• Title, Summary, Keyword: Laguerre polynomials

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q-SOBOLEV ORTHOGONALITY OF THE q-LAGUERRE POLYNOMIALS {Ln(-N)(·q)}n=0 FOR POSITIVE INTEGERS N

  • Moreno, Samuel G.;Garcia-Caballe, Esther M.
    • Journal of the Korean Mathematical Society
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    • v.48 no.5
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    • pp.913-926
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    • 2011
  • The family of q-Laguerre polynomials $\{L_n^{(\alpha)}({\cdot};q)\}_{n=0}^{\infty}$ is usually defined for 0 < q < 1 and ${\alpha}$ > -1. We extend this family to a new one in which arbitrary complex values of the parameter ${\alpha}$ are allowed. These so-called generalized q-Laguerre polynomials fulfil the same three term recurrence relation as the original ones, but when the parameter ${\alpha}$ is a negative integer, no orthogonality property can be deduced from Favard's theorem. In this work we introduce non-standard inner products involving q-derivatives with respect to which the generalized q-Laguerre polynomials $\{L_n^{(-N)}({\cdot};q)\}_{n=0}^{\infty}$, for positive integers N, become orthogonal.

GENERALIZATION OF LAGUERRE MATRIX POLYNOMIALS FOR TWO VARIABLES

  • Ali, Asad;Iqbal, Muhammad Zafar
    • Honam Mathematical Journal
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    • v.43 no.1
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    • pp.141-151
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    • 2021
  • The main object of the present paper is to introduce the generalized Laguerre matrix polynomials for two variables. We prove that these matrix polynomials are characterized by the generalized hypergeometric matrix function. An explicit representation, generating functions and some recurrence relations are obtained here. Moreover, these matrix polynomials appear as solution of a differential equation.

INTEGRALS INVOLVING LAGUERRE, JACOBI AND HERMITE POLYNOMIALS

  • Nath, B.
    • Kyungpook Mathematical Journal
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    • v.12 no.1
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    • pp.115-117
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    • 1972
  • The purpose of the present paper is to evaluate certain integrals involving Laguerre, Jacobi and Hermite polynomials. These integrals are very useful in case of expansion of any polynomial in a series of Orthogonal polynomials [1, Theo. 56].

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SOME FINITE INTEGRALS INVOLVING THE PRODUCT OF BESSEL FUNCTION WITH JACOBI AND LAGUERRE POLYNOMIALS

  • Ghayasuddin, Mohd;Khan, Nabiullah;Khan, Shorab Wali
    • Communications of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.1013-1024
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    • 2018
  • The main object of this paper is to set up two (conceivably) valuable double integrals including the multiplication of Bessel function with Jacobi and Laguerre polynomials, which are given in terms of Srivastava and Daoust functions. By virtue of the most broad nature of the function included therein, our primary findings are equipped for yielding an extensive number of (presumably new) fascinating and helpful results involving orthogonal polynomials, Whittaker functions, sine and cosine functions.

OPERATIONAL IDENTITIES FOR HERMITE-PSEUDO LAGUERRE TYPE MATRIX POLYNOMIALS AND THEIR APPLICATIONS

  • Bin-Saad, Maged G.;Pathan, M.A.
    • Honam Mathematical Journal
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    • v.41 no.1
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    • pp.35-49
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    • 2019
  • In this work, it is shown that the combination of operational techniques and the use of the principle of quasi-monomiality can be a very useful tool for a more general insight into the theory of matrix polynomials and for their extension. We explore the formal properties of the operational rules to derive a number of properties of certain class of matrix polynomials and discuss the operational links with various known matrix polynomials.

Solution of Transmission Lines Using Laguerre Polynomials in Time Domain BLT Equations (Laguerre 다항식을 이용한 전송 선로의 시간 영역 BLT 방정식 해석)

  • Lee, Youn-Ju;Chung, Young-Seek;So, Joon-Ho;Shin, Jin-Wo;Cheon, Chang-Yul;Lee, Byung-Je
    • The Journal of Korean Institute of Electromagnetic Engineering and Science
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    • v.18 no.9
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    • pp.1023-1029
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    • 2007
  • In this paper, we propose the method to solve the BLT equations using Laguerre polynomials in time domain. The solution of BLT equations is obtained by recursive, differential and integral properties of Laguerre polynomials. The verification of the proposed method is tested by applying it to the two-wired transmission line with resistors and capacitors, which is illuminated by the electromagnetic plane wave pulse. And the result is compared with the corresponding transient responses obtained from inverse fast Fourier transform(IFFT) of the frequency domain solutions of BLT equations.

THE INTEGRAL EXPRESSION INVOLVING THE FAMILY OF LAGUERRE POLYNOMIALS AND BESSEL FUNCTION

  • Shukla, Ajay Kumar;Salehbhai, Ibrahim Abubaker
    • Communications of the Korean Mathematical Society
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    • v.27 no.4
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    • pp.721-732
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    • 2012
  • The principal aim of the paper is to investigate new integral expression $${\int}_0^{\infty}x^{s+1}e^{-{\sigma}x^2}L_m^{(\gamma,\delta)}\;({\zeta};{\sigma}x^2)\;L_n^{(\alpha,\beta)}\;({\xi};{\sigma}x^2)\;J_s\;(xy)\;dx$$, where $y$ is a positive real number; $\sigma$, $\zeta$ and $\xi$ are complex numbers with positive real parts; $s$, $\alpha$, $\beta$, $\gamma$ and $\delta$ are complex numbers whose real parts are greater than -1; $J_n(x)$ is Bessel function and $L_n^{(\alpha,\beta)}$ (${\gamma};x$) is generalized Laguerre polynomials. Some integral formulas have been obtained. The Maple implementation has also been examined.

CERTAIN NEW GENERATING RELATIONS FOR PRODUCTS OF TWO LAGUERRE POLYNOMIALS

  • CHOI, JUNESANG;RATHIE, ARJUN KUMAR
    • Communications of the Korean Mathematical Society
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    • v.30 no.3
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    • pp.191-200
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    • 2015
  • Generating functions play an important role in the investigation of various useful properties of the sequences which they generate. Exton [13] presented a very general double generating relation involving products of two Laguerre polynomials. Motivated essentially by Exton's derivation [13], the authors aim to show how one can obtain nineteen new generating relations associated with products of two Laguerre polynomials in the form of a single result. We also consider some interesting and potentially useful special cases of our main findings.