Laguerre polynomials

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Problem 3.54 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 3.3 "Orthogonal and orthonormal systems and bases in Hilbert spaces"

Prove that the system of Laguerre polynomials

L_{n}(t) = \frac{e^{t}}{n!} \frac{d^{n}}{d t^{n}}\left(t^{n} e^{-t}\right), \quad n = 0,1, \ldots ,

is an orthonormal basis in the space $L_{2}\left((0, \infty ), e^{-t} d t\right)$ .

Problem I.4.8 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section I.4 "Orthonormal Sets of Vectors and Bases"

If the sequence $e^{-x/2}, xe^{-x/2}, x^2 e^{-x/2}, \ldots$ is orthogonalized in $L^2(0, \infty )$ , the sequence $e_n(x) = e^{-x/2} L_n(x)/n!$ is obtained, where

L_n(x) = e^x \left(\frac{d}{dx}\right)^n (x^n e^{-x}).

The functions $L_n$ are called Laguerre polynomials.

Problem 1228 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.5 "Higher-Order Derivatives and Differentials"

The Chebyshev--Laguerre polynomials are defined as follows:

L_{m}(x)=e^{x}(x^{m}e^{-x})^{(m)} \ (m=0,1,2,\cdots ) .

Find an explicit expression for the polynomial $L_{m}(x)$ .

Prove that $L_{m}(x)$ satisfies the equation

xL''_{m}(x)+(1-x)L'_{m}(x)+mL_{m}(x)=0 .

Problem 1242 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.6 "Rolle's, Lagrange's, and Cauchy's Theorems"

Prove that the Chebyshev--Laguerre polynomial

L_{n}(x)=e^{x}\frac{d^{n}}{dx^{n}}(x^{n}e^{-x})

has all its roots positive.

Problem 3.219 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Lagrange's Series. Applications"

(1)(2)(3)

Find the generating functions of the polynomials $P_{n}(x)$ , $P_{n}^{(\alpha ,\beta )}(x)$ , $L_{n}^{(\alpha )}(x)$ , defined by the formulas

(1)

$P_{n}(x) = \dfrac {1}{2^{n}n!}\dfrac {d^{n}}{dx^{n}}(x^{2}-1)^{n}$ (Legendre's polynomials);

(2)

$(1-x)^{\alpha }(1+x)^{\beta }P_{n}^{(\alpha ,\beta )}(x) = \dfrac {(-1)^{n}}{2^{n}n!}\dfrac {d^{n}}{dx^{n}}(1-x)^{n+\alpha }(1+x)^{n+\beta }$ , $\alpha >-1$ , $\beta >-1$ (Jacobi's polynomials);

(3)

$e^{-x}x^{\alpha }L_{n}^{(\alpha )}(x) = \dfrac {1}{n!}\dfrac {d^{n}}{dx^{n}}e^{-x}x^{n+\alpha }$ , $\alpha >-1$ (generalized Laguerre's polynomials).

Remark.

Cf.Part VI, problems 84, 98, 99. The generating function of the Legendre polynomials is the series

P_{0}(x) + P_{1}(x)w + P_{2}(x)w^{2} + \cdots + P_{n}(x)w^{n} + \cdots = 1 + xw + \frac{3x^{2}-1}{2}w^{2} + \cdots ,

the sum of which has to be found as a function of $x$ and $w$ ; similarly in the other two cases.

Problem 5.58 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 5 "Applications of Rolle's Theorem"

One may define the Legendre, the Laguerre and the Hermite polynomials by the formulae

P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2-1)^n, \qquad e^{-x} L_n(x) = \frac{1}{n!} \frac{d^n}{dx^n} e^{-x} x^n, \qquad e^{-\frac{x^2}{2}} H_n(x) = \frac{1}{n!} \frac{d^n}{dx^n} e^{-\frac{x^2}{2}}

respectively (problems 84, 99, 100 of Part Six). Deduce from this definition that these polynomials have only real and simple zeros, which lie in the interior of the intervals

(-1,+1), \qquad (0,+\infty ), \qquad (-\infty ,+\infty )

respectively (problems 97, 99, 100 of Part Six).

Problem 6.99 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 6.11 "Legendre Polynomials and Related Topics"

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 6.101 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 6.11 "Legendre Polynomials and Related Topics"

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 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 6.11 "Legendre Polynomials and Related Topics"

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 .

Problem 6.106 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 6.12 "Further Maximum-Minimum Problems on Polynomials"

Let $\alpha$ be a constant, $\alpha >-1$ . Determine the maximum of $\left|P(0)\right|$ for all polynomials $P(x)$ of $n$ th degree which satisfy the condition

\int _0^{\infty } e^{-x}x^\alpha [P(x)]^2\, dx = 1.

How does this maximum behave for large values of $n$ ?

Problem 2.4.15 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.4 "Elementary Hilbert Space Theory"

Compute

\min _{a, b, c} \int _{0}^{\infty }\left|x^{3}-a-b x-c x^{2}\right|^{2} e^{-x} d x

State and solve the corresponding maximum problem, as in Exercise 14.

References

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.

[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.