This is a nice trick to speed up calculations into the high-index realm.

The speed with which a series can be developed can be accellerated by repeated application of the mapping:

• U(a,b)F ⇒ U(a,U₂)[F²-2]

This is a base-2 accelleration (because it uses U2). Let's look at U(1,1)5 - the terius of 5 - for an example:

After the mapping U(1,1)5 ⇒ U(1,4)23 we look at the latter:

1	4	91	2089	47956	1100899	25272721	580171684	13318676011	305749376569	7018916985076
U0	U1	U2	U3	U4	U5	U6	U7	U8	U9	U10

After the mapping U(1,4)23 ⇒ U(1,91)527 we look at the latter:

1	91	47956	25272721	13318676011	7018916985076
U0	U1	U2	U3	U4	U5

After the mapping U(1,91)527 ⇒ U(1,47956)277727 we look at the latter:

1	47956	13318676011
U0	U1	U2

This, then, is the general idea base-2. It zaps down the parent series with steps that are powers of 2: 2, 4, 8, 16, ...
We can do it base-3, but that's just like base-2 really. It employs the mapping:

• U(a,b)F ⇒ U(a,U₃)[F³-3F]

Applied repeatedly, it one zaps down the parent series with steps that are powers of 3: 3, 9, 27, 81, ...

Matrices of the Factor base-n

Base-1 1             Base-2 1 -2           Base-3 1 -3           Base-4 1 -4 2         Base-5 1 -5 5         Base-6 1 -6 9 -2       Base-7 1 -7 14 -7       Base-8 1 -8 20 -16 2     Base-9 1 -9 27 -30 9     Base-10 1 -10 35 -50 25 -2   Base-11 1 -11 44 -77 55 -11   Base-12 1 -12 54 -112 105 -36 2 ...

Of course we need the coefficients matrix of the factor F up to 'base-n' so here is how to go about that: In this matrix every column consists of the subsequent differences of the next column. Using this property the matrix can be extended indefinitely. The base defines the highest power of the factor F. Powers decrease with steps of 2. The general term of the polynom base-n could be constructed, but considering the size of the factors that can be reached by extending the matrix, this would seem a bit premature.

Base-1 1                     Base-2 1 2                   Base-3 1 2 +1                 Base-4 1 2                   Base-5 1 2 -1 -2 1             Base-6 1 2 -2 -4 1 2           Base-7 1 2 -3 -6 2 4 +1         Base-8 1 2 -4 -8 4 8           Base-9 1 2 -5 -10 7 14 -2 -4 1     Base-10 1 2 -6 -12 11 22 -6 -12 1 2   Base-11 1 2 -7 -14 16 32 -13 -26 3 6 +1 Base-12 1 2 -8 -16 22 44 -24 -48 9 18   ...

In a non-zero-series U(a,b,c)F the factor F follows the above matrix just as in a zero-series. The constant c develops as c*(matrix over 1), and that one looks like this: Note that each column consists if the subsequent differences, not so much of the next column, but of the one next to that. Using this property the matrix can be extended indefinitely. The power of the factor F is one less than the base. Powers decrease with steps of 1.

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I hate to work without examples, so let's have a look at U(1,1,2)3 and accelerate it base-5.
Here's the parent series:
The matrix over a and b renders F⁵ - 5F³ + 5F.
The matrix over c renders c*(F⁴ + 2F³ - F² - 2F + 1).
For F=3 and c=2 this adds up to a new factor 123 and a new constant 242 respectively.
It would seem that the mapping we're after is: U(1,1,2)3 ⇒ U(1,100,242)123
After this mapping we look at the latter again:

1	100	12541	1542685	189737956	23336226145
U0	U1	U2	U3	U4	U5

Bingo!
Let's repeat the mapping base-5.
For F=123 and c=242 this adds up to a new factor 28143753123 and a new constant 56287506242 respectively.
It would seem that the mapping we're after is: U(1,100,242)123 ⇒ U(1,23336226145,56287506242)28143753123
After this mapping we look at the latter again:

1	23336226145	656768987503665507076
U0	U1	U2

How about that :)
U₁ and U₂ are of course U₂₅ and U₅₀ of the parent series. Repeat the base-5 mapping and U₁ and U₂ will be U₁₂₅ and U₂₅₀, repeat it again and you'll get U₆₂₅ and U₁₂₅₀ and so on.
An awesome tool!