Bernoulli polynomial

1. Introduction

The generating function of Bernoulli polynomial is defined as:

When x =0, (known as Bernoulli number)


When x =1,

2. Identities

2.1. Theorem of complement

Proof

(1)

(2)

(3)

from (1)  and (3), 

(4)

therefore,

2.2. Theorem of arguments addition

Proof

(1)

from  E.ID.1 

 (2)

the convolution of the two series is (SC.ID.2)

 

(3)

 

from (1)  and (3),   

 

(4) 

therefore, 

2.3. Theorem of multiplication

Proof

 to prove the above identity we use the  following geometric identity (GS.ID.1):

 

therefore,

(1)

 multiplying and dividing by m , therefore 

 

(2)

 

 (3)

 

(4)

 from (1)  and (4),  

 

(5)

 

therefore,

 

2.4. Theorem of hyperbolic-Bernoulli function

 

proof

 

 

 

 

 

 

therefore;

2.5. Theorem of  exponential-Bernoulli function

based on (UI.ID.1)

and based on 

therefore,

 

2.5. Theorem of  Bernoulli polynomial in term of Bernoulli number

 

Proof

 

(1)

(2)

based on (SC.ID.2)

(3)

from (1) and (3) , therefore

 

equating the similar coefficients 

2.6. Theorem of  sum of  nth powers

 

(1)

based on geometric series  of exponential  and  taylor series of exponential function (GS.ID.2) (E.ID.1)

(2)

from (1) and (2)

  (3)

from  the definitions of Bernoulli polynomials and number

 (4)

from (3) and (4)

 

 (5)

therefore,

 (6)

  (7)

 (8)

 (9)

2.6. Theorem of  recurrence 


subtracting (6) from (7) or (8) from (9) (above)