Boyd, the optimization of convergence for chebyshev polynomial methods in an unbounded domain, journal of computational physics, 45 1982, 4379. In addition, the representations of q hermite polynomials by q laguerre polynomials and their related integrals are given. Connection problem for sums of finite products of legendre. Then they are also named generalized laguerre polynomials, as will be done here alternatively associated laguerre polynomials or, rarely, sonine polynomials, after their inventor nikolay yakovlevich sonin. Transactions of the american mathematical society 350, 1998, 347393. Divisibility properties of generalized laguerre polynomials. Polynomials laguerreln,lambda,z complex characteristics 4 formulas real part 1 formula imaginary part 1 formula argument 1 formula conjugate value 1 formula. Mar 21, 2018 a variety of polynomials, their extensions and variants have been extensively investigated, due mainly to their potential of applications in diverse research areas. Expand the following derivative using the result of the exercise 10. For n l these polynomials reduce to the polynomials found in 4 and for n0 we have koornwinders generalized laguerre polynomials. I checked a few polynomials up to about order 10, and the coefficients were accurate. Combining 55 and 56, and changing order of summation, we immediately have. Operational techniques for laguerre and legendre polynomials.
It is wellknown that the laguerre polynomials are all hyperbolic i. So we study generalizations of the classical laguerre polynomials which are also further generalizations of the laguerre type and koornwinders generalized laguerre polynomials. In this sequel, we introduce a new class of generalized laguerreeuler polynomials and present certain potentially useful formulas and identities such as implicit summation formulae and symmetry identities. The qlaguerre matrix polynomials pubmed central pmc. Just asking if someome knows how to explain to me what are laguerre polynomials and generalized laguerre polynomials. Riemannhilbert analysis for laguerre polynomials with large. More generally, a laguerre function is a solution when n is not necessarily a nonnegative integer.
On the completeness of the generalized laguerre polynomials. Factorize from the sum the exponential function and the maximal possible power of the monomial. Generalized laguerre polynomial file exchange matlab central. One of the most important properties of the laguerre polynomials is that they constitute an orthogonal polynomial system with respect to the weight function e xx on the interval 0. When n is not a nonnegative integer, laguerrel returns the laguerre function.
Shorey b a department ofmathematics, eth zurich, rdmistrasse 101, 8092 ziirich, switzerland b school ofmathematics, tata institute offundamental research, homi bhabha road, 400005 mumbai, india communicated by prof. The asymptotic behavior of these polynomials is discussed in lot of books and papers for example, see 1. In the special case in which there are no restrictions on singletons. The laguerre polynomials are formally analogous to the hermite polynomials. Four generating functions of this matrix polynomials are investigated. The integrals involving multivariate q laguerre polynomials and then auxiliary ones are studied. For nonnegative integer values of n, the function returns the generalized laguerre polynomials that are orthogonal with respect to the scalar product. In mathematics, the qlaguerre polynomials, or generalized stieltjeswigert polynomials p. Pdf generalized polynomials and operational identities. The generalized laguerre polynomials, the associated bessel functions and application to propagation problems. Pdf in this paper, we give some relations between multivariable laguerre. Generalized hermite and laguerre polynomials in multiple. In this paper, certain secondorder matrix qdifference equation is investigated and solved.
Some integrals involving laguerre polynomials and applications. Generalized laguerre polynomials and quantum mechanics. Generalized polynomials, operational identities and their applications article pdf available in journal of computational and applied mathematics 11812. Differential equations for generalized jacobi polynomials. The solutions usually proceed by series expansion frobenius method, and. Generalized laguerre polynomials and rational chebyshev. Generalized laguerre function and laguerre polynomials. We show how polynomial mappings of degree k from a union of disjoint intervals onto 1,1 generate a countable number of special cases of a certain generalization of the chebyshev polynomials.
In this case we only have one endpoint of the interval of orthogonality of the classical orthogonal polynomials. Generating functions for products of special laguerre 2d and. In this paper we recover some of these results using yet another method. Many of the identities presented, involving families of different polynomials were derived by using the structure of the. Laguerre functions and differential recursion relations p. Journal of computational and applied mathematics 126, 2000, 1. Generalized polynomials, operational identities and their. A generalization of the chebyshev polynomials request pdf. This operational matrix is applied together with the modified generalized laguerre tau method for solving general linear multiterm fractional differential equations fdes. Swarttouw 2010, 14 give a detailed list of their properties. Most special functions arise as solutions to \di cult odes, meaning ones not solvable by exponentials or polynomials. These sets are less common in mathematical physics than the legendre and bessel functions of chapters 11 and 12, but hermite polynomials occur in solutions of the simple harmonic oscillator of quantum mechanics and laguerre polynomials in wave functions of the hydrogen atom.
Pdf the laguerre polynomials in several variables researchgate. At last, some generalized integrals associated with generalized q hermite polynomials are deduced. The laguerre polynomials have been extended to laguerre matrix polynomials by means of studying certain secondorder matrix differential equation. Generalized polynomials and operational identities.
Generalized laguerre polynomials collocation method for. The q laguerre matrix polynomials springerplus full text. The convolution structures of the laguerre polynomials were presented in ref. Generalized laguerre expansions of multivariate probability. Laguerre derivative and monogenic laguerre polynomials core. Section 5, is devoted to approximations by the generalized laguerre polynomials and a useful theorem.
Combining theorem 1 hyperbolicity and theorem 2 simplicity we obtain. The generalized laguerre polynomials, the associated bessel functions and application to propagation problems article pdf available in radiation physics and chemistry 593. The combinatorial interpretation of the integral of a product of charlier polynomials given by zeng 27 can be explained in terms of partition polynomials. Generalized hermite and laguerre polynomials in multiple symmetric matrix arguments and their applications yasuko chikuse department of information science faculty of economics kagawa university 21 saiwaicho takamatsushi, kagawaken, japan 760 and mcgill university montreal, quebec, canada submitted by george p. The laguerre approximation has been widely used for numerical solutions of di. I was a little surprised that no default was supplied for alpha. On generalized laguerre polynomials with real and complex. Differential equations for generalized laguerre and jacobi.
Combining sturms theorem with theorems 3 and 4 it is possi. Pdf generalized polynomials, operational identities and. Generalized rook polynomials and orthogonal polynomials 3 the same block. Oct 14, 2012 in this paper, we derived a new operational matrix of fractional integration of arbitrary order for modified generalized laguerre polynomials. Laguerrelnu,lambda,z 183 formulasprimary definition 1 formula specific values 22 formulas general characteristics 14 formulas. Generalized laguerre functions and differential recursion. The definition and properties of the generalized laguerre polynomials. The 2variable generalized laguerre polynomials m l. The classical laguerre polynomials first we state some properties of the classical laguerre polynomials. The series corresponding to this term is a negative binomial expansion, so we can use the formula, valid for any even complex number r.
On the galois group of generalized laguerre polynomials. Divisibility properties of generalized laguerre polynomials by clemens fuchsa and in. The asymptotic zero distribution for the generalized laguerre polynomials and several others with real and degree dependent parameter n nn. Hermite and laguerre polynomials college of charleston. A new class of generalized laguerreeuler polynomials. The generalized laguerre polynomials constitute a complete orthogonal set of functions on the semiinfinite interval 0. On differential equations for sobolevtype laguerre polynomials. As usual, let rfsdenote the generalized hypergeometric function with rnu. Hydrogen atom laguerre polynomials example 2 c 1 2 3 c 0 8 c 2 1 7 c 1 2 21 c 0 9 this agrees with the values above. The book also explains the laguerre polynomials, their properties, laguerre functions, and the tabulation of the laguerre polynomials and functions. Tables of laguerre polynomials and functions 1st edition. David department of chemistry university of connecticut storrs, connecticut 062693060 dated.
Primary definition 1 formula 19982020 wolfram research, inc. Thus, a set of linear equations is formed and a solution of the considered problem is introduced. Pdf operatorial methods and two variable laguerre polynomials. The generalized laguerre matrix method or solving linear. The fractional integration is described in the riemannliouville sense. Pdf we use operational identities to introduce multivariable laguerre polynomials. In 12, we derived some special generating functions for products of two laguerre 2d polynomials. The generalized laguerre series, which is defined around a gamma pdf, is suited for.
The use of generalized laguerre polynomials in spectral. Combining the multivariate laguerre approximation and bayes theorem. The corresponding generating functions with general 2d matrices u as parameters in these polynomials are fairly complicated 12. Generalizations of laguerre polynomials sciencedirect.
The method has so far been applied to various families of hermite including higher orders and multiindex cases and laguerre polynomials. A laguerre polynomial at one point can be expressed in terms of neighboring laguerre polynomials at the same point. Its solution gives a generalized of the q laguerre polynomials in matrix variable. A new modified generalized laguerre operational matrix of. Aug 16, 2007 the higher order polynomials are computable, and the code is indeed efficient.
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