1. Introduction

The generating function of Bernoulli polynomial is defined as:

2. Identities

**2.1. Theorem of complement**

Proof

therefore,

(1)

**2.3. Theorem of multiplication**

Proof

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)

(3)

from (1) and (3) , therefore

equating the similar coefficients

**2.6. Theorem of sum of**

*n*th 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)**

(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)

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