé theory automatically.
Examples¶
Canonical¶
For a series f(z), the cell indexed (m,n) contains a normalized P_m(z)/Q_n(z) whose product Q_n f minus P_m has its initial coefficients vanish through the required order.
Mapped back: source → one formal series; indices → m,n; object → rational function; criterion → coefficient matching; normalization → declared.
Applied / In Practice¶
A spreadsheet listing least-squares rational fits at sampled points is not a Padé table because entries are not obtained by local series coefficient matching.
Mapped back: input → sampled values; fit → least squares; formal series condition → absent; verdict → not Padé table.
Structural Tensions¶
T1 — Local Coefficient Agreement versus Global Approximation. Matching many terms at the expansion point can extend analytic information while introducing spurious pole-zero pairs elsewhere.
Diagnostic: Which singularities persist under degree changes?
T2 — Complete Array versus Useful Paths. The whole table organizes possibilities while theory often guarantees convergence only along selected rows, diagonals, or subsequences.
Diagnostic: Which path and theorem support the inference?
Structural–Framed Character¶
Padé Table is structural as a two-degree array of coefficient-matched rational functions and framed by complex approximation theory. One source series binds every cell.
Structural Core vs. Domain Accent¶
The broader pattern is a parameterized model family arranged in a grid. Analysis contributes numerator/denominator degrees, formal series, normalization, defects, poles, and continued fractions.
Instantiates / Related Primes¶
This entry presupposes Approximation.
-
Approved mathematical root. No the broader abstraction entails the complete two-index family of rational coefficient-matching approximants.
-
Related — Padé approximant, Taylor series, continued fraction, and rational interpolation. They are a cell, source representation, linked path, and different fitting problem.
Relationships to Other Abstractions¶
Current abstraction Padé Table Domain-specific
Parents (1) — more general patterns this builds on
-
Padé Table presupposes Approximation Prime
Padé Table presupposes Approximation because every table entry is a rational approximation matching a power series to a prescribed order.Every reviewed Padé Table instance depends on the parent role: every table entry is a rational approximation matching a power series to a prescribed order. Removing that role makes the frozen child identity undefined or changes it into a different abstraction. Approximation can occur without Padé Table, so the relation is dependency rather than subsumption.
Hierarchy path (1) — routes to 1 parentless root
- Padé Table → Approximation → Representation → Abstraction
Neighborhood in Abstraction Space¶
Padé Table sits in a crowded region of the domain-specific corpus (32nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Polynomial Rings & Rational Approximants (15 abstractions)
Nearest neighbors
- Humbert Polynomials — 0.91
- Continued Fraction — 0.91
- Peters Polynomials — 0.90
- Log-Sum Inequality — 0.88
- Grey Relational Analysis — 0.87
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
- Padé approximant. Tell: Is one rational function at a chosen degree pair.
- Taylor table. Tell: Would organize polynomial coefficients rather than rational degree pairs.
- Rational interpolation. Tell: Matches sampled values rather than necessarily a local power series.
- Continued fraction. Tell: Can generate a path of convergents but is not the whole table.
References¶
- Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Pad%C3%A9_table (revision 1343496119).
- Preserved source candidate: https://oeis.org/
- Preserved source candidate: https://archive.org/details/in.ernet.dli.2015.153147
- Preserved source candidate: https://archive.org/details/continuedfractio0000jone
- Preserved source candidate: https://archive.org/details/continuedfractio0000jone/page/185
The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.