Understanding conditional replacement

Question

I have encountered recently a replacement and I tried to look up documentation but could not quite find anything and hope to get some explanation from experts. The replacement is the following:

Gamma[2*x+c] /. Gamma[t:2*g_ + d_:0 ] -> 1/Sqrt[Pi]*2^(t-1)*Gamma[t/2]*Gamma[(t+1)/2]

I have never seen this way of conditional replacement. This gives correct replacement for any

Gamma[2*x+c].

What I want to know if anyone can explain the piece, what it does:

Gamma[t:2*g_ + d_:0 ]

What I understood is that whenever it sees argument of the type 2*x+c, it replaces and in the replaced output it does the following:

t-> 2*x+c

and +d_:0 is needed to recognise any argument of type

2*x+c

I will be grateful if anyone can explain this type of notations for replacement and what are the scopes and how it will look for multiple variable replacement.

Answer 1

Take your expression and wrap it in FullForm[HoldForm[expr]] and it will spell out the details:

FullForm[HoldForm[
  Gamma[2*x + c] /. 
   Gamma[t : 2*g_ + d_: 0] -> 
    1/Sqrt[Pi]*2^(t - 1)*Gamma[t/2]*Gamma[(t + 1)/2]]]

The construct

t:2*g_

is a shortcut for a named pattern. t represents the pattern object 2*g_ on the right hand side during the replacement.

The construct

d_:0

is a shortcut for Optional. It means that if that part of the pattern is omitted it will use 0 for d during the replacement.