Abstract: The familiar powers 1, x, x^2, ... satisfy the relation
d/dx ( x^n ) = n x^(n-1)
and can be thought of as generated by the recursion relation
p_0(x) = 1 and for n = 1,2,3, ... , d/dx ( p_n(x) ) = n p_(n-1)(x) with p_n(0) = 0
The p_n(0) = 0 part is a normalization condition to set the constant of integration when p_(n-1)(x) is integrated to form p­_n(x).

The Bernoulli polynomials are generated in the same way but with a twist to the normalization condition:
B_0(x) = 1 and for n = 1,2,3 ,... , d/dx ( B_n(x) ) = n B_(n-1)(x) where the mean value of B_n(x) on the interval 0 The Bernoulli polynomials (and the Bernoulli numbers B_n, which are just B_n(0)) pop up in many different ways. For example, the familiar sums of powers are actually Bernoulli polynomials:
1 + 2 + ... + n = n(1+n) / 2 = ( B_2(n+1) - B_2 ) / 2
1^2 + 2^2 + ... + n^2 = n(1+n)(1+2n) / 6 = ( B_3(n+1) - B_3 ) / 3
1^3 +2^3 + ... + n^3 = n^2 (1+n)^2 / 4 = ( B_4(n+1) - B_4 ) / 4
etc.

The purpose of this talk is to discuss the polynomials and their properties, explain the point of the normalization twist, and describe some of the ways they appear in mathematics. The talk uses only first year calculus, except for the very end where I will need a bit of Fourier series.