While working on this answer, I miraculously found (and proved) the following reduction formula: $$T_{2n-1}(x)=(-1)^nx+2x\sum_{k=1}^n(-1)^{k+1}T_{2(n-k)}(x)$$ This basically allows us to express an odd-order Chebyshev polynomial as a sort of combination of the even order polynomials below it. My questions:
Is this formula new? I looked at DLMF, functions.wolfram.com, Wolfram mathworld, Wikipedia, etc and didn't find it. Is it worth publication?
Are there other formulas like it? Like, can we express the even order Chebyshev polynomials as a combination of the odd order polynomials below it?
Thanks.