Special Polynomial Sequences & Identities

← Back to Domain-Specific Families

Abstractions about Bernoulli numbers, named polynomial families, special analytic functions, and recurrence or reproducing identities.

6 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Bernoulli number — Define a rational-number sequence by the exponential generating function for x divided by exp(x) minus one, yielding coefficients that govern power sums, Euler-Maclaurin correction, and zeta-value identities.
• Christoffel–Darboux formula — An identity collapsing a finite weighted sum of products of orthogonal polynomials into a quotient involving only two consecutive polynomial degrees.
• E-function — A Siegel E-function is an entire exponential-generating series with algebraic coefficients of controlled conjugate size and denominator growth that also satisfies a linear differential equation over the polynomials.
• Faber polynomials — Polynomials canonically associated with a normalized Laurent series or conformal map, defined by canceling the principal part of its powers.
• 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.
• Narumi polynomials — A parameterized Sheffer polynomial sequence defined by the exponential generating function (t/log(1+t))^a(1+t)x.