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laguerre generating function

November 14, 2020 by

Copyright © 2020 by authors and Scientific Research Publishing Inc. AMS subject classifications: 33C45, 33C65 Key words: Modified Laguerre polynomials, generating function, multilinear and multilateral >> /Filter /FlateDecode xڬ�c��]�%\�ͮS�mۮ.۶m[]�m��6�l�U_��;w����o~��gg�^�2W�8�� It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. An Academic Publisher, Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials (). (2015) Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials. %PDF-1.5 Copyright © 2006-2020 Scientific Research Publishing Inc. All Rights Reserved. The Laguerre ODE is not self-adjoint, and the Laguerre polynomials Ln(x) do not by themselves form an orthogonal set. 8 0 obj The bilinear generating function for products of two Laguerre 2D polynomials. This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License. The problem of determination of the basic generating function for simple Laguerre 2D and Hermite 2D polynomials was solved in [9{12, 18{20]. w���-�{gf�U���=\hszgЖ1)��(��������L)�r�����FI���03��3�T������; ���˫���{h��A�~����h�-�pc(1��?������`���W~|,"w��!����AB��4��Ȟ�RV߉��?^B�{ �~�����Z��|����=�+�d�z�]�`�.C�z�-�N�O��p�]]��e:9?������]��b����A�i���F/��8�N�ȼ�th��ԈZ3�٣Oz.�=��߿����L������寖�������ʿ�#6k4�VC;,E�K���;��0r�����N�aVD��s�i��Y� �b�uj� |��5+e/mF��"}*�F;�e�%7˷۟Af�߾Z���T�����j�m�-���Q�o_!�֧gd�9U�q: �oN�k�}W�����2�Ӏ�S���������P14�d_���J��ac5-?��� Ù*�ҹg��3&d��:}�����5vX6?�Mmz�;��H2KN@����p�̡֛�Nm�,��V$���b���`0��8���,���x-β�.�jh;Nq,���L�U�g�p���Y����0�;�* �>�oTxi"���h�b�;a����M�O+�M���g �Y���ƕ�B�"�r��4/�4ν7�{a��4� �2�ڔ0���s��"0��0���. \) This restriction on λ, imposed by our boundary condition, has the effect of quantifying the energy Laguerre polynomials. << �F� S�}�׮jkl�hmak�W�[ �ed`�o>s#+���. Scientific Research 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. 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 … Wünsche, A. 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�&� 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.) Now that we have the generating function, we can use it to derive the normalization condition. /Length3 0 %���� Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. %PDF-1.3 /Length 24687 A more di cult problem is the determination of generating functions for products of two Laguerre 2D polynomials or of a Laguerre … 4 0 obj Laguerre Functions and Differential Recursion Relations -p. 3/42 Laguerre Polynomials Rodrigues Formula Lα n(x) = exx−α n! /Length2 23849 stream << /Length 5 0 R /Filter /FlateDecode >> The authors declare no conflicts of interest. Institut für Physik, Nichtklassische Strahlung, Humboldt-Universit?t, Berlin, Germany, Creative Commons Attribution 4.0 International License. dn dxn (e−xxn+α) Generating Function (1−w)−α−1 exp xw Some special cases and important applications are also discussed. The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. We also get generating function relations be-tween the modified Laguerre polynomials and the generalized Lauricella functions. %��������� x���z��q���P���25��"I�%�eY��E&��g�6ɦ��%�Q�VYx�g��.���`�^ �P�ԩ{թ�1�>�1��p�dY�EX��*���.�eiZ_�j.yIW͓���*��9�� .��,�"Ƀ��\����(8�v�sw|�[.|��]X�qq�JN�lV'���e�8����l��4��+�/o�*x����˳� �����������u���yD���࿂m����MZU��w�D�v���p:q���8��77���Ǜ;s���2)^m�;8����3��c5C�ye�~l'�Σ)E\&n�q�s5C� zӠI��8,���v�Q��Q�5,��|��W�71��$p����;r}��ư��e�'5J�#�~�*8��,�;4t2��8)�8I�����_�}���x�� �Q'V���ǟGY�q�UE��y4X��\���*,,��8���_m�_����{Q&���m��kN�0�N<17/��5O��UWal��`�$�^�rs���P�[��s�z���I%ql�zְ��5����dح`0w��ӽ��P���h���������ޞ������A�Y��7�ظ�����\w���\u��ݼ��0�����.�]�w ���sȘ��xW�j\o���p��Q7K%C2.��Q��v� �����p�^�"�Dc��_?�b�¶� �?��]�v8]_�J=�/�7��# ���?�n�o�:?��*��{� �7N�����KǫG����W,���?��� stream /Length1 1616 \) 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. [�����D�2�W�)��J������j��a����(E����A�sx�2��h�8��>9&F.

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