dorsal/arxiv
View SchemaSeparation of variables and integral relations for special functions
| Authors | Vadim B. Kuznetsov, Evgueni K. Sklyanin |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9705006 |
| URL | https://arxiv.org/abs/q-alg/9705006 |
| DOI | 10.1023/A:1009880307186 |
| Journal | The Ramanujan J. 3 (1999) 5-35 |
Abstract
We show that the method of separation of variables gives a natural generalisation of integral relations for classical special functions of one variable. The approach is illustrated by giving a new proof of the ``quadratic'' integral relations for the continuous q-ultraspherical polynomials. The separating integral operator M expressed in terms of the Askey-Wilson operator is studied in detail: apart from writing down the characteristic (``separation'') equations it satisfies, we find its spectrum, eigenfunctions, inversion, invariants (invariant q-difference operators), and give its interpretation as a fractional q-integration operator. We also give expansions of the A1 Macdonald polynomials into the eigenfunctions of the separating operator M and vice versa.
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"abstract": "We show that the method of separation of variables gives a natural\ngeneralisation of integral relations for classical special functions of one\nvariable. The approach is illustrated by giving a new proof of the\n``quadratic\u0027\u0027 integral relations for the continuous q-ultraspherical\npolynomials. The separating integral operator M expressed in terms of the\nAskey-Wilson operator is studied in detail: apart from writing down the\ncharacteristic (``separation\u0027\u0027) equations it satisfies, we find its spectrum,\neigenfunctions, inversion, invariants (invariant q-difference operators), and\ngive its interpretation as a fractional q-integration operator. We also give\nexpansions of the A1 Macdonald polynomials into the eigenfunctions of the\nseparating operator M and vice versa.",
"arxiv_id": "q-alg/9705006",
"authors": [
"Vadim B. Kuznetsov",
"Evgueni K. Sklyanin"
],
"categories": [
"q-alg",
"math.CA",
"math.QA"
],
"doi": "10.1023/A:1009880307186",
"journal_ref": "The Ramanujan J. 3 (1999) 5-35",
"title": "Separation of variables and integral relations for special functions",
"url": "https://arxiv.org/abs/q-alg/9705006"
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