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I have a series $\{g_n\}$ whose values are hard to compute, but I calculated a generating function for it (I know the square root is unconventional, but it results in a nice exponential function):

$$ \sum_n \frac{x^n}{\sqrt{n!}}g_n = e^{f(x)} $$

Now, I am interested in the inner product of a sequence $h_n$ with $g_n$:

$$ Y = \sum_n g_nh_n $$

However, I don't have a generating function for $h_n$ because it's a sequence of numerical values.

Is there a way of combining the generating function $e^{f(x)}$ with $h_n$ to compute $Y$?

I have found that two generating functions could be combined via a Hadamard product (so I would need something like $F\odot G(1)$, barring the $\sqrt{n!}$ factor), but I'm not sure how that can help.

Ziofil
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  • A shot in the dark would be to guess a holnomic recurrence for both $g_n$ and $h_n$. If both are holonomic, then you can compute a recurrence for the Hadamard product (i.e. elementwise product). This rec. can be used to compute either a closed form in terms of e.g. hypergeometric sequences or to compute a differential equation for the generating function. All this can be done automatically with a CAS (e.g. gfun for Maple, HolonomicFunctions for Mathematica, ore_algebra for sage). However, since $\sqrt{n!}$ is not holonomic your nice closed form for this generating function will prob. not help. – blablablup Aug 14 '20 at 17:05
  • Thanks for the suggestion, but $h_n$ is really just a list of complex numbers (coming from the output of an algorithm), and I need to iterate this inner product many times, each time with a different $h_n$. – Ziofil Aug 14 '20 at 20:07

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My instinct is this: make a function $$ G(u)=g(e^{i u})=\sum_n \frac{e^{i u n}}{\sqrt{n!}}g_n = e^{f(e^{i u})} $$

Then, by definition, ${g_n / \sqrt{n!}}$ is the coefficients of the fourier series representation of $G(u)$. Then, define $$ H(u)=\sum_m h_m \sqrt{m!} e^{i u m}$$

which you can do because $h_n$ is a finite series of numerical values.

Now, by Parseval's theorem we have $$\sum_n \overline{(h_n\sqrt{n!})} (g_n / \sqrt{n!}) = \sum_n \overline{h_n} g_n = \int_{-\pi}^\pi G(u) \overline{H(u)} du$$

So this whole thing is a single integral which might be easier to compute in closed form. You also might be able to expand $H(u)$ in that integral so the answer is a finite sum in the form $$\sum_m h_m\sqrt{m!}\int_{-\pi}^\pi e^{f(e^{i u})} e^{-i u m} du $$

If for some reason each of those integrals has a closed form formula then this might be easier. (I'd expect they do because we know they should be related to $g_n$)

Steve132
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