The Chebyshev polynomials of the first kind $T_n(x)$ are known to form a complete orthogonal basis for functions on $[-1,1]$. I was looking for a proof of the completeness part, without any luck, when I finally bumped into this formula on Wolfram's function site: $$ \sum_{n=0}^\infty T_n(x)T_n(y)= \frac{\pi}{2}\sqrt[4]{1-x^2}\sqrt[4]{1-y^2}\delta(x-y),\quad x,y\in(-1,1). $$ Does anybody know how to prove this? Or can you please suggest a book where I can find it?
2026-03-26 18:57:49.1774551469
Completeness relation for Chebyshev polynomials
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