Bernoulli polynomials

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A quick guide to Bernoulli polynomials

- What they are: Bernoulli polynomials B_n(x) are a family of polynomials that tie together Bernoulli numbers and binomial coefficients. They show up in power series, in the Euler–Maclaurin formula, and in the study of special functions like the Riemann and Hurwitz zeta functions.

- Generating function: The Bernoulli polynomials are defined by the generating function
(t e^{x t})/(e^t − 1) = sum_{n=0}^∞ B_n(x) t^n / n!.

- Basic explicit form: You can compute B_n(x) from Bernoulli numbers B_k by
B_n(x) = sum_{k=0}^n binom(n,k) B_{n−k} x^k.

- First few Bernoulli polynomials:
B0(x) = 1
B1(x) = x − 1/2
B2(x) = x^2 − x + 1/6
B3(x) = x^3 − (3/2)x^2 + (1/2)x
B4(x) = x^4 − 2x^3 + x^2 − 1/30

- Key properties:
- B_n(0) = B_n (these are the Bernoulli numbers), and B_n(1) = B_n for n ≠ 1.
- Derivative: B_n′(x) = n B_{n−1}(x). In particular, B_n is an Appell sequence.
- Symmetry: B_n(1 − x) = (−1)^n B_n(x).
- Integrals: ∫_a^x B_n(u) du = (B_{n+1}(x) − B_{n+1}(a))/(n+1). Also ∫_x^{x+1} B_n(u) du = x^n.
- A simple forward-difference relation: Δ B_n(x) = B_n(x+1) − B_n(x) = n x^{n−1}.

- Connection to Hurwitz zeta: Bernoulli polynomials are linked to the Hurwitz zeta function by B_n(x) = −n ζ(1 − n, x). This ties the polynomials to a broader family of zeta-type functions.

- Euler polynomials (related family): Euler polynomials E_n(x) have their own generating function
2 e^{x t}/(e^t + 1) = sum_{n=0}^∞ E_n(x) t^n / n!.
They satisfy similar identities and have their own explicit formulas.

- Addition and scaling (multiplication theorems): There are formulas that express B_n(x + y) as a sum of B_k(x) times powers of y, and similar results for scaling the argument (such as B_n(m x) in terms of B_n(x) at fractional shifts). These are useful for expanding polynomials at different scales.

- Special values and relations:
- B_n(1/2) = (1/2^{n−1} − 1) B_n for n ≥ 0.
- For large n, when scaled properly, Bernoulli polynomials show a close connection to sine and cosine functions.

- How they’re used: They help expand functions into series, simplify sums, and appear in formulas that relate sums to integrals (Euler–Maclaurin formula). They also provide a bridge to zeta functions and other special functions.

- Summary: Bernoulli polynomials are a versatile tool in analysis and number theory, built from a generating function, with simple derivative and integral rules, and a rich set of identities that connect them to numbers, polynomials, and zeta functions.