Say f(i) is a function of sampling index i and f(i) is fully defined on just 15 index values.

Specifically, f(i) = 27/16(i^3 + 2i^2 + i).
Now, there is some signal s(j) where j is it’s sampling index.
I wanted to form a new signal n(j) by summing the multiplied pairs f(i)s(j+i) for i = -7 to +7.

SUM the above to give a single value of n at some value of index j.

Furthermore, I want to do this across all possible values of j for signal s to form the entire new signal n(j).
Nyquist has some filter functions but none are cubic and I don’t see a way to specify ones own filter function that would do what I want.
How do you set this thing up?

Nyquist can be extended by writing your own functions. Sorry I can’t help with that, I’ve not gone that deeply into Nyquist, but you can find some information about it in Appendix 1 of the Nyquist manual.

Oh boy, Appendix 1 does not look good.
Also, I found this section on DSP in Lisp which seems a bit easier but still more work than I intended and I don’t even know if what I want can even be done.

ASIDE: Strange that Nyquist didn’t put a general filter function that specifies forward and backward polynomials by their coefficients like FILTER(sound, a0, a1, … ,an, b0, b1, … bn).

I was going to suggest posting on the Audacity-Nyquist mailing list, but I see that you already have (and you’ve got hold of the top man for Nyquist - You may have noticed that it’s Roger Dannenberg’s name on the manual).

Also, any suggestion on a good (easy to understand) Lisp tutorial.

Nyquist is based on XLisp. I find the XLisp manual very useful as it has examples of the syntax for functions. XLisp

Note that some things are different when using Nyquist in Audacity to using the standalone version of Nyquist (notably anything to do with “warp”). Unless you specifically require the functionality of Audacity, it may be a lot easier to work with the standalone version. You will then be able to use the up to date version of Nyquist, and will avoid the confusion of where the manual is different from how it works in Audacity.