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Peters Polynomials

The polynomial sequence whose exponential generating function is (1+t)x/(1+(1+t)λ)^μ.

Version
v1 · 2026-09-28 · History
Domain-specific #
11282
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Special Functions, Umbral Calculus → Mathematics
Aliases
Peters Polynomial

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.

Structural Signature

Sig role-phrases:

  • Formal variable t — Indexes the generating series and coefficient extraction. It is series variable. Counterfactual: Treating t as the polynomial variable x confuses two roles.
  • Polynomial variable x — Appears in the generalized binomial factor (1+t)^x. It is polynomial argument. Counterfactual: Fixing x removes the sequence's polynomial character.
  • Parameter λ — Controls the inner power (1+t)^λ in the denominator. It is family parameter. Counterfactual: Changing λ defines a different member family.
  • Parameter μ — Controls the denominator exponent. It is family parameter. Counterfactual: Omitting μ loses part of the defining family.
  • Exponential normalization — Associates s_n(x) with t^n/n!. It is coefficient rule. Counterfactual: Ordinary generating-function coefficients use another normalization.
  • Coefficient extraction — Uniquely recovers each polynomial s_n(x). It is defining operation. Counterfactual: A few matching values do not establish the whole sequence.

What It Is Not

  • 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.
  • Closest near-miss. Boole polynomials are the closest family neighbor because Peters polynomials generalize them through the parameterized denominator.

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.

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

  1. Fix λ and μ and declare whether the series is formal or analytic.
  2. Expand (1+t)^x in the chosen coefficient ring.
  3. Expand and invert the parameterized denominator consistently.
  4. Multiply the series while retaining exponential normalization.
  5. Extract the t^n coefficient and multiply by n! to obtain s_n(x).
  6. 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.

Examples

Applied / In Practice

Expanding the defining quotient as a formal series and multiplying the coefficient of t^n by n! yields s_n(x).

Mapped back: input → defining quotient; normalization → t^n/n!; output → s_n(x).

Applied / In Practice

Choosing parameter values that reduce the denominator to the Boole-polynomial generating form recovers that narrower sequence.

Mapped back: general → Peters family; special case → Boole family.

Structural Tensions

T1 — Compact Definition versus Coefficient Complexity. One generating function defines the full sequence, while explicit coefficients can require nontrivial expansion or umbral identities.

Diagnostic: Is a claimed identity derived from the exact normalization?

T2 — General Family versus Specialized Sequence. Parameters unify related polynomials but special-case properties need not hold throughout the family.

Diagnostic: Which λ and μ are fixed?

Structural–Framed Character

The structural content is coefficient extraction from one parameterized exponential generating function. The names Peters and Boole, together with umbral-calculus conventions for λ, μ, x, and n!, supply the specialized frame.

Structural Core vs. Domain Accent

The invariant core is a formal-series definition of a polynomial sequence. Its domain accent lies in the exact quotient (1+t)x/(1+(1+t)λ)^μ, the exponential rather than ordinary normalization, and its relation to Boole polynomials.

This entry presupposes Formal power series.

  • Approved root. The frozen graph retains Peters polynomials without a parent edge.

  • Related — Boole polynomials and Ordinary generating function. A specialized family generalized by Peters polynomials. Uses t^n rather than t^n/n! coefficients.

Relationships to Other Abstractions

Local relationship map for Peters 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.Peters PolynomialsDOMAINDomain-specific abstraction: Formal power series — presupposesFormalpower seriesDOMAIN

Current abstraction Peters Polynomials Domain-specific

Parents (1) — more general patterns this builds on

  • 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

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

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

Not to Be Confused With

  • Boole polynomials. Tell: A specialized family generalized by Peters polynomials.
  • Ordinary generating function. Tell: Uses t^n rather than t^n/n! coefficients.
  • Sheffer sequence. Tell: A broader generating-function framework requiring its own conditions.
  • Petersen polynomial. Tell: An unrelated name resemblance.

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Peters_polynomials (revision 1363887256).
  • Preserved source candidate: http://www.ams.org/journal-getitem?pii=S0002-9904-1956-09972-0
  • Preserved source candidate: http://www.ams.org/journal-getitem?pii=S0002-9904-1956-10046-3
  • Preserved source candidate: https://books.google.com/books?id=eihMuwkh4DsC
  • Preserved source candidate: https://books.google.com/books?id=JpHjkhFLfpgC

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.