Reciprocal series and digamma function relationship

 DI.ID.3: Reciprocal series and digamma function relationship 

Given

(1)

(2)

(3)

(4)

Proof

(5)

(6)

(7)

(8)

(9)

(10)

from (4) , A=B , 

(11)

(12)

from (1)

(13)

(14)

Bell series

Definition of Bell series

BS.ID.1: Exponential representation

Given

 

Proof

 

 (1)

 

 (2)

 

 (3)

 

 (4)

Using binomial expansion

 (5)

Combining (4) and (5), 

 (6)

 

 (7)

 

 (8)

 (9)

Polylogarithm

Definition of Polylogarithm

PL.ID.1: Functional Equation of Polylogarithm

Proof

 (1)

 (2)

summing (1) and (2)

 (3)

 (4)

 (5) 

using the definition of polylogarithm  , the function equation is:

Useful Exponential Identity

UEI.ID.1: Series

 

Proof

(1)

(2)

(3)

(4)

 

 (5)

 

Generating Function

GF.ID.1: Chebyshev polynomial

 

Given:

 (1.a)

(1.b)

(1.c)

Expanding of Chebyshev polynomial as following:

(2.a)

(2.b)

rewriting (1.c) as :

(3)

from (2.b) and (3)

(4.a)

(4.b)

(4.c)

(4.d)

similar to (3) , Expanding of Chebyshev polynomial as following:

(5.a)

 

 (5.b)

From (4.d) and (5.b) , 

 (6.a)

 (6.b)

 rearranging (6.b)