Mott polynomials

A polynomial sequence defined by an exponential generating function involving the Catalan-series expression (sqrt(1−t²)−1)/t, introduced in connection with electron theory.

Version

v1 · 2026-09-08 · History

Domain-specific #

5677

Origin domain

special functions

Subdomain

polynomial sequences

Core Idea

Mott polynomials s_n(x) are the coefficients in exp[x(sqrt(1−t²)−1)/t]=sum s_n(x)t^n/n factorial. Expanding the Catalan-related exponent and then the exponential determines each polynomial and its recurrence and convolution identities. 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.

The load-bearing residual is not the broad topic of special functions. It is named binomial-type polynomial family arising from a Catalan-composed exponential. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that generating-function normalization and coefficient convention are fixed, including signs and factorial scaling fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test.

Scope of Application

Mott polynomials belongs to special functions and is useful where the analyst can specify polynomial index n and variable x, exponential generating function, formal power-series variable t, square-root expansion, Catalan coefficients and binomial-type identities, then evaluate generating-function normalization and coefficient convention are fixed, including signs and factorial scaling. The scope is broad within that domain but bounded by the need for generating-function normalization and coefficient convention are fixed, including signs and factorial scaling. 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 generating-function normalization and coefficient convention are fixed, including signs and factorial scaling 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 Mott polynomials 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 Mott polynomials. Mott polynomials 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: polynomial index n and variable x, exponential generating function, formal power-series variable t, square-root expansion, Catalan coefficients and binomial-type identities. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express generating-function normalization and coefficient convention are fixed, including signs and factorial scaling independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of special functions because they reuse polynomial index n and variable x, exponential generating function, formal power-series variable t, square-root expansion, Catalan coefficients and binomial-type identities, Expanding the Catalan-related exponent and then the exponential determines each polynomial and its recurrence and convolution identities., and type the carrier, state every parameter and convention in the definition, test that generating-function normalization and coefficient convention are fixed, including signs and factorial scaling, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Mott polynomials Domain-specific

Parents (1) — more general patterns this builds on

• Mott polynomials is a kind of Representation Prime

  The proposed strict upward parent is prime:representation.

Hierarchy path (1) — routes to 1 parentless root

• Mott polynomials → Representation → Abstraction

Neighborhood in Abstraction Space

Mott polynomials sits in a moderately populated region (44^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Special Polynomial Sequences & Identities (6 abstractions)

Nearest neighbors

• Narumi polynomials — 0.90
• Christoffel–Darboux formula — 0.89
• Bernoulli number — 0.89
• E-function — 0.89
• Lommel polynomial — 0.89

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