Working definition
The Bernoulli numbers are defined at DLMF 24.2.1 by the generating function t over the quantity e to the t minus one, expanded as a sum of B sub n times t to the n over n factorial, converging for the modulus of t less than two pi. The polynomials follow at 24.2.3 from the same function multiplied by e to the x t, in the same disk, and 24.2.4 records that B sub n is the polynomial evaluated at zero. The Euler numbers are defined at 24.2.6 by two e to the t over the quantity e to the two t plus one, converging only for the modulus of t less than pi over two. The vanishing rules at 24.2.2 and 24.2.7 say that odd-indexed Bernoulli numbers vanish apart from the first, and that odd-indexed Euler numbers vanish.
Notation
B sub n for the Bernoulli numbers and B sub n of x for the polynomialsE sub n for the Euler numbers and E sub n of x for the polynomialst for the generating variableAssumptions
- The Bernoulli generating function at 24.2.1 converges only for the modulus of t less than two pi, which is the distance to the nearest pole of the generating expression.
- The Euler generating function at 24.2.6 converges only for the modulus of t less than pi over two, a strictly smaller disk than the Bernoulli one.
- The vanishing rule at 24.2.2 exempts the first Bernoulli number, so the odd indices are not uniformly zero.
- The polynomials at 24.2.3 reduce to the numbers only at argument zero, by 24.2.4.
Invariants
- Bernoulli generating function with its radius, DLMF 24.2.1
- Odd-indexed Bernoulli numbers vanish apart from the first, and the even ones alternate in sign, DLMF 24.2.2
- Euler generating function with its smaller radius, DLMF 24.2.6
Reproducible procedure
- Take the numbers from the generating function at 24.2.1 rather than from a recurrence, so the convergence radius stays visible.
- Check the index parity before assuming a coefficient is present: by 24.2.2 the odd Bernoulli terms are absent above the first.
- Where an expansion is indexed by even numbers only, look for a Bernoulli coefficient behind it rather than treating the gap as a convention.
Error and boundary controls
- The generating series say nothing outside their disks. Beyond two pi for Bernoulli, or pi over two for Euler, the definition still holds but the series is not a usable expansion.
- The even-indexed Bernoulli numbers grow rapidly and alternate in sign, so a series carrying them is a candidate for catastrophic cancellation rather than a safe summation.
- The chapter defines the coefficients and makes no claim about the stability of any recurrence used to generate them numerically.
What this does not establish
These are definitions with stated convergence radii. They do not establish that any recurrence for computing the coefficients is numerically stable, and they carry no claim about series that use them.
Explicit applications
0 cross-domain bridges
This foundational concept currently supports related concepts; a direct domain bridge is scheduled for a later registry version.
Authoritative references
- [1]DLMF Chapter 24: Bernoulli and Euler Polynomials · National Institute of Standards and Technology
Establishes: Generating-function definitions for the Bernoulli and Euler numbers and polynomials, each with the radius in which the generating series converges, and the vanishing and sign rules for the odd and even indices.
Boundary: The generating functions converge only inside the stated radii, so they define the coefficients without providing a usable expansion outside those disks, and the chapter makes no claim about the numerical stability of any recurrence used to generate them.
Direct answer
- The Bernoulli numbers are defined at DLMF 24.2.1 by the generating function t over the quantity e to the t minus one, expanded as a sum of B sub n times t to the n over n factorial, converging for the modulus of t less than two pi. The polynomials follow at 24.2.3 from the same function multiplied by e to the x t, in the same disk, and 24.2.4 records that B sub n is the polynomial evaluated at zero. The Euler numbers are defined at 24.2.6 by two e to the t over the quantity e to the two t plus one, converging only for the modulus of t less than pi over two. The vanishing rules at 24.2.2 and 24.2.7 say that odd-indexed Bernoulli numbers vanish apart from the first, and that odd-indexed Euler numbers vanish.
Mechanism and method
- Take the numbers from the generating function at 24.2.1 rather than from a recurrence, so the convergence radius stays visible.
- Check the index parity before assuming a coefficient is present: by 24.2.2 the odd Bernoulli terms are absent above the first.
- Where an expansion is indexed by even numbers only, look for a Bernoulli coefficient behind it rather than treating the gap as a convention.
What is measured
- Bernoulli generating function with its radius, DLMF 24.2.1
- Odd-indexed Bernoulli numbers vanish apart from the first, and the even ones alternate in sign, DLMF 24.2.2
- Euler generating function with its smaller radius, DLMF 24.2.6
Limitations
- The generating series say nothing outside their disks. Beyond two pi for Bernoulli, or pi over two for Euler, the definition still holds but the series is not a usable expansion.
- The even-indexed Bernoulli numbers grow rapidly and alternate in sign, so a series carrying them is a candidate for catastrophic cancellation rather than a safe summation.
- The chapter defines the coefficients and makes no claim about the stability of any recurrence used to generate them numerically.
- The Bernoulli generating function at 24.2.1 converges only for the modulus of t less than two pi, which is the distance to the nearest pole of the generating expression.
- The Euler generating function at 24.2.6 converges only for the modulus of t less than pi over two, a strictly smaller disk than the Bernoulli one.
- The vanishing rule at 24.2.2 exempts the first Bernoulli number, so the odd indices are not uniformly zero.
- The polynomials at 24.2.3 reduce to the numbers only at argument zero, by 24.2.4.
What this does not establish
- These are definitions with stated convergence radii. They do not establish that any recurrence for computing the coefficients is numerically stable, and they carry no claim about series that use them.