The Laguerre polynomials are orthogonal with respect to the scalar product $$ \langle f, g\rangle = \int_0^\infty f(x) g(x) \exp(-x)\,\text{d}x. $$ Is there a class of polynomials that is orthogonal w.r.t. the integral over the entire axis, i.e., $$ \langle f, g\rangle = \int_{-\infty}^\infty f(x) g(x) \exp(-|x|)\,\text{d}x? $$ I could certainly create those using Stieltjes procedure, but I'm looking for recurrence coefficients in explicit form.
2026-02-23 10:45:14.1771843514
Laguerre polynomials with integration over entire axis
41 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in ORTHOGONAL-POLYNOMIALS
- Is there something like "associated" Chebyshev polynomials?
- What is the difference between Orthogonal collocation and Weighted Residual Methods
- Calculate Stieltjes Polynomial
- How do I show this :$\int_{-\infty}^{+\infty} x^n 2\cosh( x)e^{-x^2}=0$ if it is true with $n$ odd positive integer?
- Gegenbauer functions and applications (esp. circular envelope special case)?
- Calculating coefficient of approximation polynomial which is expanded in to a series of Legendre Polynomials
- If $P_n(1)=1$ calculate $P'_n(1)$ in Legendre polynomials
- Linear Functional and Orthogonal polynomial sequence relation
- Show that if $\{P_n\}$,$ n\geq0$ is orthogonal with respect to a linear functional $L$ then the following two are equivalent.
- Orthogonality and norm of Hermite polynomials
Related Questions in LAGUERRE-POLYNOMIALS
- Laguerre polynomials with integration over entire axis
- about the Laguerre square expansion Sin(x)
- Double series with Beta reciprocals $\sum_{j=0}^\infty \sum_{k=0}^\infty \frac{x^j}{j!}\frac{y^k}{k!} \frac{1}{\boldsymbol{B}(j+1,k+1)} = ? $
- Generating function of the Confluent Hypergeometric Function of the First Kind
- Integration of Gaussian $\times$ Laguerre functions
- The Correct Change of Variable of an nth Order Derivative
- How should I prove $\int_0^\infty\frac{d^j}{dx^j}(x^je^{-x})dx=0$?
- How to modified the Gauss–Laguerre quadrature in the case of $\int\limits_{10}^{\infty}{{e^{-x}}f(x)}\mathrm{d}x$?
- Alternative roots of generalized Laguerre polynomials
- Fractional Laguerre function $L_{n-\frac{1}{2}}(x)$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?