A question about the implementation of Jacobi basis

[fhuang233]

Dear authors,

Thanks for sharing the code of the excellent work.

I notice that the implementation of Jacobi basis is different from the gold form of recurrence relations of Jacobi polynomials, as it introduces extra two parameters l and r. Although the two versions are equivalent when fixing l=-1 and r=1, I wonder what the source of the recurrence formula corresponding to the implementation.

Besides, I aslo notice that in line 148 of the file impl/PolyConv.py, the variable tmp2_2 has a negative before it. If simply considering the a parameter as beta of Jacobi polynomials and the b parameter as alpha, the coef1 (line 134) should have a negative also. I wonder if this is my misunderstanding on the implementation.

Very thanks for your attention and reply!

[Xi-yuanWang]

Hi,

Here, l, r determine the orthogonal interval of bases. For example, ordinary jacobi basis is orthogonal in [-1, 1]. However, if we use $\hat L$ as basis, the interval should be changed to [0, 2]. We derive the formula on our own and use the l=-1, r=1 case only.

line 134. I think so. I will repair it soon.

[fhuang233]

Thanks, I get it. I find my statement was wrong. It should be "If simply considering the a parameter as beta of Jacobi polynomials and the b parameter as alpha, the first term (a-b)/2 of coef1 (line 134) should have a negative also". Thanks your attention much again.