Christoffel–Darboux formula

An identity collapsing a finite weighted sum of products of orthogonal polynomials into a quotient involving only two consecutive polynomial degrees.

Version

v1 · 2026-09-08 · History

Domain-specific #

3680

Origin domain

orthogonal polynomials

Subdomain

kernel identities

Core Idea

The Christoffel–Darboux formula rewrites the degree-n kernel sum of products of orthogonal polynomials as an antisymmetric expression in the nth and next polynomial divided by the point difference. Terms generated by the three-term recurrence cancel telescopically, leaving only a boundary pair; taking coincident points yields the confluent derivative form. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application

Christoffel–Darboux formula belongs to orthogonal polynomials and is useful where the analyst can specify an orthogonal-polynomial sequence, an inner-product measure, squared norms and leading coefficients, two evaluation points, a finite reproducing-kernel sum, and the three-term recurrence, then evaluate polynomial normalization, norm weights and leading-coefficient convention are consistent on both sides of the identity. The scope is broad within that domain but bounded by the need for polynomial normalization, norm weights and leading-coefficient convention are consistent on both sides of the identity. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making polynomial normalization, norm weights and leading-coefficient convention are consistent on both sides of the identity the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Christoffel–Darboux formula can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Christoffel–Darboux formula. Christoffel–Darboux formula compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: an orthogonal-polynomial sequence, an inner-product measure, squared norms and leading coefficients, two evaluation points, a finite reproducing-kernel sum, and the three-term recurrence. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express polynomial normalization, norm weights and leading-coefficient convention are consistent on both sides of the identity independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of orthogonal polynomials because they reuse an orthogonal-polynomial sequence, an inner-product measure, squared norms and leading coefficients, two evaluation points, a finite reproducing-kernel sum, and the three-term recurrence, Terms generated by the three-term recurrence cancel telescopically, leaving only a boundary pair; taking coincident points yields the confluent derivative form., and type the carrier, state every parameter and convention in the definition, test that polynomial normalization, norm weights and leading-coefficient convention are consistent on both sides of the identity, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Christoffel–Darboux formula Domain-specific

Parents (1) — more general patterns this builds on

• Christoffel–Darboux formula is a kind of Compression Prime

  The proposed strict upward parent is prime:compression.

Hierarchy paths (3) — routes to 3 parentless roots

• Christoffel–Darboux formula → Compression → Abstraction

Show alternative paths (2)

Neighborhood in Abstraction Space

Christoffel–Darboux formula sits in a moderately populated region (43^rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Special Polynomial Sequences & Identities (6 abstractions)

Nearest neighbors

• Bochner's theorem (orthogonal polynomials) — 0.90
• Mott polynomials — 0.89
• Bombieri norm — 0.89
• Stable polynomial — 0.89
• Constant-recursive sequence — 0.89

Computed from structural-signature embeddings · 2026-09-08