A new approach to Bernoulli polynomials. (English) Zbl 0663.10009
Author’s abstract: “Beginning with Jacob Bernoulli’s discovery before 1705 of the polynomials that bear his name, there have been five approaches to the theory of Bernoulli polynomials. These can be associated with the names of Bernoulli, Euler (1738), Lucas (1891), Appell (1882), and Hurwitz (personal communication via George Pólya). Each mathematician chose to define the Bernoulli polynomials in a different way, and from his definition derived as theorems one or more of the four other definitions. The present article introduces a sixth definition from which the other five are derived.
Reviewer: L.Skula
MSC:
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
| 05A99 | Enumerative combinatorics |
Online Encyclopedia of Integer Sequences:
Triangle giving coefficients of (n+1)!*B_n(x), where B_n(x) is a Bernoulli polynomial, ordered by falling powers of x.Triangle T(n,k) giving numerator of coefficient of x^(n-k) in Bernoulli polynomial B(n, x), n >= 0, 0<=k<=n.
Triangle T(n,k) giving denominator of coefficient of x^(n-k) in Bernoulli polynomial B(n, x), n >= 0, 0 <= k <= n.
Numerators of coefficients of Bernoulli polynomials with rising powers of the variable.
Triangle of denominators of the coefficient of x^m in the n-th Bernoulli polynomial, 0 <= m <= n.
