laguerre generating function
Published by on November 13, 2020
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A more di cult problem is the determination of generating functions for products of two Laguerre 2D polynomials or of a Laguerre … << Institut für Physik, Nichtklassische Strahlung, Humboldt-Universit?t, Berlin, Germany, Creative Commons Attribution 4.0 International License. We also get generating function relations be-tween the modified Laguerre polynomials and the generalized Lauricella functions. The Laguerre ODE is not self-adjoint, and the Laguerre polynomials Ln(x) do not by themselves form an orthogonal set. Now that we have the generating function, we can use it to derive the normalization condition. 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 … Copyright © 2020 by authors and Scientific Research Publishing Inc. 4 0 obj xڬ�c��]�%\�ͮS�mۮ.۶m[]�m��6�l�U_��;w����o~��gg�^�2W�8�� Some special cases and important applications are also discussed. /Length1 1616 The generating function for associated Laguerre polynomials is: e xz=(1 z) (1 kz) +1 = ¥ å n=0 Lk n(x)zn (13) (As I said, it’s not something you can pull out of a hat.) stream /Length2 23849 Scientific Research /Length3 0 �F� S�}�jkl�hmak�W�[ �ed`�o>s#+���. stream %���� /Filter /FlateDecode The authors declare no conflicts of interest. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. dn dxn (e−xxn+α) Generating Function (1−w)−α−1 exp xw The problem of determination of the basic generating function for simple Laguerre 2D and Hermite 2D polynomials was solved in [9{12, 18{20]. \) This is necessary because the Laguerre function of nonintegral n would diverge, which is unacceptable for our physical problem, in which \( \lim_{r\to\infty} R(r) =0. (2015) Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. [�����D�2�W�)��J������j��a����(E����A�sx�2��h�8��>9&F. ʴ��v�&bv�δ�t\ U%ukkc;Z%;�_3+)������������ @�� bb`b0rrr���=-�̝1(��i���O���?=o:Y����~��X��ۘ�:������lbp67�ZX� ��~Jʉ(��T�&�&�� Ck#������� %���`�������?�9���t ��M�,�^3q72���E�7q��pr�� �p�9�:�큳���������v��w��a���L����������7�����t67p�'���_7���o�����?%������l`a�p6qw�'�� ��������o�`����pq��5�/4 G3Gck'��0����� �_�7�������ݿQ��������)#�ߜF�s�Y����3*���v F������������Q�33�I��Z{ �MLa�����P�ߩL��N���?����������/����{���b.��r6&�^��� �Y2��[�������)��G�����#�����Vښ������?�Nb�&� << /Length 5 0 R /Filter /FlateDecode >> %PDF-1.3 Laguerre Functions and Differential Recursion Relations -p. 3/42 Laguerre Polynomials Rodrigues Formula Lα n(x) = exx−α n! 8 0 obj Wünsche, A. Copyright © 2006-2020 Scientific Research Publishing Inc. All Rights Reserved. As is seen from the form of the generating function, from the form of Laguerre’s ODE, or from Table 13.2, the Laguerre polynomials have neither odd nor even symmetry under the parity transformation x →−x. The bilinear generating function for products of two Laguerre 2D polynomials. The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. %PDF-1.5 Laguerre polynomials. >> \) This restriction on λ, imposed by our boundary condition, has the effect of quantifying the energy %��������� AMS subject classifications: 33C45, 33C65 Key words: Modified Laguerre polynomials, generating function, multilinear and multilateral
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