Editing Bernoulli polynomials (section)

==Periodic Bernoulli polynomials==
A '''periodic Bernoulli polynomial''' {{math|''P''<sub>''n''</sub>(''x'')}} is a Bernoulli polynomial evaluated at the [[fractional part]] of the argument {{math|''x''}}.  These functions are used to provide the [[remainder term]] in the [[Euler–Maclaurin formula]] relating sums to integrals.  The first polynomial is a [[Sawtooth wave|sawtooth function]].

Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions, and {{math|''P''<sub>0</sub>(''x'')}} is not even a function, being the derivative of a sawtooth and so a [[Dirac comb]].

The following properties are of interest, valid for all <math> x </math>:
* <math>P_k(x)</math> is continuous for all <math> k > 1 </math>
* <math>P_k'(x)</math> exists and is continuous for <math> k > 2 </math>
* <math>P'_k(x) = k P_{k-1}(x)</math> for <math> k > 2 </math>