Chebyshev Polynomials - Explicit Expressions

Explicit Expressions

Different approaches to defining Chebyshev polynomials lead to different explicit expressions such as:

T_n(x) = \begin{cases} \cos(n\arccos(x)), & \ x \in \\ \cosh(n \, \mathrm{arccosh}(x)), & \ x \ge 1 \\ (-1)^n \cosh(n \, \mathrm{arccosh}(-x)), & \ x \le -1 \\ \end{cases} \,\!


\begin{align} T_n(x) & = \frac{(x-\sqrt{x^2-1})^n+(x+\sqrt{x^2-1})^n}{2} \\ & = \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{2k} (x^2-1)^k x^{n-2k} \\ & = x^n \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{2k} (1 - x^{-2})^k \\ & = \frac{n}{2}\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k \frac{(n-k-1)!}{k!(n-2k)!}~(2x)^{n-2k} \quad (n>0) \\ & = n \sum_{k=0}^{n}(-2)^{k} \frac{(n+k-1)!} {(n-k)!(2k)!}(1 - x)^k \quad (n>0)\\ & = \, _2F_1\left(-n,n;\frac 1 2; \frac{1-x} 2 \right) \\ \end{align}


\begin{align} U_n(x) & = \frac{(x+\sqrt{x^2-1})^{n+1} - (x-\sqrt{x^2-1})^{n+1}}{2\sqrt{x^2-1}} \\ & = \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n+1}{2k+1} (x^2-1)^k x^{n-2k} \\ & = x^n \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n+1}{2k+1} (1 - x^{-2})^k \\ & =\sum_{k=0}^{\lfloor n/2\rfloor} \binom{2k-(n+1)}{k}~(2x)^{n-2k} \quad (n>0)\\ & =\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k \binom{n-k}{k}~(2x)^{n-2k} \quad (n>0)\\ & = \sum_{k=0}^{n}(-2)^{k} \frac{(n+k+1)!} {(n-k)!(2k+1)!}(1 - x)^k \quad (n>0)\\ & = (n+1) \, _2F_1\left(-n,n+2; \tfrac{3}{2}; \tfrac{1}{2}\left \right) \end{align}

where is a hypergeometric function.

Read more about this topic:  Chebyshev Polynomials

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