Math Problems Vault

Problem 6.1

The zeros of $T_n(x)$ and $U_n(x)$ are all real and distinct and lie in the interior of the interval $-1, 1$ . Determine these zeros.

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Problem 6.2

Prove that the following relations hold:

T_{n+1}(x) = xT_n(x) - (1-x^2)U_{n-1}(x),

U_n(x) = xU_{n-1}(x) + T_n(x), \qquad n = 1, 2, 3, \ldots .

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Problem 6.3

The polynomials $T_n(x)$ and $U_n(x)$ satisfy the following differential equations:

(1-x^2)T_n''(x) - xT_n'(x) + n^2 T_n(x) = 0,

(1-x^2)U_n''(x) - 3xU_n'(x) + n(n+2)U_n(x) = 0.

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Problem 6.4

$T_n(x)$ and $U_n(x)$ satisfy the following orthogonality relations:

\left. \begin{aligned} \int _{-1}^{1} \frac{1}{\sqrt{1-x^2}} T_m(x) T_n(x) \, dx & = 0, \\ \int _{-1}^{1} \sqrt{1-x^2} \, U_m(x) U_n(x) \, dx & = 0, \end{aligned} \right\} \qquad m, n = 0, 1, 2, \ldots ; \quad m \geqq n.

Evaluate these integrals for $m = n$ .

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Problem 6.5

$T_n(x)$ and $U_n(x)$ satisfy the identities

\frac{T_n(x)}{\sqrt{1-x^2}} = \frac{(-1)^n}{1 \cdot 3 \cdot 5 \cdots (2n-1)} \frac{d^n}{dx^n} (1-x^2)^{n - \frac{1}{2}},

\sqrt{1-x^2} \, U_n(x) = \frac{(-1)^n (n+1)}{1 \cdot 3 \cdot 5 \cdots (2n+1)} \frac{d^n}{dx^n} (1-x^2)^{n + \frac{1}{2}}.

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Problem 6.6

Let $f(x)$ be a function defined in the interval $-1 \leqq x \leqq 1$ with a continuous $n$ th derivative in this interval. Then we have

\int _0^{\pi } f(\cos \vartheta ) \cos n\vartheta \, d\vartheta = \frac{1}{1 \cdot 3 \cdot 5 \cdots (2n-1)} \int _0^{\pi } f^{(n)}(\cos \vartheta ) \sin ^{2n} \vartheta \, d\vartheta .

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Problem 6.7

For $n = 1, 2, 3, \ldots$ , $-1 \leqq x \leqq 1$ , we have the inequalities

\left|T_n(x)\right| \leqq 1, \qquad \left|U_n(x)\right| \leqq n+1.

In the first inequality we have equality for precisely $n+1$ values of $x$ , namely at the $n-1$ zeros of $U_{n-1}(x)$ and, in addition, at $x = -1$ and $x = 1$ . In the second inequality we have equality only for $x = -1$ and $x = 1$ .

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