# Generalized-Hypergeometric Solutions of the General Fuchsian Linear ODE Having Five Regular Singularities

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## Abstract

**:**

## 1. Introduction

## 2. The Approach

## 3. Explicit Solutions of Lowest Orders $N=0,1,2$

## 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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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Ishkhanyan, 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