チェビシェフ多項式

チェビシェフ多項式の母関数

by nomura · 2020年10月13日

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チェビシェフ多項式の母関数

通常型母関数

(1)

\sum_{k = 0}^{\infty} T_{k} (x) t^{k} = \frac{1 - t x}{1 - 2 t x + t^{2}}

(2)

\sum_{k = 0}^{\infty} U_{k} (x) t^{k} = \frac{1}{1 - 2 t x + t^{2}}

指数型母関数

(3)

\sum_{k = 0}^{\infty} T_{k} (x) \frac{t^{k}}{k!} = e^{t x} \cos (t \sqrt{1 - x^{2}})

(4)

\sum_{k = 0}^{\infty} U_{k} (x) \frac{t^{k}}{k!} = e^{t x} {\cos (t \sqrt{1 - x^{2}}) + \frac{x}{\sqrt{1 - x^{2}}} \sin (t \sqrt{1 - x^{2}})}

(1)

\begin{aligned} \sum_{k = 0}^{\infty} T_{k} (\cos y) t^{k} & = \sum_{k = 0}^{\infty} \cos (k y) t^{k} \\ = ℜ (\sum_{k = 0}^{\infty} {(e^{i y} t)}^{k}) \\ = ℜ (\frac{1}{1 - e^{i y} t}) \\ = \frac{1}{2} (\frac{1}{1 - e^{- i y} t} + \frac{1}{1 - e^{i y} t}) \\ = \frac{1}{2} (\frac{1 - e^{i y} t + 1 - e^{- i y} t}{(1 - e^{i y} t) (1 - e^{- i y} t)}) \\ = \frac{1}{2} (\frac{2 - (e^{i y} + e^{- i y}) t}{1 - (e^{i y} + e^{- i y}) t + t^{2}}) \\ = \frac{1 - t \cos y}{1 - 2 t \cos y + t^{2}} \end{aligned}

これより、

\sum_{k = 0}^{\infty} T_{k} (x) t^{k} = \frac{1 - t x}{1 - 2 t x + t^{2}}

(1)-2

\begin{aligned} \sum_{k = 0}^{\infty} T_{k} (x) t^{k} & = T_{0} (x) + T_{1} (x) t + \sum_{k = 2}^{\infty} T_{k} (x) t^{k} \\ = 1 + x t + \sum_{k = 0}^{\infty} T_{k + 2} (x) t^{k + 2} \\ = 1 + x t + \sum_{k = 0}^{\infty} (2 x T_{k + 1} (x) - T_{k} (x)) t^{k + 2} \\ = 1 + x t + 2 x t \sum_{k = 0}^{\infty} T_{k + 1} (x) t^{k + 1} - t^{2} \sum_{k = 0}^{\infty} T_{k} (x) t^{k} \\ = 1 + x t + 2 x t {\sum_{k = 0}^{\infty} T_{k} (x) t^{k} - T_{0} (x) t^{0}} - t^{2} \sum_{k = 0}^{\infty} T_{k} (x) t^{k} \\ = 1 + x t - 2 x t + (2 x t - t^{2}) LHS \\ = \frac{1 - t x}{1 - 2 t x + t^{2}} \end{aligned}

(2)

\begin{aligned} \sum_{k = 0}^{\infty} U_{k} (\cos y) t^{k} & = \frac{1}{\sin y} \sum_{k = 0}^{\infty} \sin ((k + 1) y) t^{k} \\ = \frac{1}{\sin y} \sum_{k = 0}^{\infty} (\sin (k y) \cos y + \cos (k y) \sin y) t^{k} \\ = \frac{\cos y}{\sin y} \sum_{k = 0}^{\infty} (\sin (k y) t^{k}) + \sum_{k = 0}^{\infty} (\cos (k y) t^{k}) \\ = \frac{\cos y}{\sin y} ℑ (\sum_{k = 0}^{\infty} {(e^{i y} t)}^{k}) + \sum_{k = 0}^{\infty} (T_{k} (\cos y) t^{k}) \\ = \frac{\cos y}{\sin y} ℑ (\frac{1}{1 - e^{i y} t}) + \sum_{k = 0}^{\infty} (T_{k} (\cos y) t^{k}) \\ = \frac{\cos y}{2 i \sin y} (\frac{1}{1 - e^{i y} t} - \frac{1}{1 - e^{- i y} t}) + \sum_{k = 0}^{\infty} (T_{k} (\cos y) t^{k}) \\ = \frac{\cos y}{2 i \sin y} \frac{(1 - e^{- i y} t) - (1 - e^{i y} t)}{(1 - e^{i y} t) (1 - e^{- i y} t)} + \sum_{k = 0}^{\infty} (T_{k} (\cos y) t^{k}) \\ = \frac{\cos y}{2 i \sin y} \frac{(e^{i y} - e^{- i y}) t}{1 - (e^{i y} + e^{- i y}) t + t^{2}} + \sum_{k = 0}^{\infty} (T_{k} (\cos y) t^{k}) \\ = \frac{t \cos y}{1 - 2 t \cos y + t^{2}} + \frac{1 - t \cos y}{1 - 2 t \cos y + t^{2}} \\ = \frac{1}{1 - 2 t \cos y + t^{2}} \end{aligned}

これより、

\sum_{k = 0}^{\infty} U_{k} (x) t^{k} = \frac{1}{1 - 2 t x + t^{2}}

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(2)-2

\begin{aligned} \sum_{k = 0}^{\infty} U_{k} (x) t^{k} & = U_{0} (x) + U_{1} (x) t + \sum_{k = 2}^{\infty} U_{k} (x) t^{k} \\ = 1 + 2 x t + \sum_{k = 0}^{\infty} U_{k + 2} (x) t^{k + 2} \\ = 1 + 2 x t + \sum_{k = 0}^{\infty} (2 x U_{k + 1} (x) - U_{k} (x)) t^{k + 2} \\ = 1 + 2 x t + 2 x t \sum_{k = 0}^{\infty} U_{k + 1} (x) t^{k + 1} - t^{2} \sum_{k = 0}^{\infty} U_{k} (x) t^{k} \\ = 1 + 2 x t + 2 x t {\sum_{k = 0}^{\infty} U_{k} (x) t^{k} - U_{0} (x) t^{0}} - t^{2} \sum_{k = 0}^{\infty} U_{k} (x) t^{k} \\ = 1 + 2 x t - 2 x t + (2 x t - t^{2}) LHS \\ = \frac{1}{1 - 2 t x + t^{2}} \end{aligned}

(3)

\begin{aligned} \sum_{k = 0}^{\infty} T_{k} (x) \frac{t^{k}}{k!} & = \sum_{k = 0}^{\infty} \cos (k \cos^{∙} x) \frac{t^{k}}{k!} \\ = \frac{1}{2} \sum_{k = 0}^{\infty} (e^{i k \cos^{∙} x} + e^{- i k \cos^{∙} x}) \frac{t^{k}}{k!} \\ = \frac{1}{2} \sum_{k = 0}^{\infty} ({(e^{i \cos^{∙} x})}^{k} + {(e^{- i \cos^{∙} x})}^{k}) \frac{t^{k}}{k!} \\ = \frac{1}{2} \sum_{k = 0}^{\infty} ({(x + i \sin \cos^{∙} x)}^{k} + {(x - i \sin \cos^{∙} x)}^{k}) \frac{t^{k}}{k!} \\ = \frac{1}{2} {\sum_{k = 0}^{\infty} {(x + i \sqrt{1 - x^{2}})}^{k} \frac{t^{k}}{k!} + \sum_{k = 0}^{\infty} {(x - i \sqrt{1 - x^{2}})}^{k} \frac{t^{k}}{k!}} \\ = \frac{1}{2} {e^{t (x + i \sqrt{1 - x^{2}})} + e^{t (x - i \sqrt{1 - x^{2}})}} \\ = \frac{e^{t x}}{2} (e^{+ i t \sqrt{1 - x^{2}}} + e^{- i t \sqrt{1 - x^{2}}}) \\ = e^{t x} \cos (t \sqrt{1 - x^{2}}) \end{aligned}

(4)

\begin{aligned} \sum_{k = 0}^{\infty} U_{k} (x) \frac{t^{k}}{k!} & = \sum_{k = 0}^{\infty} \frac{\sin ((k + 1) \cos^{∙} x)}{\sin \cos^{∙} x} \frac{t^{k}}{k!} \\ = \frac{1}{2 i \sqrt{1 - x^{2}}} \sum_{k = 0}^{\infty} (e^{i (k + 1) \cos^{∙} x} - e^{- i (k + 1) \cos^{∙} x}) \frac{t^{k}}{k!} \\ = \frac{1}{2 i \sqrt{1 - x^{2}}} \sum_{k = 0}^{\infty} ({(e^{i \cos^{∙} x})}^{k + 1} - {(e^{- i \cos^{∙} x})}^{k + 1}) \frac{t^{k}}{k!} \\ = \frac{1}{2 i \sqrt{1 - x^{2}}} \sum_{k = 0}^{\infty} ({(x + i \sin \cos^{∙} x)}^{k + 1} - {(x - i \sin \cos^{∙} x)}^{k + 1}) \frac{t^{k}}{k!} \\ = \frac{1}{2 i \sqrt{1 - x^{2}}} {\sum_{k = 0}^{\infty} {(x + i \sqrt{1 - x^{2}})}^{k + 1} \frac{t^{k}}{k!} - \sum_{k = 0}^{\infty} {(x - i \sqrt{1 - x^{2}})}^{k + 1} \frac{t^{k}}{k!}} \\ = \frac{1}{2 i \sqrt{1 - x^{2}}} {(x + i \sqrt{1 - x^{2}}) e^{t (x + i \sqrt{1 - x^{2}})} - (x - i \sqrt{1 - x^{2}}) e^{t (x - i \sqrt{1 - x^{2}})}} \\ = \frac{e^{t x}}{2 i \sqrt{1 - x^{2}}} ((x + i \sqrt{1 - x^{2}}) e^{+ i t \sqrt{1 - x^{2}}} - (x - i \sqrt{1 - x^{2}}) e^{- i t \sqrt{1 - x^{2}}}) \\ = e^{t x} {\cos (t \sqrt{1 - x^{2}}) + \frac{x}{\sqrt{1 - x^{2}}} \sin (t \sqrt{1 - x^{2}})} \end{aligned}

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第2種チェビシェフ多項式の因数分解

U_{2 n - 1} (x) = 2 U_{n - 1} (x) T_{n} (x)

(*)チェビシェフ多項式のロドリゲス公式

T_{n} (x) = \frac{(- 1)^{n} \sqrt{π} \sqrt{1 - x^{2}}}{2^{n} Γ (n + \frac{1}{2})} \frac{d^{n}}{d x^{n}} {(1 - x^{2})}^{n - \frac{1}{2}}

第1種・第2種と第3種チェビシェフ多項式同士の関係

V (- x) = (- 1)^{n} W_{n} (x)

チェビシェフの微分方程式

(1 - x^{2}) T_{n}^{″} (x) - x T_{n}^{'} (x) + n^{2} T_{n} (x) = 0