Skip to content

Secondary Polynomials

Secondary Polynomials is a recurring orthogonal polynomials, analysis identity in which a difference quotient of an orthogonal polynomial is integrated against its density to generate an associated polynomial sequence.

Version
v1 · 2026-09-28 · History
Domain-specific #
7754
Origin domain
Orthogonal Polynomials

Core Idea

Given polynomials pₙ orthogonal with respect to a density ρ, their secondary polynomials qₙ are obtained by integrating the divided difference qₙ(x) = ∫ℝ [(pₙ(t) − pₙ(x))/(t − x)] ρ(t) dt. The construction compares the value of pₙ at the integration variable t with its value at x, divides by their separation, and averages the result against the same density that defines the original orthogonality. Because pₙ(t) − pₙ(x) vanishes at t = x, it is divisible by t − x.

Scope of Application

Secondary polynomials apply wherever a polynomial sequence is orthogonal with respect to a declared density and the density-weighted divided-difference integral exists. The construction is literal only when the same density governs orthogonality and integration, the numerator is p_n(t) − p_n(x), and the required moments converge; another companion sequence or transform is a different object.

  • General orthogonal-polynomial families — each admissible sequence {p_n} and its governing density ρ generate an associated sequence {q_n} through the fixed integral construction.
  • Divided-difference calculations — polynomial divisibility removes the apparent singularity at t = x and exposes the coefficients to be integrated.
  • Moment-functional analysis — powers of the integration variable are replaced by moments of ρ, making convergence of the needed moments an explicit existence condition.
  • Degree and polynomiality proofs — algebraic cancellation and finite moment integration establish that the output is a polynomial in x rather than a singular integral expression.

Clarity

Secondary polynomials are not an arbitrary companion sequence to an orthogonal family. The name identifies a particular divided-difference integral using both the original polynomial pₙ and the same density ρ that supplies its orthogonality. Keeping ρ visible explains why identical displayed input polynomials can yield different secondary sequences when their underlying measures differ. The quotient’s apparent singularity at t = x is also misleading: polynomial divisibility removes it before integration.

Manages Complexity

An orthogonal-polynomial family carries degree-by-degree coefficients, a measure or density, and an expanding collection of moments. The secondary-polynomial construction packages their interaction into one operator: take the divided difference of pₙ at t and x, then integrate the t-dependence against ρ. For each n, the apparent two-variable quotient collapses to a polynomial in x whose coefficients are a finite set of moments of the same density.

Abstract Reasoning

Secondary-polynomial reasoning begins by removing the apparent singularity algebraically. Because pₙ(t) − pₙ(x) vanishes at t = x, division by t − x gives a polynomial divided difference. Expanding that quotient and integrating each t-coefficient against ρ yields input polynomial + density moments → polynomial qₙ(x). For p(x) = x³, for example, t² + tx + x² leads directly to coefficients given by the second, first, and zeroth moments.

Knowledge Transfer

Within orthogonal-polynomial analysis, secondary polynomials transfer literally across admissible polynomial families and densities by applying the same density-weighted divided-difference transform. The cargo that carries intact is the orthogonal sequence, its measure or density, the quotient formed from values at t and x, the removable diagonal singularity, and the moments required to integrate the coefficients. Diagnostics transfer by checking divisibility, moment existence, degree, and sensitivity to changing the measure while holding the displayed polynomial fixed.

Relationships to Other Abstractions

Local relationship map for Secondary PolynomialsParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Secondary PolynomialsDOMAINPrime abstraction: Pattern — is a kind ofPatternPRIME

Current abstraction Secondary Polynomials Domain-specific

Parents (1) — more general patterns this builds on

  • Secondary Polynomials is a kind of Pattern Prime

    The typed carrier is an orthogonal-polynomial sequence together with its governing density, and the granularity is the indexed member n evaluated through variables t and x.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Secondary Polynomials sits in a sparse region of the domain-specific corpus (88th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Multivariate & Spectral Signal Analysis (10 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08