Math Problems Vault

Problem 6.84

$P_n(x)$ is, apart from a constant factor, equal to the $n$ th derivative of $(x^2-1)^n$ :

P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}(x^2-1)^n \qquad (\textit{Rodrigues' formula}).

?

Problem 6.85

We have

(1-t)^n P_n\left(\frac{1+t}{1-t}\right) = 1 + \binom {n}{1}^2 t + \binom {n}{2}^2 t^2 + \binom {n}{3}^2 t^3 + \cdots + \binom {n}{n-1}^2 t^{n-1} + t^n.

?

Problem 6.86

$P_n(x)$ may be represented in the following integral form:

P_n(x) = \frac{1}{\pi }\int _0^{\pi } (x + \sqrt{x^2-1}\cos \varphi )^n \, d\varphi \qquad (\textit{Laplace's formula}).

?

Problem 6.87

Between three successive Legendre polynomials there holds the following recurrence relation:

P_n(x) = \frac{2n-1}{n}xP_{n-1}(x) - \frac{n-1}{n}P_{n-2}(x), \qquad n = 2, 3, 4, \ldots .

?

Problem 6.88

There exists a unique polynomial $S_n(x)$ of $n$ th degree with real coefficients such that the equation

\int _{-1}^{1} S_n(x)K(x) \, dx = K(1)

is satisfied for all polynomials $K(x)$ of $n$ th degree. Express $S_n(x)$ and $(1-x)S_n(x)$ as a linear combination of Legendre polynomials.

?

Remark.

?

Express $K(x)$ similarly.

Problem 6.89

(Continuation of problem 88.) The polynomials

S_0(x), \quad S_1(x), \quad S_2(x), \quad \ldots , \quad S_n(x), \ldots

satisfy the orthogonality condition

\int _{-1}^{1} (1-x)S_m(x)S_n(x)\, dx = \begin{cases} 0, & \text{if } m \neq n, \\ \dfrac {n+1}{2}, & \text{if } m=n; \quad m,n=0,1,2,\ldots . \end{cases}

?

Problem 6.90

$P_n(x)$ satisfies a homogeneous linear differential equation of second order

(1-x^2)P_n''(x) - 2xP_n'(x) + n(n+1)P_n(x) = 0.

?

Problem 6.91

For sufficiently small $w$ , we have identically in $w$ and $x$

\frac{1}{\sqrt{1-2xw+w^2}} = P_0(x) + P_1(x)w + P_2(x)w^2 + \cdots + P_n(x)w^n + \cdots .

(Generating function for the Legendre polynomials.)

?

Problem 6.92

Deduce problems 84, 86, 87, 90 directly from problem 91. (Problem 91 may also be regarded as offering a definition of the Legendre polynomials, and in fact a ``centrally located'' definition: from here there are particularly convenient paths leading to most of the properties of the Legendre polynomials.)

?

Problem 6.93

$P_n(\cos \vartheta )$ is a cosine polynomial with non-negative coefficients. Determine these coefficients.

Deduce from this the following inequality:

\left|P_n(x)\right| \leqq 1, \qquad -1 \leqq x \leqq 1.

For $n>0$ the equality sign holds only if $x=1$ or $x=-1$ .

?

Problem 6.94

Let $x>1$ . The sequence

P_0(x), \quad P_1(x), \quad P_2(x), \quad \ldots , \quad P_n(x), \ldots

increases monotonically.

?

Problem 6.95

The sum of the first $n$ Legendre polynomials is non-negative in the interval $-1 \leqq x \leqq 1$ :

P_0(x)+P_1(x)+P_2(x)+\cdots +P_n(x) \geqq 0, \qquad -1 \leqq x \leqq 1.

The equality sign holds only if $n$ is odd and $x=-1$ .

?

Remark.

?

See Problem 157 of Part III.

Problem 6.96

The sum of the first $n$ Legendre polynomials

P_0(x)+P_1(x)+P_2(x)+\cdots +P_n(x)

is positive for every value of $x$ if $n$ is even; it has its only change of sign at the point $x=-1$ if $n$ is odd.

?

Remark.

?

See problem 94.

Problem 6.97

The $n$ th Legendre polynomial $P_n(x)$ has only real, simple zeros all of which lie in the interior of the interval $-1, 1$ .

?

Remark.

?

See Problem 140 of Part II.

Problem 6.98

Generalize the theorems in problems 84--91, 97 for the Jacobi (hypergeometric) polynomials

P_0^{(\alpha ,\beta )}(x), \quad P_1^{(\alpha ,\beta )}(x), \quad P_2^{(\alpha ,\beta )}(x), \quad \ldots , \quad P_n^{(\alpha ,\beta )}(x), \ldots , \qquad \alpha , \beta > -1,

defined by the following conditions: (1), (2) $P_n^{(\alpha ,\beta )}(x)$ is of degree $n$ , has real coefficients and orthogonality property

\int _{-1}^{1} (1-x)^\alpha (1+x)^\beta P_m^{(\alpha ,\beta )}(x)P_n^{(\alpha ,\beta )}(x)\, dx

= \begin{cases} 0, & \text{if } m \neq n, \\ ]4pt] \dfrac {2^{\alpha +\beta +1}}{2n+\alpha +\beta +1}\dfrac {\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{\Gamma (n+1)\Gamma (n+\alpha +\beta +1)}, & \text{if } m=n; \quad m,n=0,1,2,\ldots \end{cases} \[ (for $n=0$ the above expression is to be taken as $2^{\alpha +\beta +1}\dfrac {\Gamma (\alpha +1)\Gamma (\beta +1)}{\Gamma (\alpha +\beta +2)}$); (3) the coefficient of $x^n$ in $P_n^{(\alpha ,\beta )}(x)$ is positive. Some cases already encountered, corresponding to special values of $\alpha $ and $\beta $, are collected in the following table: \] \begin{array}{lccccc} \alpha = & 0, & 1, & -\tfrac {1}{2}, & \tfrac {1}{2}, \\ \beta = & 0, & 0, & -\tfrac {1}{2}, & \tfrac {1}{2}, \\ \end{array} \[ \] P_n^{(\alpha ,\beta )}(x) = P_n(x), \quad \frac{2}{n+1}S_n(x), \quad \frac{1\cdot 3\cdots (2n-1)}{2\cdot 4\cdots 2n}T_n(x), \quad 2\frac{1\cdot 3\cdots (2n+1)}{2\cdot 4\cdots (2n+2)}U_n(x). \[ Cf.\ p.\ 85, problem 89, problem 4, problem 4. (The coefficient of $T_n(x)$ must be replaced by $1$ for $n=0$.) \]

?

Problem 6.99

Prove the analogues of problems 84, 85, 87--91, 97 for the generalized Laguerre polynomials

L_0^{(\alpha )}(x), \quad L_1^{(\alpha )}(x), \quad L_2^{(\alpha )}(x), \quad \ldots , \quad L_n^{(\alpha )}(x), \ldots , \qquad \alpha > -1,

defined by the following conditions: (1), (2) $L_n^{(\alpha )}(x)$ is of degree $n$ , has real coefficients and orthogonality property

\int _0^{\infty } e^{-x}x^\alpha L_m^{(\alpha )}(x)L_n^{(\alpha )}(x)\, dx = \begin{cases} 0, & \text{for } m \neq n, \\ ]4pt] \Gamma (\alpha +1)\dbinom {n+\alpha }{n}, & \text{for } m=n; \quad m,n=0,1,2,\ldots ; \end{cases} \[ (3) the coefficient of $x^n$ in $L_n^{(\alpha )}(x)$ has the sign $(-1)^n$. \]

?

Problem 6.100

Prove the analogues of problems 84, 87, 88, 90, 91, 97 for the Hermite polynomials

H_0(x), \quad H_1(x), \quad H_2(x), \quad \ldots , \quad H_n(x), \ldots ,

defined by the following conditions: (1), (2) $H_n(x)$ is of degree $n$ , has real coefficients and orthogonality property

\int _{-\infty }^{\infty } e^{-\frac{x^2}{2}}H_m(x)H_n(x)\, dx = \begin{cases} 0, & \text{for } m \neq n, \\ ]4pt] \sqrt{2\pi }\, n!, & \text{for } m=n; \quad m,n=0,1,2,\ldots ; \end{cases} \[ (3) the coefficient of $x^n$ in $H_n(x)$ has the sign $(-1)^n$. \]

?

Problem 6.101

If $P_n^{(\alpha ,\beta )}(x)$ and $L_n^{(\alpha )}(x)$ are the functions defined in problems 98 and 99, respectively, then we have

\lim _{\beta \to +\infty } P_n^{(\alpha ,\beta )}(1-\varepsilon ) = L_n^{(\alpha )}(x),

if $\varepsilon$ converges to $0$ as $\beta$ increases in such a manner that

\lim _{\beta \to +\infty } \varepsilon \beta = 2x.

?

Problem 6.102

If $L_n^{(\alpha )}(x)$ and $H_n(x)$ are the functions defined in problems 99 and 100, respectively, then we have

H_{2q}(x) = \frac{(-1)^q}{1\cdot 3\cdot 5\cdots (2q-1)}L_q^{(-1/2)}\left(\frac{x^2}{2}\right),

H_{2q+1}(x) = \frac{(-1)^{q+1}}{1\cdot 3\cdot 5\cdots (2q+1)}xL_q^{(1/2)}\left(\frac{x^2}{2}\right), \qquad q=0,1,2,\ldots .

?