User blog comment:MilkyWay90/Help with understanding Veblen array notation/@comment-30754445-20180811202716/@comment-30754445-20180817080104

Don't have a link, but I can post a definition myself right here:

(Small letters represent ordinals. A capital N represents a positive integer.)

(1) leading zeros can be omitted.

(for example, φ(0,0,0,1,3) = φ(1,3))

(2) φ(a) = ωa

(3) If c∈Lim then φ(a,b, ... ,c,0, ... ,0) = sup (φ(a,b, ... ,x,0, ... ,0) with x<c)

(3a) Definition 3 also works without the trailing zeros:

If c∈Lim then φ(a,b, ... ,c) = sup (φ(a,b, ... ,x) with x<c)

(4) φ(a,b, ... ,c+1,0,0, ... ,0) = sup (xN) where:

x1 = 0

xN+1 = φ(a,b, ... ,c,xN,0, ..., 0)

(4a) Definition 4 also works with a single trailing zero:

φ(a,b, ... ,c+1,0) = sup (xN) where:

x1 = 0

xN+1 = φ(a,b, ... ,c,xN)

(5) φ(a,b, ... ,c+1,d+1) = sup (xN) where:

x1 = φ(a,b, ... ,c+1,d)

xN+1 = φ(a,b, ... ,c,xN)

(6) If c∈Lim then φ(a,b, ... ,c,d+1) = sup ( φ(a,b, ... ,x,y) with x<c and y≤φ(a,b, ... ,c,d) )

(7) φ(a,b+1,0, ... ,0,d+1) = sup (xN) where:

x1 = φ(a,b+1, ... ,0,d)

xN+1 = φ(a,b,xN,0, ... ,0)

(8) If b∈Lim then φ(a,b,0 ... 0,c+1) = sup ( φ(a,x,y,0, ... ,0) with x<b and y≤φ(a,b,0 ... ,0,c) )

And that's it, I think. Looks like I've covered all the cases.