Talk:Binomial transform

message: Hi Linas. Good luck with you edit. Note that the term self-conjugate is unknown to WP. The proper term is involution. Bo Jacoby 23:01, 28 January 2006 (UTC)
replies: text: Thanks. If you are good at terminology, what is the name for a matrix/operator B when it has the property that MB=M. I keep wanting to say its "in the kernel of M", or "M kills B" or "its nilpotent", or "B a homomorphism of the kernel of M" but none of these, except for the last, are the right term. Any idea? linas 00:36, 31 January 2006 (UTC)

text: Oh, in this example, M is the Hankel transform, and B is the binomial transform. Think of these two transforms not as transforms, but as inf-dimensional matrices. Applying the binomial transform and more generally the k-binomial transform to a series leaves the Hankel transform of that series invariant.  Clearly, M has a kernel, and B twists the kernel around. In the finest tradition of invariants in mathematics, whenever one has a kernel, one should mod by it, and so I am also wondering how to best describe the cosets of M modulo B, and/or make other interesting conclusions about it.  linas 05:57, 31 January 2006 (UTC)
thread_title: Involution

message: If an is a polynomial in n, then sn is finally zero. So you can extend a vector (like 1,4,9) by transforming into (1,-3,1), extending with zeroes: (1,-3,1,0,0,0), and transforming again: (1,4,9,16,25,36). This is polynomial extrapolation. 
 Transform
  1   4   9
 -3  -5
  2
 Extend and transform
  1  -3   2   0   0   0
  4  -5   2   0   0
  9  -7   2   0
 16  -9   2
 25 -11
 36
Bo Jacoby 16:58, 30 January 2006 (UTC)
replies: text: Huh. Never heard of polynomial extrapolation. linas 00:36, 31 
January 2006 (UTC)
Then see Extrapolation. Bo Jacoby 04:35, 31 January 2006 (UTC)

text: Oh. Duhh. OK, you just mean "use finite differences to fit a polynomial to the data"; yes, that is a cute trick. I didn't recognize it in the form you presented it in above; I'm reminded again that Newton must've been very smart, because the things he did would leave me scratching my head.  I'll add a brief note to article on extrapolation. linas 05:15, 31 January 2006 (UTC)

text: Done. The article on finite differences should also be modified to bring in some of the theory from Newton series, which should be expanded. In fact, I just added the section on Newton series a few days ago, its pretty thin. linas 05:26, 31 January 2006 (UTC)
thread_title: Polynomial extrapolation

message: Is there some extra condition needed for the pair of sequences a_n, b_n to be unique inverses of each other? Because it seems that this sequence is not unique.
Let a_n = n, then both b_k=-\delta_{1k} and b_k=n2^{n-k} satisfy the inverse relation. But I think only the former is the correct forward transform of a_k. —The preceding unsigned comment was added by 154.20.149.5 (talk) 09:09, 8 January 2007 (UTC)
thread_title:  Inverse sequence not unique? 

message: This article are hard de decipher for the beginner because two variables without anything in between have several meanings: The formula
:(Ta)_n = \sum_{k=0}^\infty T_{nk} a_k
might be written
:(Ta)_n = \sum_{k=0}^\infty T_{n,k}\cdot a_k
to explain where it means multiplication as in 
:\ T_{nk} a_k
and where it means separator as in 
:\ _{nk}
and where it means function value as in 
:\ Ta.
Bo Jacoby 20:16, 25 July 2007 (UTC)
thread_title:  Explicite multiplication signs help 

message: The section should be self-contained. There is no explanation what {\Delta^n a_0} is.--84.189.98.228 (talk) 13:36, 28 October 2009 (UTC)
thread_title:  Euler transform not clear 

message: Isn't the example wrong? I certainly don not get the same answer if I use either of the definitions given in the text.  —Preceding unsigned comment added by MathHisSci (talk • contribs) 20:55, 15 September 2010 (UTC)
replies: text: Nothing wrong with the example, the problem was with me as could be expected. MathHisSci (talk) 10:03, 8 November 2010 (UTC)
thread_title:  Example wrong? 

message: I see this page hasn't been edited for a while. The main formula appears to contain a typo. (-1) should be to the power n-k. A clue perhaps is in the fact that the inverse has the same formula, although that appears to be correct. See for instance Wolfram. SamCardioNgo (talk)
:s_n = \sum_{k=0}^n (-1)^{n-k} {n\choose k} a_k.
Just not sure if I'm missing something SamCardioNgo (talk) (date supposedly 22:22, 8 October 2015 (UTC)
thread_title: Main Formula