关于伯努利数存在两种符号约定的原因咨询

Great question—this is one of those historical math quirks that feels frustratingly arbitrary until you dig into the context of how different fields evolved. Let’s break down the key reasons:

• Historical Origins & Generating Function Choices
  The split traces back to how early mathematicians formalized Bernoulli numbers. The "classical" convention (where ( B_1 = -1/2 )) comes from the generating function ( \frac{x}{e^x - 1} = \sum_{n=0}^\infty B_n \frac{x^n}{n!} ), tied directly to Jacob Bernoulli’s original work on sums of powers. Later, some mathematicians (notably in combinatorics and calculus) adopted a modified generating function ( \frac{x}{2} \coth\left(\frac{x}{2}\right) = \sum_{n=0}^\infty B_n^+ \frac{x^n}{n!} ), where ( B_1^+ = 1/2 ). This choice eliminated extra negative signs in formulas like the Euler-Maclaurin summation rule, making intermediate steps cleaner.

• Field-Specific Convenience
  Different disciplines prioritized different simplifications based on their core problems:

  • Number Theory & Analytic Number Theory: The ( B_1 = -1/2 ) convention aligns seamlessly with results involving the Riemann zeta function (e.g., ( \zeta(-n) = -\frac{B_{n+1}}{n+1} ) for positive integers ( n )) and Dirichlet L-functions. It avoids extra sign adjustments when working with congruences or zeta values at negative integers.
  • Combinatorics & Calculus: The ( B_1 = 1/2 ) convention makes formulas for sums of powers or trigonometric Taylor expansions (like tangent/cotangent) more symmetric. For example, the sum ( \sum_{k=1}^m k^p = \frac{1}{p+1} \sum_{k=0}^p \binom{p+1}{k} B_k^+ m^{p+1 -k} ) lacks the awkward negative sign on ( B_1 ) that the classical convention would require.

• Institutional Inertia
  Once a convention took hold in a field—often because a prominent textbook or foundational paper used it—it became self-perpetuating. Number theorists learned the classical convention in coursework and stuck with it; combinatorists did the same with the modified version. There’s no "correct" convention, just ones that fit the needs of the problem at hand.

The confusion isn’t your fault—it’s a product of math’s fragmented historical development across subfields. Always check the generating function or an explicit definition when encountering Bernoulli numbers in a new source; that’s the fastest way to resolve which convention is being used.

内容的提问来源于stack exchange，提问作者Kellen O