Probabilistic derivation of a bilinear summation formula for the Meixner-Pollaczek polynominals
Using the technique of canonical expansion in probability theory, a bilinear summation formula is derived for the special case of the Meixner-Pollaczek polynomials {λn(k)(x)} which are defined by the generating function ∑n=0∞λn(k)(x)zn/n!=(1+z)12(x−k)/(1−z)12(x+k), |z|<1. These polynomials...
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| Format: | Article |
| Language: | English |
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Hindawi Publishing Corporation
1980
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| Online Access: | http://eprints.um.edu.my/17452/1/LeePA_%281980%29.pdf |
| _version_ | 1796947150314143744 |
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| author | Lee, P.A. |
| author_facet | Lee, P.A. |
| author_sort | Lee, P.A. |
| collection | UM |
| description | Using the technique of canonical expansion in probability theory, a bilinear summation formula is derived for the special case of the Meixner-Pollaczek polynomials {λn(k)(x)} which are defined by the generating function
∑n=0∞λn(k)(x)zn/n!=(1+z)12(x−k)/(1−z)12(x+k), |z|<1.
These polynomials satisfy the orthogonality condition
∫−∞∞pk(x)λm(k)(ix)λn(k)(ix)dx=(−1)nn!(k)nδm,n, i=−1
with respect to the weight function
p1(x)=sech πx
pk(x)=∫−∞∞…∫−∞∞sech πx1sech πx2 … sech π(x−x1−…−xk−1)dx1dx2…dxk−1, k=2,3,… |
| first_indexed | 2024-03-06T05:42:33Z |
| format | Article |
| id | um.eprints-17452 |
| institution | Universiti Malaya |
| language | English |
| last_indexed | 2024-03-06T05:42:33Z |
| publishDate | 1980 |
| publisher | Hindawi Publishing Corporation |
| record_format | dspace |
| spelling | um.eprints-174522017-07-07T02:29:47Z http://eprints.um.edu.my/17452/ Probabilistic derivation of a bilinear summation formula for the Meixner-Pollaczek polynominals Lee, P.A. QA Mathematics Using the technique of canonical expansion in probability theory, a bilinear summation formula is derived for the special case of the Meixner-Pollaczek polynomials {λn(k)(x)} which are defined by the generating function ∑n=0∞λn(k)(x)zn/n!=(1+z)12(x−k)/(1−z)12(x+k), |z|<1. These polynomials satisfy the orthogonality condition ∫−∞∞pk(x)λm(k)(ix)λn(k)(ix)dx=(−1)nn!(k)nδm,n, i=−1 with respect to the weight function p1(x)=sech πx pk(x)=∫−∞∞…∫−∞∞sech πx1sech πx2 … sech π(x−x1−…−xk−1)dx1dx2…dxk−1, k=2,3,… Hindawi Publishing Corporation 1980 Article PeerReviewed application/pdf en http://eprints.um.edu.my/17452/1/LeePA_%281980%29.pdf Lee, P.A. (1980) Probabilistic derivation of a bilinear summation formula for the Meixner-Pollaczek polynominals. International Journal of Mathematics and Mathematical Sciences, 3 (4). pp. 761-771. ISSN 0161-1712, DOI https://doi.org/10.1155/S0161171280000555 <https://doi.org/10.1155/S0161171280000555>. http://dx.doi.org/10.1155/S0161171280000555 doi:10.1155/S0161171280000555 |
| spellingShingle | QA Mathematics Lee, P.A. Probabilistic derivation of a bilinear summation formula for the Meixner-Pollaczek polynominals |
| title | Probabilistic derivation of a bilinear summation formula for the Meixner-Pollaczek polynominals |
| title_full | Probabilistic derivation of a bilinear summation formula for the Meixner-Pollaczek polynominals |
| title_fullStr | Probabilistic derivation of a bilinear summation formula for the Meixner-Pollaczek polynominals |
| title_full_unstemmed | Probabilistic derivation of a bilinear summation formula for the Meixner-Pollaczek polynominals |
| title_short | Probabilistic derivation of a bilinear summation formula for the Meixner-Pollaczek polynominals |
| title_sort | probabilistic derivation of a bilinear summation formula for the meixner pollaczek polynominals |
| topic | QA Mathematics |
| url | http://eprints.um.edu.my/17452/1/LeePA_%281980%29.pdf |
| work_keys_str_mv | AT leepa probabilisticderivationofabilinearsummationformulaforthemeixnerpollaczekpolynominals |