Friday, April 4, 2014

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)

Wednesday, July 24, 2013

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)








Monday, February 11, 2013

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:









Tuesday, January 1, 2013

Saturday, December 1, 2012

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)








Saturday, November 24, 2012

Dirichlet Series of Divisor Function




 DF.ID.1: Dirichlet series of first and second order  divisor function:




 

proof
Euler product representation of Dirichlet series of divisor function:


 (1.a)

 (1.b)
 (2.a)

 (2.b)

according to 
from (1.a) and (2.a):


 
 (3.a)
from (1.b) and (2.b):  

 


  (3.b)
The derivative of geometric series (GS.ID.1):


 
 (4)
 
 (5.a)
 

 
 (5.b)
from ( 3.a) and (4) , the Dirichlet series of first order divisor function:


(6)
from (3.b) and (5.b) , the Dirichlet series of second order divisor function:


 
 (7)

DF.ID.2: Dirichlet series of first  order  divisor function: of squared number


 

proof
 (8)
 
  (8.b)
 (8.c)
given that :
 (9)
The derivative of previous series:
 (10)
 (11)
from (8.c)(10) and (11), 
 (12)
 
 (13)








 DF.ID.3: Dirichlet series of  sum of higher order  divisor function:


Proof

 (1)



 (2)


 (3)



 (4)


 (5)



 (6)


 (7)
replacing (7) in (6) gives:
 (8)


 (9)



 (10)

 (11)


 (12)

 (13)