Monday, August 8, 2011

Central Binomial Coeffecient



(1)
CBD.ID.1:   Recurrence Relation
 
(2)

 
 (3)
 
(4)
where
(5)
CBD.ID.2:  Generating Function
 (6)
Differentiating both sides with respect to x
 
(7)
(8)
using the recurrence relation  (4)
 (9)
 
(10)

(11)
(12)

 
 (13)

integrating both sides with respect to x ,
(14)
(15)
since



therefore,


 
(16)

CBD.ID.3:   Trigonometric Identity

 
proof
replacement  in (16) by :
 
 
therefore

 
 (17)
 
(18)
from (17) and (18) , therefore, 
 
(19)
integrate (19) from 0 to t

(20)
therefore, 
(21)



CBD.ID.4:   Catalan Number  Identity

 

Proof

integrating (16) from 0 to t



(22)


replacing 1-4x by u in (23)
(23)


 


(24)


(25)
divide both sides by t

 (26)


(27)



(28)
CBD.ID.5:   Square  Identity of Central Binomial Coefficient


Proof
squaring both sides of (16)


(29)
applying geometric series (1) identity on the left side , and Cauchy product method  (2) on the right side;


 
(30)

equating the coefficients of    from both sides  , we get

(31)