Peters Polynomials¶
The polynomial sequence whose exponential generating function is (1+t)x/(1+(1+t)λ)^μ.
Core Idea¶
Peters polynomials s_n(x) are a parameterized polynomial sequence defined coefficientwise by the exponential generating function (1+t)^x divided by (1+(1+t)λ)μ, generalizing the Boole polynomials.
Expanding the defining quotient as a formal series and multiplying the coefficient of t^n by n! yields s_n(x). Choosing parameter values that reduce the denominator to the Boole-polynomial generating form recovers that narrower sequence.
Scope of Application¶
- Umbral calculus. Manipulates the sequence through generating functions.
- Special polynomials. Places the family among parameterized polynomial sequences.
- Combinatorics. Extracts coefficient identities and recurrences.
- Symbolic computation. Expands the formal generating series.
Clarity¶
Include polynomial sequences obtained from the stated exponential generating function with declared λ and μ and t^n/n! normalization. Exclude arbitrary Peters-named polynomials, ordinary generating functions, Boole polynomials presented without the relevant specialization, and coefficient lists lacking the defining series. Inclusion test: Include polynomial sequences obtained from the stated exponential generating function with declared λ and μ and t^n/n! normalization. Exclusion test: Exclude arbitrary Peters-named polynomials, ordinary generating functions, Boole polynomials presented without the relevant specialization, and coefficient lists lacking the defining series. Nearest boundary: Boole polynomials are the closest family neighbor because Peters polynomials generalize them through the parameterized denominator. Exit condition: The identity changes when the generating function, parameter placement, or exponential coefficient normalization changes. Common misclassifications: It is not any polynomial studied by an author named Peters. It is not an ordinary generating function sequence. It is not identical to Boole polynomials for arbitrary parameters. It is not defined by numerical coefficients without λ, μ, and normalization. Nearest named distinctions: Boole polynomials: A specialized family generalized by Peters polynomials. Ordinary generating function: Uses t^n rather than t^n/n! coefficients. Sheffer sequence: A broader generating-function framework requiring its own conditions. Petersen polynomial: An unrelated name resemblance.
Manages Complexity¶
One generating function defines the full sequence, while explicit coefficients can require nontrivial expansion or umbral identities. Parameters unify related polynomials but special-case properties need not hold throughout the family.
Abstract Reasoning¶
- Fix λ and μ and declare whether the series is formal or analytic.
- Expand (1+t)^x in the chosen coefficient ring.
- Expand and invert the parameterized denominator consistently.
- Multiply the series while retaining exponential normalization.
- Extract the t^n coefficient and multiply by n! to obtain s_n(x).
- Test any Boole specialization by substituting parameters into the original generating function.
Knowledge Transfer¶
Generating-function reasoning transfers coefficient identities, recurrences, and umbral manipulations only with the same normalization and parameters λ and μ; Boole-polynomial specializations cannot be generalized without checking the denominator and parameter values.
Relationships to Other Abstractions¶
Current abstraction Peters Polynomials Domain-specific
Parents (1) — more general patterns this builds on
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Peters Polynomials presupposes Formal power series Domain-specific
Peters Polynomials presupposes Formal power series because the polynomial sequence is defined coefficientwise by one exponential generating series.
Hierarchy path (1) — routes to 1 parentless root
- Peters Polynomials → Formal power series → Representation → Abstraction
Neighborhood in Abstraction Space¶
Peters Polynomials sits in a crowded region of the domain-specific corpus (33rd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Polynomial Rings & Rational Approximants (15 abstractions)
Nearest neighbors
- Humbert Polynomials — 0.98
- Padé Table — 0.90
- Wagstaff Prime — 0.87
- Continued Fraction — 0.87
- Semidirect Product — 0.86
Computed from structural-signature embeddings · 2026-10-08