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Fractional Integral and Generalized Stieltjes Transforms for Hypergeometric Functions as Transmutation Operators
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- Відділення математики
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2015, том 11
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Fractional Integral and Generalized Stieltjes Transforms for Hypergeometric Functions as Transmutation Operators
Інший ID:
2010 Mathematics Subject Classification: 33C05; 44A15; 44A20; 26A33
DOI:10.3842/SIGMA.2015.074
DOI:10.3842/SIGMA.2015.074
Посилання:
Fractional Integral and Generalized Stieltjes Transforms for Hypergeometric Functions as Transmutation Operators / T.H. Koornwinder // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 26 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of
Luc Vinet. The full collection is available at http://www.emis.de/journals/SIGMA/ESSA2014.html.
I am very grateful to Sergei Sitnik for his comments, in particular about Letnikov’s paper [14]
from 1874. Thanks also to Dmitry Karp for helpful comments. Furthermore, the paper took
profit from comments and lists of typos in referees’ reports.
Дата:
2015
Переглядів:
466
Завантажень:
225
Анотація:
For each of the eight n-th derivative parameter changing formulas for Gauss hypergeometric functions a corresponding fractional integration formula is given. For both types of formulas the differential or integral operator is intertwining between two actions of the hypergeometric differential operator (for two sets of parameters): a so-called transmutation property. This leads to eight fractional integration formulas and four generalized Stieltjes transform formulas for each of the six different explicit solutions of the hypergeometric differential equation, by letting the transforms act on the solutions. By specialization two Euler type integral representations for each of the six solutions are obtained.
