To achieve recursion in lambda calculus, a fixed-point function is required.
The fixed-point function is generally referred to as Y, and must by definition satisfy Y`f`=`f`(Y`f`).

The function used for Y is λ`f`.(λ`g`.`f`(`g``g`))(λ`g`.`f`(`g``g`)).
Y`f` can be beta reduced to (λ`g`.`f`(`g``g`))(λ`g`.`f`(`g``g`)), which in turn can be beta reduced to `f`(λ`g`.`f`(`g``g`))(λ`g`.`f`(`g``g`)), satisfying Y`f`=`f`(Y`f`).

Using Y, a function has access to a bound copy of itself.
If lambda expressions were named, you might want to write `f`=λ`x _{1}`...

`x`

_{n}`E`, where

`E`is some expresion refering to

`f`. With the Y combinator, the function on the right becomes Yλ

`f`

`x`...

_{1}`x`

_{n}`E`.