The $q-$Onsager algebra and multivariable $q-$special functions
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| Format: | Preprint |
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2016
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| _version_ | 1866909113369231360 |
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| author | Baseilhac, Pascal Vinet, Luc Zhedanov, Alexei |
| author_facet | Baseilhac, Pascal Vinet, Luc Zhedanov, Alexei |
| contents | Two sets of mutually commuting $q-$difference operators $x_i$ and $y_j$, $i,j=1, ...,N$ such that $x_i$ and $y_i$ generate a homomorphic image of the $q-$Onsager algebra for each $i$ are introduced. The common polynomial eigenfunctions of each set are found to be entangled product of elementary Pochhammer functions in $N$ variables and $N+3$ parameters. Under certain conditions on the parameters, they form two `dual' bases of polynomials in $N$ variables. The action of each operator with respect to its dual basis is block tridiagonal. The overlap coefficients between the two dual bases are expressed as entangled products of $q-$Racah polynomials and satisfy an orthogonality relation. The overlap coefficients between either one of these bases and the multivariable monomial basis are also considered. One obtains in this case entangled products of dual $q-$Krawtchouk polynomials. Finally, the `split' basis in which the two families of operators act as block bidiagonal matrices is also provided. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1611_09250 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | The $q-$Onsager algebra and multivariable $q-$special functions Baseilhac, Pascal Vinet, Luc Zhedanov, Alexei Mathematical Physics Quantum Algebra Exactly Solvable and Integrable Systems Two sets of mutually commuting $q-$difference operators $x_i$ and $y_j$, $i,j=1, ...,N$ such that $x_i$ and $y_i$ generate a homomorphic image of the $q-$Onsager algebra for each $i$ are introduced. The common polynomial eigenfunctions of each set are found to be entangled product of elementary Pochhammer functions in $N$ variables and $N+3$ parameters. Under certain conditions on the parameters, they form two `dual' bases of polynomials in $N$ variables. The action of each operator with respect to its dual basis is block tridiagonal. The overlap coefficients between the two dual bases are expressed as entangled products of $q-$Racah polynomials and satisfy an orthogonality relation. The overlap coefficients between either one of these bases and the multivariable monomial basis are also considered. One obtains in this case entangled products of dual $q-$Krawtchouk polynomials. Finally, the `split' basis in which the two families of operators act as block bidiagonal matrices is also provided. |
| title | The $q-$Onsager algebra and multivariable $q-$special functions |
| topic | Mathematical Physics Quantum Algebra Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/1611.09250 |