§ 7   Bernoulli numbers and Bernoulli polynomials

 

1.        Bernoulli number

 

function expansion

   

In , the coefficients of , are called Bernoulli numbers, and are sometimes expanded as functions

       

To define the relationship between Bernoulli numbers and

                            

For a table of numbers see Chapter V §3 , IX .

                                 

               

                              

 

2.        Bernoulli polynomial

 

function

                 (where is the Bernoulli number)

is called a Bernoulli polynomial, which can also be expanded by the generating function

                           

to define .

       

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The Bernoulli polynomial formula is as follows: