Questions pertaining to certain sets of polynomials that satisfy an orthogonality criterion with respect to some specified inner product.
Questions tagged [orthogonal-polynomials]
726 questions
12
votes
3 answers
Is there a representation of an inner product where monomials are orthogonal?
There are plenty of examples of inner products on special sequences of polynomials such that they are orthogonal. I can't quite wrap my head around the inner product s.t. monomials are orthogonal. Say we have polynomials defined on the unit…
muaddib
- 8,267
10
votes
4 answers
How does one prove Rodrigues' formula for Legendre Polynomials?
I am trying to prove that $\frac{1}{n!\space2^n}\frac{d^n}{dx^n}\{(x^2-1)^n\}=P_n(x)$, where $P_n(x)$ is the Legendre Polynomial of order n.
I've been told that the proof uses complex analysis, of which I know nothing, isn't there a proof with…
Alubeixu
- 472
5
votes
1 answer
Interesting summation of Laguerre polynomials
I have discovered two interesting summation identities for Laguerre polynomials (although my derivation is obscure) and can't find them mentioned anywhere in my usual online resources. My background is physics, so I don't know better than that. As…
The Vee
- 3,071
4
votes
1 answer
Arbitrary Constants in Orthogonal Polynomials?
The generalized Rodrigues formula (Hassani Mathematical Physics P174) is of the form
$$K_n\frac{1}{w}(\frac{d}{dx})^n(wp^n)$$
The constant $K_n$ is seemingly chosen completely arbitrarily, & I really need to be able to figure out a quick way to…
bolbteppa
- 4,389
4
votes
0 answers
L4 norm of Legendre polynomials
The classical Legendre polynomials are the sequence of polynomials given by the recurrence
$$(k+1)P_{k+1}(x) = (2k+1)x P_k(x) - k P_{k-1}(x)$$
with initial conditions $P_0(x) = 1$ and $P_1(x) = 1$.
They satisfy
$$\int_{-1}^1 P_k(x) P_\ell(x)…
Alf
- 2,597
3
votes
1 answer
Constructing sequence of orthogonal polynomials for arbitrary scalarproduct?
I need to construct a sequence of orthogonal polynomial $(P_i)_{i=0}^{\infty}$ for a family of scalarproducts. I want to look at different scalar products $\langle P_n(x),P_m(x) \rangle_{\mu(x)}=\int_a^b P_n(x)P_m(x) \, d\mu(x)$ with differing…
ckrk
- 235
3
votes
2 answers
Why should we care about orthogonal polynomials?
I know the mathematical definition but I'm having a hard time understanding the utility of orthogonal polynomials. I'm not saying they are useless, far from that! It is just that I like understanding thinks from a higher level than its mathematical…
Felipe Aguirre
- 173
3
votes
1 answer
Orthogonality of Laguerre polynomials...
Laguerre polynomials is a kind of orthogonal polynomials whose inner product is zero. (Is this correct?)
To show that two Laguerre polynomials $L_n(x)$ and $L_m(x)$ are orthogonal, they must satisfy the integral
$\int\limits_0^\infty e^{-x} L_m (x)…
Glenio Rosario
- 331
3
votes
1 answer
Prove Integral representation of Laguerre polynomials
Let $(L_n^{(\alpha)}(x))_n $ a sequence of Laguerre polynomials, for $n=0,1,..., $ and ${\alpha>-1}$, prove that :
$$ n!L_n^{(\alpha)}(x)=x^{-\frac{\alpha}{2}}\int_0^{\infty}e^{x-
y}y^{n+\frac{\alpha}{2}}J_{\alpha}(2\sqrt…
EBS
- 67
3
votes
1 answer
Why does orthogonalizing the monomials give Legendre polynomials?
It's pretty well known that performing a Gram-Schmidt process on the monomials,
$$
p_j(x) = x^j - \sum_{i=0}^{j-1} \frac{\langle x^j|p_i\rangle}{\langle p_i|p_i \rangle}p_i(x),
$$
gives (scaled) Legendre polynomials when $p_0(x) = 1$, $p_1(x) = x$,…
rayhem
- 133
3
votes
0 answers
For an even weight function prove that the orthogonal polynomial is even or odd function depending on the grade.
Let $w(x)$ be an even weight function and [a,b] is a symmetric region with respect to $0$. Prove that the orthogonal polynomial satisfies $p_{n}(-x)=(-1)^{n}p_{n}(x)$ for $n=0,1,2..$
It says that an orthogonal polynomial with respect to an even…
3
votes
1 answer
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 integers)
$$ \tilde L^{(2l+1)}_n(x) = \frac{1}{n!}…
Fabian
- 23,360
3
votes
3 answers
Showing Hermite polynomials are orthogonal
I need to show that two Hermite polynomials are orthogonal, but I'm a little confused.
I have: $H_2(x) = 4x^2-2$ and $H_3(x) = 8x^3-12x$
I know I need to integrate $$\int_{-L}^L (4x^2-2)(8x^3-12x) dx=0,$$ because it says I need to show it's…
user48148
- 215
3
votes
1 answer
How to calculate nodes and weights of Legendre Gauss Lobatto rule?
In the book of "Implementing Spectral Methods for Partial Differential Equations" by David A. Kopriva it is written on page no. 34 (ch. # 1):
For problems with Legendre weighted integrals, the abscissas and weights for the Gauss-Lobatto rule are…
Abbeha
- 113
- 1
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- 7
3
votes
2 answers
orthogonal polynomials and weight functions
how are orthogonal polynomials related to their weight function? is there an algebraic relationship other than the defining integral
$$\int_a^b w(x)P_n(x)P_m(x)\,dx$$?
thanks for the help!
mj_indefinite
- 393