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Change of basis using lah numbers

Posted 3 years ago

I am looking to effect the change of basis of a polynomial, q, with respect to the falling power basis to one with respect to the rising factorial power basis, p.

My attempt is to generate a 4 x 4 matrix, unsigned Lah numbers, then take the Inverse of that matrix and subsequently take the Transpose of the Inverse, giving me the matrix:

{{1&-2&6&-24},{0&1&-6&36},{0&0&1&-12},{0&0&0&1}}

Then I multiplied this matrix on the left with the coordinate vector of the polynomial under question, q, namely:

{3, -1, 2, 5}

and ended up with the polynomial, p, now with respect to the rising basis:

p = -57 - 31\alpha - 22\alpha^2 + 5\alpha^3

I am not too confident on the procedure that I have just outlined. The reason being that the Lah numbers are described on Wikipedia to be coefficients expressing rising factorial in terms of falling factorials.

Any thoughts as to whether this is the right approach?

or

Any suggestions or links to some documentation on this subject would be helpful.

Edit:

The matrix listed above started at the second column, second row indices and I think it should have started at the first column, first row indices, so the matrix changes to:

{{1&0&0&0},{0&1&-2&6},{0&0&1&-6},{0&0&0&1}}

and all else remains the same except now we have the polynomial p:

= 3 - 5\alpha - 22\alpha^2 + 5\alpha^3

I think that does it.

POSTED BY: Thomas Vermaak
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