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Questions tagged [orthogonal-polynomials]

Questions pertaining to certain sets of polynomials that satisfy an orthogonality criterion with respect to some specified inner product.

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Associated Laguerre polynomials of half-integer parameters

It appears that associated Laguerre polynomials $L_n^{(\alpha)}(x^2)$ for half integer parameters in $n$ and $\alpha$ may be expressed in terms of an exponential function, an imaginary error function ...
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Integral involving a generalized Laguerre Polynomial

I want to evaluate the following integral: $$ \int_0^\infty z^{1/2}e^{-a\space z}L_{m}^{1/2}(z)\space dz, $$ where $L_{m}^{1/2}$ is a generalized Laguerre polynomial. I found a certain indefinite ...
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Density of polynomials in $L^2$ on the unit ball

Are polynomials dense in the Lebesgue space $L^2$ of the unit disk? References are very welcome, thank you.
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Demonstration problem on orthogonal polynomials

Consider the set of polynomials {${\phi_n}$}built using the Gramm-Schmidt method with respect to the inner product: $$(f,g) = \int_{a}^{b} f(x)g(x)w(x)dx$$ with $w(x)>0$. Prove that if $x_0$ is ...
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Integrals involving Legendre polynomials and exponentials

I found the following integrals in the article here: $$I_1(x,\lambda)=\int_{-1}^{1}P_n(x')\,\mathrm{e}^{-(x-x')\,\lambda}\,\mathrm{d}x=\mathrm{e}^{-\lambda\,x}\sum_{k=0}^n\frac{(n+k)!\,(-1)^n}{\...
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Development of a 2-dimensional function on $[0,\infty)\times[0,2\pi]$ in a complete set

Actually I would like to develop function $g(r,\phi)$ on $[0,\infty)\times[0,2\pi]$ in a complete set of functions. This does not seem to very difficult: $$g(r,\phi) = \sum_{m=-\infty}^{\infty} R_m(...
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Szegő's method of finding the generating function of the Jacobi polynomials

In Orthogonal Polynomials (4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, 1975), Szegő starts off section 4.4 by giving the following integral representation of the Jacobi polynomials: $$P_n^{(\...
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Are there degrees or categories of orthogonality?

The vectors $\begin{bmatrix}a \\ 0 \\ 0\end{bmatrix}$, $\begin{bmatrix}0 \\ b \\ 0\end{bmatrix}$, and $\begin{bmatrix}0 \\ 0 \\ c\end{bmatrix}$ are orthogonal under the dot product. The vectors $1$, $...
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Jacobi polynomials and Gram determinants

On page 294, Andrews, Askey and Roy - Special functions. For sequences of (independent) functions $\lbrace \phi(x) \rbrace_{n=0}^{\infty}$ and $\lbrace \psi(x) \rbrace_{n=0}^{\infty}$, a sequence $\...
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1answer
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Do orthogonal polynomials determine the moments of their orthogonality measure?

I am currently learning about the inverse problem for orthogonal polynomials for orthogonality measures supported on the real line. My question is not about finding the orthogonality measure from the ...
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Chebyshev Polynomials of the Second Kind from Orthogonality

I am tasked with finding the degree 5 Chebyshev-II polynomial, using the fact that it's orthogonal to those preceding it w.r.t the Chebyshev-II inner product. I am told to use the normalisation that ...
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Orthogonal polynomials in the complex domain

A source tells me that a set $P_n: \mathbb{C} \to \mathbb{C}$ of polynomials are orthogonal over a contour if $$ \frac{1}{2\pi} \int_{\Gamma} P_n(z) \overline{P_m(z)}h(z)d|z|=\delta_{nm}, $$ where h ...
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Indefinite integral involving the product of two generalized Laguerre polynomials

I am trying to find the indefinite integral \begin{align} \int{x^{\alpha +1}e^{-x}\left(L_{m}^{\alpha}(x)\right)^{2}dx} \end{align} where $L_{m}^{\alpha}(x)$ is the generalized Laguerre Polynomial, ...
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Completeness relation for Jacobi Polynomials

I was wondering if there exists a completeness relation for Jacobi Polynomials, $P^{\alpha,\beta}_{n}(x)$ as in the case of Hermite polynomials, $H_{n}(x)$ such that $$ \sum^{\infty}_{n=0} \psi_n(x) \...
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1answer
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Relation involving generalized Laguerre polynomials

Playing around with different approaches to solve the radial part of the Schrodinger equation for the hydrogen-like atom, I have obtained the following expression ($l$ and $n$ are non-negative ...
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1answer
57 views

Solve Gegenbauer integral $\int\limits_{-1}^{1} x^k \cdot (1-x^2)^{\alpha-1/2} C_n^{\alpha}(x) dx$

I am looking for an analytic solution of the integral \begin{align} \int\limits_{-1}^{1} x^k \cdot (1-x^2)^{\alpha-1/2} C_n^{\alpha}(x) dx \end{align} where $C_n^{\alpha}(x)$ is a given Gegenbauer ...
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1answer
44 views

Gaussian quadrature: Orthogonal polynomial for chi distribution

I have a problem involving numerical integration of the form: $$I = \int_0^\infty \!dx \, w(x) f(x)$$ where the weighting function is a chi distribution of degree 2, i.e., $$w(x) = x \, e^{\frac{-...
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Finding $\int_{0}^{1} \frac{T_{i}^{*''}(x) T_{j}^{*}(x)}{\sqrt{x-x^2}} dx$

Shifted Chebyshev polynomials $$T_{i}^{*}(x) = \cos(i \arccos(2x-1))$$ We want to calculate $$I=\int_{0}^{1} \frac{T_{i}^{*''}(x) T_{j}^{*}(x)}{\sqrt{x-x^2}} dx$$ Which is equal to $$\sum_{\substack{...
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completeness of Laguerre polynomials

Can you help me to prove that system of Laguerre polynomials $$ L_n = \dfrac{e^t}{n!}\dfrac{d^n}{dt^n} (t^n e^{-t})$$ is completeness in space $L_2((0, \infty),e^{-t}dt)$ ? i have idea of proof: ...
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Is the following matrix defined by the roots of Chebyshev polynomial invertible?

Let $x_0, \dots , x_n$ the roots of the Chebyshev polynomial, $T_{n+1}(x)$. We define: $\begin{pmatrix} \frac{1}{\sqrt2}T_0(x_0) & \cdots & \frac{1}{\sqrt2}T_0(x_0) \\ T_1(x_0) & \...
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Convergence of sum $f_n(x)=\sum_{l,k} w_{l,n} w_{l,k} x^k$ , with $w$ expansion coefs of an orthonormal system

Good day, Let $\{P_k\}$ be a complete orthonormal system (Fourier series, Legendre-Fourier series, etc..) on interval $(a,b)$ which can be expanded into powers : $$ P_n = \sum_{k=0}^\infty w_{n,k}x^...
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Derive 3-term recursion of polynomials from some recursion OR are my polynomials orthogonal

considering a sequence of polynomials $(Q_n)$ on $\mathbb{R}$ given by $$(q_1 - x)Q_0 + p_1 Q_1 =0$$ and for $n\geq 2$ by $$(q_n-x)Q_{n} +p_n Q_{n+1}+\sum_{i=0}^{n-1} c_{n,i} Q_i =0$$ where all ...
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68 views

Sum and Integration with Legendre Polynomial

What is the value of the following infinite sum after integrating the product of two Legendre Polynomials $P_m^0,~P_n^1$, $$\sum_{n=1}^{\infty}\sum_{m=0}^{\infty} \frac{1}{\sqrt{n(n+1)}} A_m\, A_n \...
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2answers
108 views

Prove that these sets of polynomials have real and distinct roots.

Can anyone tell me if the following set of polynomials have a special name? $$P_{0}(x)=1,P_{1}(x)=x$$ $$P_{n}(x)=xP_{n-1}-P_{n-2}$$ The above gives: $$P_{2}(x)=x^2-1;P_{3}(x)=x^3-2x;P_{4}(x)=x^4-3x^2+...
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1answer
36 views

On the construction of orthogonal polynomials

In the following proof, argument goes on based on considering $C_n$ to be nonzero then it finishes the proof for $C_n=0$ : Also if we set $C_n=0$ in Eq. (6.10) then must $m=0,1,2,...,n- 2$ in Eq. (6....
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1answer
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Can I form a complete set of functions using $e^{-nx}$?

If I start with the set of functions $e^{-nx}$ for all integers $n>1$, can I use them as basis to create a complete set of orthogonal functions on the interval $(0,+\infty)$? By complete I mean ...
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31 views

Let $S$ be the subspace of $\Bbb R^3$ spanned by the vector $x= (x_1,x_2,x_3)^T$ and $y= (y_1,y_2,y_3)^T$

(a) Let $S$ be the subspace of $\Bbb R^3$ Spanned by the vector $x= (x_1,x_2,x_3)^T$ and $y= (y_1,y_2,y_3)^T$, let $A =\begin{bmatrix}x_1 &x_2 & x_3 \\ y_1 &y_2 &...
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Can we refine this asymptotic for Laguerre polynomials?

I just found an interesting and useful limit for Laguerre polynomials: $$\lim_{n \to \infty} L_n \left( \frac{2r}{n+1/2} \right)=J_0(2 \sqrt{2r})$$ I'm using specifically this form of the argument ...
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Orthogonal polynomials on triangles: Connection with quadrature rules?

In a 1D interval $[a, b]$, quadrature rules and orthogonal polynomials are tightly interconnected. For example, given the $n$ roots $t_j$ of the $n$th orthogonal polynomial $p_n$, one has the ...
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Evaluating an integral using Gegenbauer polynomials

I want to evaluate the following integral $$\int \frac{(r-r'\cos\theta')^2r'^2\,dr'\sin\theta'\,d\theta'\,d\phi'}{(r^2+r'^2-2rr'\cos\theta')^{3/2}}$$ Working that a little bit i end up with this ...
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Orthogonality of generalized Newton symbol

Consider the functions $P_{n}(x)={x \choose n}.$ My question is, if there exists a measure $\mu$ with support being a subset of $(0,\infty)$ such that the family $\{P_{n}\}$ is orthogonal in $L^{2}(\...
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119 views

Define on $P3$ the inner product $<f,g>=\int_{-1}^1 f(t)g(t)dt$, find orthogonal projection

Define on $P3$ the inner product $\langle f,g \rangle=\int_{-1}^1 f(t)g(t)dt$. a) find the orthogonal projection of $p(x)=x^3$ onto $P2$ I know the orthogonal projection formula, but how do I solve ...
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What is this generalization of the Chebyshev polynomials?

For $\varepsilon>0$ consider the tridiagonal matrix $$L_{\varepsilon}=\begin{bmatrix} 0 & 1 & \ & \ & \ & \ & \ & \ \\ 1 & \varepsilon & 1 & \ & \ &...
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Integral involving the Associated Laguerre polynomials

I'm trying to solve this integral $\int_{0}^{\infty} L^n_p L^n_{p'} e^{-x} x^{n-1} dx = \dfrac{1}{n} \dfrac{(p!)^3}{(p-n)!} \delta_{pp'}$ I started with integration by parts where $u = $ $L^n_p L^...
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30 views

Projection on a subspace

An inner product is defined on $P_3$ such that $<f, g>$ $=$ $\int _{-1}^1\:f\left(t\right)g\left(t\right)dt$. What is the orthogonal projection of $p(x)$ $=$ $x^3$ onto the subspace $S$ $=$ $\...
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An closed-form expression of an integral of Chebyshev series and exponential function

Does the following integral has a closed-form expression? $\int_{-1}^{1} T_n(x)\exp(i\pi x)dx,$ where, $T_n$ is the Chebyshev polynomial of degree n.
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Orthogonal Projection on a Polynomial Space

An inner product is defined on $P3$ such that $<f, g>$ $=$ $\int _{-1}^1\:f\left(t\right)g\left(t\right)dt$. What is the orthogonal projection of $p(x)$ $=$ $x^3$ onto $P2$? So I got that $f_1\...
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Identifying the orthogonal polynomial from the recurrence relation

I am trying to see if this recursion relation can be solved in terms of standard orthogonal polynomials : $$ p_{n+1}(x)=2 x p_{n}(x)-2(n+1) p_{n-1}(x), $$ with $p_0(x)=1$ and $p_1(x)=2x$. These ...
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1answer
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The generalized Laguerre polynomials: Are there any expressions valid for any case?

There are general expressions of the generalized Laguerre polynomials. For example: $$ L_n^{(\alpha)}(x) = {}_{n+\alpha}\mathrm{C}_n\ {}_1F_1(-n, \alpha+1, x), \hspace{20pt}(1) $$ $$ L_n^{(\alpha)}(x)...
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1answer
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Use Gram-Schmidt process on Chebychev polynomials

I need help using the Gram-Schmidt orthogonalization process to derive the first four orthonormal Chebychev polynomials. Using the range $[-1,1]$ and the weight function $w(x)=(1-x^2)^\frac{1}{2}$. ...
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1answer
107 views

Orthogonal polynomials with respect to discrete probability distributions

The polynomials from the Askey scheme are orthogonal with respect to a standard probability distribution. For example, Hermite polynomials (in a suitable form) are orthogonal with respect to a ...
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Polynomials orthogonal wrt Rayleigh distribution

As per the title, have the class of polynomials that are orthogonal with respect to Rayleigh distribution been studied? Do they have a name?
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1answer
63 views

Algorithm for orthogonalizing polynomials with specific inner product

I am attempting to generate a as big as possible collection of orthogonal polynomials $p_1, p_2, ..., p_n$, $\left\langle p_i, q_i\right \rangle = \delta_{ij}$ where the inner product is with respect ...
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$L_2$ scalar product between Hermite polynomials

I am trying to compute the $L_2$ scalar product between (probabilists’) Hermite polynomials (defined as in Wiki) with Gaussian weight and different scales, i.e. for some constants $c, d$: $$\frac{1}{\...
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Orthogonal polynomials as eigenfunctions of a second-order difference operator

I am Reading the theorem 6.1.3 of this book https://books.google.com.mx/books?id=RusIDAAAQBAJ&pg=PA146&lpg=PA146&dq=up+to+normalization,+the+charlier,+krawtchouk,+meixner,+and+chebyshev%E2%...
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3answers
142 views

Weird Integral Involving Hermite Polynomials

I have stumbled upon the following integral involving the Hermite polynomials: $$ I(m) = \int_\mathbb{R} e^{i m x} \left[ e^{-\frac{x^2}{2}} H_m(x) \right] dx \, , \quad m \in \mathbb{N} \cup \{0\} \...
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1answer
79 views

An Addition formula for Hermite polynomials

My question concerns an addition formula that can be found on the Wikipedia page of Hermite Polynomials but I can't find it anywhere else. The well-known formula that can be found in many books is the ...
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1answer
45 views

Sum involving Hermite polynomials

I am wondering if there is a simpler form of the following summation involving the (physicists') Hermite polynomials: $$\sum_{k=0}^{n}\frac{H_k(x)}{(2i)^k},$$ where $i=\sqrt{-1}$ is the imaginary ...
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1answer
152 views

Find the Fourier-Bessel Series for $f(x)$ With Respect to the Orthogonal Set: How Was $w(x)$ Found?

I have the following problem: If $f(x) = x$, $0 < x < 2$, find the Fourier-Bessel series for $f(x)$ with respect to the orthogonal set $\{ J_1 (k_n x) \}$, where $k_n$ is the $n$th positive ...
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0answers
81 views

Integral of a triple product of Laguerre polynomials

I would like to know if there's an exact expression for this integral in terms of known elementary or special functions: $$\int_0^\infty \exp \left(-\frac{a+b+c}{2}x \right) L_j (a x) L_k (b x) ...