Bernoulli numbers & power sums — flashcards
Every card the map can derive about Bernoulli numbers & power sums — definitions, statements, attribution, and the edges it sits on.
Theorems 5
State: Faulhaber's formula.
The sum of the kth powers of the first n integers is a polynomial in n of degree k+1 whose coefficients are the Bernoulli numbers. The defining system is lower triangular and dense: each Bernoulli number depends on every earlier one.
derived from st/faulhaber by the statement rule
State: Negative power sums have no closed form.
The finite-difference argument that produces a polynomial for positive powers fails for negative ones, because the falling factorials no longer span. The discrete and continuous cases fail for different reasons, which is why the harmonic numbers have an asymptotic expansion but no closed form.
derived from st/no-closed-form-negative by the statement rule
State: Newton's identities.
Power sums and elementary symmetric polynomials determine each other by a triangular system. This is how information moves between a polynomial's roots and its coefficients without any root ever being computed.
derived from st/newton-identities by the statement rule
State: Poisson summation.
The sum of a function over a lattice equals the sum of its Fourier transform over the dual lattice. It is the bridge back from harmonic analysis to the Bernoulli and zeta material, and the reason the functional equation of ζ looks like a symmetry rather than a coincidence.
derived from st/poisson-summation by the statement rule
State: The Gumbel distribution's skewness constant and Apéry's constant are the same identity, seen from moment-generating and log-Gamma sides.
The freezing transition (st/freezing-transition-log-correlated-extremes) produces Gumbel-distributed extreme-value statistics; the Gumbel distribution's skewness constant, 12√6·ζ(3)/π³, was derived here from first principles rather than looked up, by identifying the Gumbel distribution's moment generating function as literally Γ(1−t) — so its cumulant generating function is the same log-Gamma Taylor series whose coefficients are zeta values at integers. This ties Apéry's constant ζ(3)'s appearance in the freezing transition's maximum-value statistics directly to the digamma/zeta coefficient identity already established for the completed zeta function's log-derivative (st/xi-log-derivative-local-global-split).
derived from st/gumbel-apery-freezing-link by the statement rule
Names and dates 4
Faulhaber's formula — who, and when?
Leonhard Euler, 1735.
derived from st/faulhaber by the attribution rule
Negative power sums have no closed form — who, and when?
Leonhard Euler, 1735.
derived from st/no-closed-form-negative by the attribution rule
Newton's identities — who, and when?
Isaac Newton, c. 1666, published 1707.
derived from st/newton-identities by the attribution rule
Poisson summation — who, and when?
Siméon Denis Poisson, 1823.
derived from st/poisson-summation by the attribution rule
Connections 5
Under the Information cost lens, how do Bernoulli numbers & power sums and Peano arithmetic relate? [Information cost]
Every Bernoulli number depends on all the earlier ones: the defining system is lower triangular and dense. This is the worst case, and it is the baseline the sparse recurrences get compared against.
derived from information-cost:bernoulli-numbers→peano by the connection rule
Under the Information cost lens, how do Generating functions and Bernoulli numbers & power sums relate? [Information cost]
The partition recurrence needs only a handful of earlier terms, and the reason is Franklin's involution: almost everything cancels in pairs. Sparse memory is not a lighter problem, it is a heavier theorem — the cancellation has to be proved.
derived from information-cost:generating-functions→bernoulli-numbers by the connection rule
Under the Degeneracy lens, how do Symmetric functions and Bernoulli numbers & power sums relate? [Degeneracy]
Power sums are the symmetric functions you get by fixing a particular basis. Newton's identities convert between the choices, which is what makes the choice visible as a choice.
derived from degeneracy:symmetric-functions→bernoulli-numbers by the connection rule
Under the Correspondence test lens, how do Bernoulli numbers & power sums and Graph theory & combinatorial proof relate? [Correspondence test]
Partial. No direct interpretation the way binomial coefficients count something outright, but one step removed — through the Genocchi and zigzag numbers, both of which do have direct combinatorial meanings. A correspondence that holds once you're willing to go via an intermediate object.
Recorded as conjectural: expected, not proved.
derived from correspondence-test:bernoulli-numbers→graph-theory by the connection rule
Under the Correspondence test lens, how do Fourier analysis and Bernoulli numbers & power sums relate? [Correspondence test]
Held, and closed a loop rather than opening a new one: Poisson summation is the bridge back from Fourier analysis to the Bernoulli/zeta material, and it's the reason the functional equation of ζ looks like a symmetry instead of a coincidence.
derived from correspondence-test:fourier→bernoulli-numbers by the connection rule
Concepts 1
What is Bernoulli numbers & power sums, in one sentence?
The coefficients that turn a sum of kth powers into a polynomial in n. The defining system is lower triangular and dense — every Bernoulli number depends on all the earlier ones — which makes them the standard against which sparser recurrences get measured.
derived from bernoulli-numbers by the concept rule