laguerre generating function

dn dxn (e−xxn+α) Generating Function (1−w)−α−1 exp xw >> �F� S�}�׮jkl�hmak�W�[ �ed`�o>s#+���. Scientific Research \) This is necessary because the Laguerre function of nonintegral n would diverge, which is unacceptable for our physical problem, in which \( \lim_{r\to\infty} R(r) =0. << /Length 5 0 R /Filter /FlateDecode >> A more di cult problem is the determination of generating functions for products of two Laguerre 2D polynomials or of a Laguerre … Laguerre Functions and Differential Recursion Relations -p. 3/42 Laguerre Polynomials Rodrigues Formula Lα n(x) = exx−α n! Wünsche, A. xڬ�c��]�%\�ͮS�mۮ.۶m[]�m��6�l�U_��;w����o~��gg�^�2W�8�� These sets are less common in mathematical physics than the Legendre and Bessel functions of Chapters 11 and 12, but Hermite polynomials occur in solutions of the simple harmonic oscillator of quantum mechanics and Laguerre … As is seen from the form of the generating function, from the form of Laguerre’s ODE, or from Table 13.2, the Laguerre polynomials have neither odd nor even symmetry under the parity transformation x →−x. \) This restriction on λ, imposed by our boundary condition, has the effect of quantifying the energy /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. An Academic Publisher, Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials (). Laguerre polynomials. 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�&� Some special cases and important applications are also discussed. The Laguerre ODE is not self-adjoint, and the Laguerre polynomials Ln(x) do not by themselves form an orthogonal set. 4 0 obj %PDF-1.3 The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. /Length1 1616 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,���?��� The bilinear generating function for products of two Laguerre 2D polynomials. Copyright © 2020 by authors and Scientific Research Publishing Inc. 8 0 obj Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. The problem of determination of the basic generating function for simple Laguerre 2D and Hermite 2D polynomials was solved in [9{12, 18{20]. Copyright © 2006-2020 Scientific Research Publishing Inc. All Rights Reserved. This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License. %PDF-1.5 %���� 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.) (2015) Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials. %��������� 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���. We also get generating function relations be-tween the modified Laguerre polynomials and the generalized Lauricella functions.

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