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)

Binomial theorem

BT.ID.1:   Statement of the  theorem

 

(1)

(2)

if x = 0 , then

(3)

BT.ID.2: Trigonometric Identity

if  n is replaced by 2n in (3) , then: 

(4)

 breaking down the left hand side

(5)

 

 (6)

(7)

So

Raabe's integral

This post aims to show the derivation of Raabe's integral 

 Duplication formula of gamma function is :

replacing t=2t

 

(1)

(2)

(3)

(4)

(5)

(6)

from (5) and (6) ,

 (7)

(8)

from (6) and (8) ,

therefore