Skip to main content
SpringerLink
Log in

Generalized-hypergeometric solutions of the biconfluent Heun equation

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

Infinitely many cases for which two independent fundamental solutions of the biconfluent Heun equation can each be presented as an irreducible linear combination of two confluent generalized hypergeometric functions are identified. The involved hypergeometric functions, which in general do not reduce to polynomials, are such that each numerator parameter (except one) exceeds a corresponding denominator parameter by unity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. The Heun project: Heun functions, their generalizations and applications, https://theheunproject.org/bibliography.html (last updated 23 Sep, 2019).

  2. Hortaçsu, M.: Heun functions and some of their applications in physics. Adv. High Energy Phys. 2018, 8621573 (2018)

    Article  MathSciNet  Google Scholar 

  3. Ronveaux, A.: Heun’s Differential Equations. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  4. Yu Slavyanov, S., Lay, W.: Special functions. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  5. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010)

    MATH  Google Scholar 

  6. Poole, E.G.C.: Introduction to the Theory of Linear Differential Equations. Dover Publications, New York (1960)

    MATH  Google Scholar 

  7. Lebedev, N.N., Silverman, R.R.: Special Functions and Their Applications. Dover Publications, New York (1972)

    Google Scholar 

  8. Hautot, A.: Sur les solutions polynomiales de l’équation différentielle. Bull. Soc. Ro. Sci. Liège 38, 660–663 (1969)

    MathSciNet  MATH  Google Scholar 

  9. Hautot, A.: Sur des combinaisons linéaires d’un nombre fini de fonctions transcendantes comme solutions d’équations différentielles du second ordre. Bull. Soc. Roy. Sci. Liège 40, 13–23 (1971)

    MathSciNet  MATH  Google Scholar 

  10. Ishkhanyan, T.A., Ishkhanyan, A.M.: Solutions of the bi-confluent Heun equation in terms of the Hermite functions. Ann. Phys. 383, 79–91 (2017)

    Article  MathSciNet  Google Scholar 

  11. Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966)

    MATH  Google Scholar 

  12. Prudnikov, A.P., Brichkov, Yu.A., Marichev, O.I.: Integrals and Series. Additional Chapters. Nauka, Moscow (1986). ((in Russian))

    Google Scholar 

  13. Svartholm, N.: Die Lösung der Fuchs’schen Differentialgleichung zweiter Ordnung durch Hypergeometrische Polynome. Math. Ann. 116, 413–421 (1939)

    Article  MathSciNet  Google Scholar 

  14. Erdélyi, A.: Certain expansions of solutions of the Heun equation. Q. J. Math. 15, 62–69 (1944)

    Article  MathSciNet  Google Scholar 

  15. Schmidt, D.: Die Lösung der linearen Differentialgleichung 2. Ordnung um zwei einfache Singularitäten durch Reihen nach hypergeometrischen Funktionen. J. Reine Angew. Math. 309, 127–148 (1979)

    MathSciNet  MATH  Google Scholar 

  16. Ishkhanyan, T.A., Ishkhanyan, A.M.: Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions. AIP Adv. 4, 087132 (2014)

    Article  Google Scholar 

  17. Leaver, E.W.: Solutions to a generalized spheroidal wave equation: Teukolsky equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27, 1238–1265 (1986)

    Article  MathSciNet  Google Scholar 

  18. El-Jaick, L.J., Figueiredo, B.D.B.: Solutions for confluent and double-confluent Heun equations. J. Math. Phys. 49, 083508 (2008)

    Article  MathSciNet  Google Scholar 

  19. El-Jaick, L.J., Figueiredo, B.D.B.: Integral relations for solutions of the confluent Heun equation. Appl. Math. Comput. 256, 885–904 (2015)

    MathSciNet  MATH  Google Scholar 

  20. El-Jaick, L.J., Figueiredo, B.D.B.: Convergence and applications of some solutions of the confluent Heun equation. Appl. Math. Comput. 284, 234–259 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Leroy, C., Ishkhanyan, A.M.: Expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta and the Appell generalized hypergeometric functions. Integral Transform. Spec. Funct. 26, 451–459 (2015)

    Article  MathSciNet  Google Scholar 

  22. 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, 4263678 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Ishkhanyan, A.M.: Series solutions of confluent Heun equations in terms of incomplete Gamma-functions. J. Appl. Anal. Comput. 9, 118–139 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Ishkhanyan, A.M.: Appell hypergeometric expansions of the solutions of the general Heun equation. Constr. Approx. 49, 445–459 (2019)

    Article  MathSciNet  Google Scholar 

  25. Letessier, J.: Co-recursive associated Jacobi polynomials. J. Comp. Appl. Math. 57, 203–213 (1995)

    Article  MathSciNet  Google Scholar 

  26. Letessier, J.: Co-recursive associated Laguerre polynomials. J. Comput. Appl. Math. 49, 127–136 (1993)

    Article  MathSciNet  Google Scholar 

  27. Letessier, J.: Some results on co-recursive associated Laguerre and Jacobi polynomials. SIAM J. Math. Anal. 25, 528–548 (1994)

    Article  MathSciNet  Google Scholar 

  28. Letessier, J., Valent, G., Wimp, J.: Some differential equations satisfied by hypergeometric functions. Int. Ser. Numer. Math. 119, 371–381 (1994)

    MathSciNet  MATH  Google Scholar 

  29. Maier, R.S.: P-symbols, Heun identities, and identities. Contemp. Math. 471, 139–159 (2008)

    Article  MathSciNet  Google Scholar 

  30. Takemura, K.: Heun’s equation, generalized hypergeometric function and exceptional Jacobi polynomial. J. Phys. A 45, 085211 (2012)

    Article  MathSciNet  Google Scholar 

  31. Ishkhanyan, A.M.: Generalized hypergeometric solutions of the Heun equation. Theor. Math. Phys. 202, 1–10 (2020)

    Article  MathSciNet  Google Scholar 

  32. Ishkhanyan, T.A., Ishkhanyan, A.M.: Generalized confluent hypergeometric solutions of the Heun confluent equation. Appl. Math. Comput. 338, 624–630 (2018)

    MathSciNet  MATH  Google Scholar 

  33. Miller, A.R., Paris, R.B.: Transformation formulas for the generalized hypergeometric function with integral parameter differences. Rocky Mount. J. Math. 43, 291–327 (2013)

    Article  MathSciNet  Google Scholar 

  34. Maier, R.S.: Extensions of the classical transformations of the hypergeometric function 3F2. Adv. Appl. Math. 105, 25–47 (2019)

    Article  Google Scholar 

  35. Y.M. Chiang, C.K. Law, G.F. Yu, Invariant subspaces of biconfluent Heun operators and special solutions of Painlevé IV. arXiv:1905.10046 (2019)

  36. Ferreira, E.M., Sesma, J.: Global solutions of the biconfluent Heun equation. Numer. Algorithms 71, 797–808 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the referee for very important comments. In particular, the idea that the number of the generalized hypergeometric functions involved in the solution can be reduced to two (instead of four) is solely due to the referee.

Author information

Authors and Affiliations

  1. Russian-Armenian University, 0051, Yerevan, Armenia

    D. Yu. Melikdzhanian & A. M. Ishkhanyan

  2. Vekua Institute of Applied Mathematics, Tbilisi State University, 0186, Tbilisi, Georgia

    D. Yu. Melikdzhanian

  3. Institute for Physical Research, 0203, Ashtarak, Armenia

    A. M. Ishkhanyan

Authors
  1. D. Yu. Melikdzhanian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. M. Ishkhanyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Yu. Melikdzhanian.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was supported by the Russian-Armenian (Slavonic) University at the expense of the Ministry of Education and Science of the Russian Federation, the Armenian Science Committee (SC Grants 20RF-171 and 21SC-BRFFR-1C021), and the Armenian National Science and Education Fund (ANSEF Grant No. PS-2520).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Melikdzhanian, D.Y., Ishkhanyan, A.M. Generalized-hypergeometric solutions of the biconfluent Heun equation. Ramanujan J 57, 37–53 (2022). https://doi.org/10.1007/s11139-021-00504-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-021-00504-w

Keywords

Mathematical Subject Classification

Search

Navigation