butterworth maximal flat odd polynomial

Hello , I am trying to get this exression, in the book its called "maximal flat" odd polynomial i think its a butterworth polynomial where we put to zero the even powers, we have a butter command. how can we reproduce the experession bellow of P_3(x) with the butter or any other command ?

Thanks

[Walter Roberson]

This appears to be a continuation of https://www.mathworks.com/matlabcentral/answers/342428-binomial-odd-mode-polynomial-coefficients

[John D'Errico]

Why not just add this information to your last question?

[fima v]

yes i thought the use of butter command would get the result because the there is a difference in power with what i was suggested. we got the same coefficients but different powers. the purpose is to get coupling tables on equal ripple like shown in the attached article. if you could help me with that

[John D'Errico]

Anyway, what are you trying to do here? Are you asking how to evaluate a polynomial of that form?

Note that P_3 is NOT actually a Legendre polynomial, which would have been my first guess. So there is insufficient information provided to be able to know how to generate that polynomial.

I would also point out that this is not apparently a third order Butterworth polynomial, as far as I can find.

[fima v]

i would like to have "maximal flat" polinomials

[Walter Roberson]

The paper goes through all the mathematics of how to select the polynomials, it appears to me.

[John D'Errico]

Yes, I imagine the paper shows what to do. So the OP should read the paper supplied, and make an effort to write some code, since it is still not at all clear what they want to do.

Is fima looking for code to produce an nth order polynomial of this form as a symbolic function? As a function handle that one can evaluate?

And what makes this a question about MATLAB, as opposed to a doit4me? Here is the paper, I've made no attempt to do anything, but please supply code for me so that I need not bother to learn enough about MATLAB to do it myself?

syms x
L = 1 + (1.5*x - x^3/2)^2;