Generalized-Hypergeometric Solutions of the General Fuchsian Linear ODE Having Five Regular Singularities
Abstract
:1. Introduction
2. The Approach
3. Explicit Solutions of Lowest Orders
4. Discussion
Author Contributions
Funding
Conflicts of Interest
References
- The Heun Project: Heun Functions, Their Generalizations and Applications. Available online: https://theheunproject.org/bibliography.html (accessed on 30 August 2019).
- Hortaçsu, M. Heun functions and some of their applications in physics. Adv. High Energy Phys. 2018, 2018, 8621573. [Google Scholar] [CrossRef]
- Fuchs, L. Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coefficienten. J. Reine Angew. Math. 1866, 66, 121–160. [Google Scholar]
- Fuchs, L. Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coefficienten. (Ergänzungen zu der im 66sten Bande dieses Journals enthaltenen Abhandlung). J. Reine Angew. Math. 1868, 68, 354–385. [Google Scholar]
- Heun, K. Zur Theorie der Riemann’schen Functionen Zweiter Ordnung mit Verzweigungspunkten. Math. Ann. 1889, 33, 161–179. [Google Scholar] [CrossRef]
- Ronveaux, A. Heun’s Differential Equations; Oxford University Press: London, UK, 1995. [Google Scholar]
- Slavyanov, S.Y.; Lay, W. Special Functions; Oxford University Press: Oxford, UK, 2000. [Google Scholar]
- Olver, F.W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W. (Eds.) NIST Handbook of Mathematical Functions; Cambridge University Press: New York, NY, USA, 2010. [Google Scholar]
- Slater, L.J. Generalized Hypergeometric Functions; Cambridge University Press: Cambridge, UK, 1966. [Google Scholar]
- Bailey, W.N. Generalized Hypergeometric Series; Stechert-Hafner Service Agency: London, UK, 1964. [Google Scholar]
- Ishkhanyan, T.A.; Ishkhanyan, A.M. Generalized confluent hypergeometric solutions of the Heun confluent equation. Appl. Math. Comput. 2018, 338, 624–630. [Google Scholar] [CrossRef] [Green Version]
- Ishkhanyan, A.M. Generalized hypergeometric solutions of the Heun equation. arXiv 2018, arXiv:1802.04263. [Google Scholar]
- Frobenius, F.G. Ueber die Integration der linearen Differentialgleichungen durch Reihen. J. Reine Angew. Math. 1873, 76, 214–235. [Google Scholar]
- Letessier, J. Co-recursive associated Jacobi polynomials. J. Comput. Appl. Math. 1995, 57, 203–213. [Google Scholar] [CrossRef] [Green Version]
- Letessier, J.; Valent, G.; Wimp, J. Some differential equations satisfied by hypergeometric functions. Intern. Ser. Numer. Math. 1994, 119, 371–381. [Google Scholar]
- Maier, R.S. P-symbols, Heun Identities, and 3F2 Identities. Contemp. Math. 2008, 471, 139–159. [Google Scholar]
- Ishkhanyan, T.A.; Shahverdyan, T.A.; Ishkhanyan, A.M. Expansions of the solutions of the general Heun equation governed by two-term recurrence relations for coefficients. Adv. High Energy Phys. 2018, 2018, 4263678. [Google Scholar] [CrossRef]
- Takemura, K. Heun’s equation, generalized hypergeometric function and exceptional Jacobi polynomial. J. Phys. A 2012, 45, 085211. [Google Scholar] [CrossRef]
- Scheffé, H. Linear differential equations with two-term recurrence formulas. J. Math. Phys. 1942, 21, 240–249. [Google Scholar] [CrossRef]
- Crowson, H.L. An analysis of a second order linear ordinary differential equation with five regular singular points. J. Math. Phys. 1964, 43, 38–44. [Google Scholar] [CrossRef]
- Crowson, H.L. Hypergeometric solutions of a second-order linear ordinary differential equation with n-regular singular points. J. Math. Phys. 1965, 44, 384–390. [Google Scholar] [CrossRef]
- Van Hoeij, M.; Kunwar, V.J. Finding 2F1 type solutions of differential equations with 5 singularities. ACM Commun. Comput. Algebra 2012, 46, 96–97. [Google Scholar] [CrossRef]
- Kunwar, V.J. Hypergeometric Solutions of Linear Differential Equations with Rational Function Coefficients. Ph.D. Thesis, Florida State University, Tallahassee, FL, USA, 2014. [Google Scholar]
- Kruglov, V.E. Solution of the linear differential equation of nth-order with four singular points. Ann. Univ. Sci. Bp. Sect. Comp. 2010, 32, 23–35. [Google Scholar]
- Ince, E.L. Ordinary Differential Equations; Dover: New York, NY, USA, 1944. [Google Scholar]
- Redkov, V.M.; Ovsiyuk, E.M. Quantum Mechanics in Spaces of Constant Curvature; Nova Science Publishers: New York, NY, USA, 2011. [Google Scholar]
- Marin, M. Cesaro means in thermoelasticity of dipolar bodies. Acta Mech. 1997, 122, 155–168. [Google Scholar] [CrossRef]
- Ishkhanyan, A.M. The third exactly solvable hypergeometric quantum-mechanical potential. EPL 2016, 115, 20002. [Google Scholar] [CrossRef] [Green Version]
- Ishkhanyan, A.M. Schrödinger potentials solvable in terms of the general Heun functions. Ann. Phys. 2018, 388, 456–471. [Google Scholar] [CrossRef]
- Exton, H. Multiple Hypergeometric Functions and Applications; Halsted Press: Chichester, UK, 1976. [Google Scholar]
- Dattoli, G.; Cesarano, C. On a new family of Hermite polynomials associated to parabolic cylinder functions. Appl. Math. Comput. 2003, 141, 143–149. [Google Scholar] [CrossRef]
- Cesarano, C. Generalized special functions in the description of fractional diffusive equations. Commun. Appl. Ind. Math. 2019, 10, 1–19. [Google Scholar] [CrossRef]
- Ishkhanyan, A.M. Appell hypergeometric expansions of the solutions of the general Heun equation. Constr. Approx. 2019, 49, 445–459. [Google Scholar] [CrossRef]
- Slavyanov, S.Y. Relations between linear equations and Painlevé’s equations. Constr. Approx. 2014, 39, 75–83. [Google Scholar] [CrossRef]
- Iwasaki, K.; Kimura, H.; Shimomura, S.; Yoshida, M. From Gauss to Painlevé: A Modern Theory of Special Functions, Aspects of Mathematics; Vieweg: Braunschweig, Germany, 1991; Volume 16. [Google Scholar]
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Ishkhanyan, A.; Cesarano, C. Generalized-Hypergeometric Solutions of the General Fuchsian Linear ODE Having Five Regular Singularities. Axioms 2019, 8, 102. https://doi.org/10.3390/axioms8030102
Ishkhanyan A, Cesarano C. Generalized-Hypergeometric Solutions of the General Fuchsian Linear ODE Having Five Regular Singularities. Axioms. 2019; 8(3):102. https://doi.org/10.3390/axioms8030102
Chicago/Turabian StyleIshkhanyan, Artur, and Clemente Cesarano. 2019. "Generalized-Hypergeometric Solutions of the General Fuchsian Linear ODE Having Five Regular Singularities" Axioms 8, no. 3: 102. https://doi.org/10.3390/axioms8030102