# Generating function of laguerre differential equation

In mathematicsthe associated Legendre polynomials are the canonical solutions of the general Legendre equation. When in addition m is even, the function is a polynomial. In that case the parameters are usually labelled with Greek letters. The Legendre ordinary differential equation is frequently encountered in physics and other technical fields.

In particular, it occurs when solving Laplace's equation and related partial differential equations in spherical coordinates. Associated Legendre polynomials play a vital role in the definition of spherical harmonics.

Some authors omit it. Moreover, since by Rodrigues' formula. Indeed, equate the coefficients of equal powers on the left and right hand side of. The following alternative notations are also used in literature: [2]. The associated Legendre polynomials are not mutually orthogonal in general. However, some subsets are orthogonal. The functions for negative m were shown above to be proportional to those of positive m :. This followed from the Rodrigues' formula definition.

This definition also makes the various recurrence formulas work for positive or negative m. From their definition, one can verify that the Associated Legendre functions are either even or odd according to. The first few associated Legendre functions, including those for negative values of mare:. The integral over the product of three associated Legendre polynomials with orders matching as shown below is a necessary ingredient when developing products of Legendre polynomials into a series linear in the Legendre polynomials.

## Laguerre polynomials

For instance, this turns out to be necessary when doing atomic calculations of the Hartree—Fock variety where matrix elements of the Coulomb operator are needed. For this we have Gaunt's formula [3]. Dong and Lemus [4] generalized the derivation of this formula to integrals over a product of an arbitrary number of associated Legendre polynomials.

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They are called the Legendre functions when defined in this more general way. They satisfy the same differential equation as before:. In many occasions in physicsassociated Legendre polynomials in terms of angles occur where spherical symmetry is involved. Together, they make a set of functions called spherical harmonics. These functions express the symmetry of the two-sphere under the action of the Lie group SO 3. When the partial differential equation. The solutions are usually written in terms of complex exponentials :.

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Recalling the relation between the associated Legendre functions of positive and negative mit is easily shown that the spherical harmonics satisfy the identity [5]. The spherical harmonic functions form a complete orthonormal set of functions in the sense of Fourier series. Workers in the fields of geodesy, geomagnetism and spectral analysis use a different phase and normalization factor than given here see spherical harmonics.Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

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Solution to Laguerre differential equation using generating function Ask Question. Asked 6 years ago.

## Hermite polynomials

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Mathematics Stack Exchange works best with JavaScript enabled.Bessel functionsfirst defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Besselare canonical solutions y x of Bessel's differential equation. Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials.

For example:. Bessel functions also appear in other problems, such as signal processing e. Because this is a second-order linear differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient. Different variations are summarized in the table below and described in the following sections.

Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by N n and n nrespectively, rather than Y n and y n. On the other hand, for integer order nthe following relationship is valid the gamma function has simple poles at each of the non-positive integers : [4]. This means that the two solutions are no longer linearly independent. In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.

Another definition of the Bessel function, for integer values of nis possible using an integral representation: [5]. Another integral representation is: [5]. This was the approach that Bessel used, and from this definition he derived several properties of the function. The Bessel functions can be expressed in terms of the generalized hypergeometric series as [10]. This expression is related to the development of Bessel functions in terms of the Bessel—Clifford function.

In terms of the Laguerre polynomials L k and arbitrarily chosen parameter tthe Bessel function can be expressed as [11]. These are sometimes called Weber functionsas they were introduced by H. If n is a nonnegative integer, we have the series [13]. See also the subsection on Hankel functions below. These linear combinations are also known as Bessel functions of the third kind ; they are two linearly independent solutions of Bessel's differential equation.

They are named after Hermann Hankel. These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. The Hankel functions are used to express outward- and inward-propagating cylindrical-wave solutions of the cylindrical wave equation, respectively or vice versa, depending on the sign convention for the frequency.

The Bessel functions are valid even for complex arguments xand an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions or occasionally the hyperbolic Bessel functions of the first and second kind and are defined as [19]. These are chosen to be real-valued for real and positive arguments x. Bessel functions can be described as Fourier transforms of powers of quadratic functions.

It can be proven by showing equality to the above integral definition for K 0.

This is done by integrating a closed curve in the first quadrant of the complex plane.In mathematicsthe Laguerre polynomialsnamed after Edmond Laguerre —are solutions of Laguerre's equation:. This equation has nonsingular solutions only if n is a non-negative integer. Then they are also named generalized Laguerre polynomialsas will be done here alternatively associated Laguerre polynomials or, rarely, Sonine polynomialsafter their inventor [1] Nikolay Yakovlevich Sonin.

More generally, a Laguerre function is a solution when n is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form.

They are orthogonal polynomials with respect to an inner product. The sequence of Laguerre polynomials n! L n is a Sheffer sequence. The rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the Tricomi—Carlitz polynomials. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space.

They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator. Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of n!

Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials. The generating function for them likewise follows. One can also define the generalized Laguerre polynomials recursively, defining the first two polynomials as.

The Rodrigues formula for them is. The generating function for them is. Given the generating function specified above, the polynomials may be expressed in terms of a contour integral. The addition formula for Laguerre polynomials: [8].

Differentiating the power series representation of a generalized Laguerre polynomial k times leads to. In Sturm—Liouville form the differential equation is. The associated, symmetric kernel polynomial has the representations Christoffel—Darboux formula [ citation needed ]. Moreover, [ clarification needed Limit as n goes to infinity? The following integral is needed in the quantum mechanical treatment of the hydrogen atom. The incomplete gamma function has the representation.

The radial part of the wave function is a generalized Laguerre polynomial. Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials.

The generalized Laguerre polynomials are related to the Hermite polynomials :. Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator. The Laguerre polynomials may be defined in terms of hypergeometric functionsspecifically the confluent hypergeometric functionsas.

The generalized Laguerre polynomials satisfy the Hardy—Hille formula [14] [15]. Using the identity. This formula is a generalization of the Mehler kernel for Hermite polynomialswhich can be recovered from it by using the relations between Laguerre and Hermite polynomials given above. From Wikipedia, the free encyclopedia.In mathematicsthe Hermite polynomials are a classical orthogonal polynomial sequence.

Hermite polynomials were defined by Pierre-Simon Laplace in[1] [2] though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in Like the other classical orthogonal polynomialsthe Hermite polynomials can be defined from several different starting points.

Noting from the outset that there are two different standardizations in common use, one convenient method is as follows:. These equations have the form of a Rodrigues' formula and can also be written as. These are Hermite polynomial sequences of different variances; see the material on variances below. The notation He and H is that used in the standard references. The n th-order Hermite polynomial is a polynomial of degree n.

The probabilist's version He n has leading coefficient 1, while the physicist's version H n has leading coefficient 2 n. These polynomials are orthogonal with respect to the weight function measure. The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function. The Hermite polynomials probabilist's or physicist's form an orthogonal basis of the Hilbert space of functions satisfying.

An orthogonal basis for L 2 Rw x dx is a complete orthogonal system. Since the linear span of Hermite polynomials is the space of all polynomials, one has to show in physicist case that if f satisfies. One possible way to do this is to appreciate that the entire function.

Variants of the above completeness proof apply to other weights with exponential decay. In the Hermite case, it is also possible to prove an explicit identity that implies completeness see section on the Completeness relation below. An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for L 2 Rw x dx consists in introducing Hermite functions see belowand in saying that the Hermite functions are an orthonormal basis for L 2 R.

Rewriting the differential equation as an eigenvalue problem. This eigenvalue problem is called the Hermite equationalthough the term is also used for the closely related equation. The general solutions to the above second-order differential equations are in fact linear combinations of both Hermite polynomials and confluent hypergeometric functions of the first kind. For example, for the physicist's Hermite equation. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions, see below.

The sequence of probabilist's Hermite polynomials also satisfies the recurrence relation. The Hermite polynomials constitute an Appell sequencei. These umbral identities are self-evident and included in the differential operator representation detailed below. In consequence, for the m th derivatives the following relations hold:.

It follows that the Hermite polynomials also satisfy the recurrence relation. These last relations, together with the initial polynomials H 0 x and H 1 xcan be used in practice to compute the polynomials quickly. Moreover, the following multiplication theorem holds:. These two equations may be combined into one using the floor function :. The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials He are.

The corresponding expressions for the physicist's Hermite polynomials H follow directly by properly scaling this: [6]. The Hermite polynomials are given by the exponential generating function. One can also derive the physicist's generating function by using Cauchy's integral formula to write the Hermite polynomials as.Ask below and we'll reply. ACT Vocabulary You Must Know ACT Writing: 15 Tips to Raise Your Essay Score How to Get Into Harvard and the Ivy League Is the ACT easier than the SAT.

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The Clarets succumbed to a 1-0 defeat at Leicester City last weekend with Demarai Gray netting in just the sixth minute. Meanwhile, Watford are also winless in the last two outings. The previous meeting between the sides was at Vicarage Road which Watford won by a 2-1 score. Our betting tips and predictions anticipate a Burnley win.

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Goal odds and predictions: Over 2. The New Zealand international has been rather quiet of late with just one goal in five outings, but that happened to be the opener in the 2-1 win over Bournemouth last week. Watford, though, are just behind their Lancashire counterparts with a three-point deficit.

In the previous match, the team had. This was the third loss in four. We also offer a number of betting articles and advice on different topics. This was the fourth loss in five matches for Benitez and his team. Leicester City beat Burnley 1-0 at the King Power stadium to make it back-to-back wins in the league. The victory comes justNewcastle lost 3-1 against Chelsea in the latest league match, as the team could not capitalise on a great start.

The victory comes just a few days after an impressive 2-1 win over Tottenham at home. Leicester City were the hosts in the recent game against Newcastle and they were able to get a 1-0 win in the game. Our betting tips and predictions anticipate a victory for Leicester City. Yet, this would not be our pick in the game. Even though Rafa Benitez and his team are at home, Leicester City are on a strong run of form.

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Four of those matches have finished as a draw, but Leicester have been able to win the last four games against Newcastle.Follow me either on PredictWise or Twitter.

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The main prime time speech went over well, when it happened, but it turned out to be (partially) plagiarized. The big nights will be Wednesday and Thursday, when PredictWise (i. The expectations for the convention are pretty low, meaning that it could be a gain for him, if he does well. I will be back with daily updates in the Musing section as well as regular Tweets and Facebook updates. Site last updated at 8 PM ET yesterday, but I hope to have the data flowing in the next few hours.

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Part-2 Generating Function In Laguerre Polynomials in Special function / Laguerre Polynomial

But, should not be a huge worry. UPDATE: all fixed by 11:30 AM ET today. Sorry for any issues Are prediction markets now too stable, because people trust them too much.

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