F0(x,y) = x + y,
Fn+1(x,0) = x, for n ≥ 0,
Fn+1(x,y+1) = Fn( Fn+1(x,y), Fn+1(x,y)+y+1), for n ≥ 0.
F0(x,0) = x
F0(0,y) = y
F0(x,x) = 2x,
e.g F0(4,4) = 2.4 = 8 ü
Properties of F1(x,y)
F1(x,0) = x,
F1(0,y) = (x + 2)(2x - 1),
F1(x,y) = F1(0,y) + x.2y,
e.g. F1(3,4) = F1(0,4) + 3.24 = 26 + 48 = 74 ü
F1(x,y) = (x + 2).2y - (y + 2), (explicit solution)
e.g. F1(8,8) = (x+2).2y - (y+2) = 10.28 - 10 = 2560 - 10 = 2550 ü
Graphical Representation of F1(x,y)
Properties of F2(x,y)
F2(x,0) = x,
F2(x,y+1) = F1( F2(x,y), F2(x,y)+y+1).
History of Fn(x,y)
In the theory of computation, the Sudan function is an example of a function that is recursive, but not primitive recursive. This is also true of the better-known Ackermann function but the Sudan function was the first function having this property to be published. It was discovered in 1927 by Gabriel Sudan, a Romanian mathematician who was a student of David Hilbert.
Applications of Fn(x,y)
Higher order Sudan functions with n ≥ 2 can be used to test the ability of computers to handle recursion efficiently.