Result: Separation of Variables and Integral Relations for Special Functions: Separation of variables and integral relations for special functions

Title:
Separation of Variables and Integral Relations for Special Functions: Separation of variables and integral relations for special functions
Authors:
Vadim B. Kuznetsov, Evgueni K. Sklyanin
Source:
The Ramanujan Journal. 3:5-35
Publication Status:
Preprint
Publisher Information:
Springer Science and Business Media LLC, 1999.
Publication Year:
1999
Subject Terms:
product formulas, Mathematics - Classical Analysis and ODEs, Lamé, Mathieu, and spheroidal wave functions, Other basic hypergeometric functions and integrals in several variables, Mathematics - Quantum Algebra, 0103 physical sciences, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Quantum Algebra (math.QA), method of separation of variables, 0101 mathematics, 01 natural sciences
Document Type:
Academic journal Article<br />Other literature type
File Description:
application/xml
Language:
English
ISSN:
1572-9303
1382-4090
DOI:
10.1023/a:1009880307186
DOI:
10.48550/arxiv.q-alg/9705006
Access URL:
http://arxiv.org/pdf/q-alg/9705006
http://arxiv.org/abs/q-alg/9705006
https://arxiv.org/abs/q-alg/9705006
https://arxiv.org/pdf/q-alg/9705006
https://link.springer.com/content/pdf/10.1023/A:1009880307186.pdf
http://ui.adsabs.harvard.edu/abs/1997q.alg.....5006K/abstract
https://link.springer.com/article/10.1023/A%3A1009880307186
Rights:
Springer Nature TDM
arXiv Non-Exclusive Distribution
Accession Number:
edsair.doi.dedup.....525d2a809733bef5cbaf12dd89c0597f
Database:
OpenAIRE

Further Information

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.
32 pages, LaTex, no figures