# Which Hadamard Products of Generating Functions Are Known?

The Hadamard product, Schur product, or entrywise product of two generating functions is computed as follows:

The Hadamard Product, H(x), given two generating functions f(x) and g(x) where

$$f(x) = c_0 + c_1x + c_2x^2 + c_3x^3 + \dots + c_nx^n + \dots$$ $$g(x) = d_0 + d_1x + d_2x^2 + d_3x^3 + \dots + d_nx^n + \dots$$

is defined as

$$H(x) = c_0d_0 + c_1d_1x + c_2d_2x^2 + c_3d_3x^3 + \dots + c_nd_nx^n + \dots$$

Simply put, it is the result of multiplying individual coefficients of two generating functions.

Results detailing certain types of functions are known. I would like to compile a list of results that are known.

For example, if we have two generating functions of the form $(1-x)^a$ and $(1-x)^b$, we obtain: ${}\_2F_1[-a,-b;1;x]$ where ${}\_2F_1$ represents a hypergeometric function of Gauss, according to

"Singularity Analysis, Hadamard Products, and Tree Recurrences", by Jim Fill, Philippe Flajolet, and Nevin Kapur In Journal of Computational and Applied Mathematics, volume 174 (February 2005), pages 271--313 (around page 289)

Again, which Hadamard products involving generating functions are known or solved? Additionally, solutions to the equations would be greatly appreciated.

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I'm not sure what you mean by "known." There are lots of closure results about classes of generating functions closed under Hadamard product - for example, rational and holonomic (D-finite) generating functions have this property. Is this the kind of result you're looking for? – Qiaochu Yuan Feb 18 '10 at 7:07
A note to anyone who wants to write an answer: based on my correspondence with the OP, what he is really interested in is ways to compute the Hadamard product mod 2. – Qiaochu Yuan Feb 19 '10 at 2:56
Not only is the holonomic class 'known', it is algorithmic with polynomial-time algorithms (essentially all linear algebra). They all have, for example, Maple implementations. So, if you can write down 2 generating functions, their Hadamard product can be computed, at least to the point of giving an explicit linear recurrence for the coefficients, which can be (very efficiently) unwound. That is as 'solved' as one could ask for, no? – Jacques Carette Feb 19 '10 at 4:22
Again from my correspondence with the OP, he is actually only interested in the Hadamard products of (large) polynomials with 0-1 coefficients. – Qiaochu Yuan Feb 19 '10 at 17:13
Explicit formulas for Hadamard products of some rational functions, proved combinatorially, can be found in Jong Hyung Kim, Hadamard Products and Tilings, Journal of Integer Sequences, Vol. 12 (2009), Article 09.7.4, cs.uwaterloo.ca/journals/JIS/VOL12/Kim/kim18.html – Ira Gessel Nov 17 '10 at 19:27