Gamma-Catalan Relation

GC.ID.1:   Catalan Number  Identity

Proof

based on recurrence relation of gamma

(1)

 

 (2)

 

  (3)

 

  (4)

 (5)

 (6)

 

  (7)

 

  (8)

 (9)

 (10)

 (11)

  (12)

  (13)

based on the Catalan number  identity , therefore ,

  (14)

  (15)

  (16)

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)