Lommel polynomial

A polynomial in the reciprocal argument that expresses shifted-order Bessel functions through a two-term basis of neighboring Bessel orders.

Version

v1 · 2026-09-08 · History

Domain-specific #

5411

Origin domain

special functions

Subdomain

special functions

Core Idea

R m,nu of z is indexed by an integer shift and order parameter, follows the Bessel recurrence and has an explicit finite gamma-coefficient sum; normalization conventions must be declared. Iterating the three-term recurrence for Bessel functions eliminates successive shifted orders, and the accumulated coefficients form reciprocal-argument polynomials multiplying the two initial Bessel functions. 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

Lommel polynomial belongs to special functions and is useful where the analyst can specify the typed special functions carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the integer index and complex order and argument, Bessel-function kind, recurrence and initial polynomials, normalization, explicit finite sum and gamma singularities, polynomial variable and domain restrictions are explicit. The scope is broad within that domain but bounded by the need for the integer index and complex order and argument, Bessel-function kind, recurrence and initial polynomials, normalization, explicit finite sum and gamma singularities, polynomial variable and domain restrictions are explicit. 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 the integer index and complex order and argument, Bessel-function kind, recurrence and initial polynomials, normalization, explicit finite sum and gamma singularities, polynomial variable and domain restrictions are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

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 Lommel polynomial. Lommel polynomial 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: the typed special functions carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the integer index and complex order and argument, Bessel-function kind, recurrence and initial polynomials, normalization, explicit finite sum and gamma singularities, polynomial variable and domain restrictions are explicit independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of special functions because they reuse the typed special functions carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, Iterating the three-term recurrence for Bessel functions eliminates successive shifted orders, and the accumulated coefficients form reciprocal-argument polynomials multiplying the two initial Bessel functions., and type the carrier, state every parameter and convention in the definition, test that the integer index and complex order and argument, Bessel-function kind, recurrence and initial polynomials, normalization, explicit finite sum and gamma singularities, polynomial variable and domain restrictions are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Lommel polynomial Domain-specific

Parents (1) — more general patterns this builds on

• Lommel polynomial is a kind of Recurrence Prime

  The proposed strict upward parent is prime:recurrence.

Hierarchy path (1) — routes to 1 parentless root

• Lommel polynomial → Recurrence

Neighborhood in Abstraction Space

Lommel polynomial sits in a crowded region of the domain-specific corpus (37^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Polynomial Algebra & Field Structure (25 abstractions)

Nearest neighbors

• Pidduck polynomials — 0.91
• Sombrero function — 0.90
• Recurrence relation — 0.89
• K-function — 0.89
• Constant-recursive sequence — 0.89

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