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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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