Poly-Bernoulli number¶
A doubly indexed integer sequence defined by an exponential generating function involving the polylogarithm and generalizing Bernoulli numbers.
Core Idea¶
Kaneko's poly-Bernoulli numbers B_n^(k) have positive and negative index regimes, dualities and combinatorial interpretations such as lonesum matrices; multiple generalized parameter conventions exist. Expanding the polylogarithmic generating function around zero determines coefficients, and algebraic identities translate those coefficients into finite sums using Stirling numbers and related counts. 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 enumerative combinatorics and number theory. It is the domain-specific identity determined by the integer indices and sign convention, exact exponential generating function and polylogarithm branch, coefficient normalization and any generalized parameters are explicit.
Scope of Application¶
Poly-Bernoulli number belongs to enumerative combinatorics and number theory and is useful where the analyst can specify the typed enumerative combinatorics and number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the integer indices and sign convention, exact exponential generating function and polylogarithm branch, coefficient normalization and any generalized parameters are explicit. The scope is broad within that domain but bounded by the need for the integer indices and sign convention, exact exponential generating function and polylogarithm branch, coefficient normalization and any generalized parameters 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 indices and sign convention, exact exponential generating function and polylogarithm branch, coefficient normalization and any generalized parameters 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. A bare label is insufficient because the name Poly-Bernoulli number 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 Poly-Bernoulli number. Poly-Bernoulli number 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¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed enumerative combinatorics and number theory carrier, defining objects and 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 indices and sign convention, exact exponential generating function and polylogarithm branch, coefficient normalization and any generalized parameters are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of enumerative combinatorics and number theory because they reuse the typed enumerative combinatorics and number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Expanding the polylogarithmic generating function around zero determines coefficients, and algebraic identities translate those coefficients into finite sums using Stirling numbers and related counts., and type the carrier, state every parameter and convention in the definition, test that the integer indices and sign convention, exact exponential generating function and polylogarithm branch, coefficient normalization and any generalized parameters are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Poly-Bernoulli number Domain-specific
Parents (1) — more general patterns this builds on
-
Poly-Bernoulli number is a kind of Function (Mapping) Prime
The proposed strict upward parent is
prime:function_mapping.
Hierarchy path (1) — routes to 1 parentless root
- Poly-Bernoulli number → Function (Mapping)
Neighborhood in Abstraction Space¶
Poly-Bernoulli number sits in a crowded region of the domain-specific corpus (11th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Arithmetic Functions & Number Sequences (16 abstractions)
Nearest neighbors
- Faulhaber's formula — 0.95
- Schröder number — 0.92
- Hyperharmonic number — 0.92
- Piecewise syndetic set — 0.92
- Stirling transform — 0.92
Computed from structural-signature embeddings · 2026-09-08