### The Heun Project

## Status:

__Active__

### Currently:

Gathering participants

This page aims to create a bibliography of the texts published on the Heun functions troughout the years.

Note that the list is not ordered in any way and is far from complete.

If you see an error in the citations or wish to add something to the list, please contact the site administrator.

**Last updated: 23.09.2019.**

### Sections:

- Textbooks on the Heun functions
- Articles on the theory of the Heun functions
- Articles with applications of the Heun functions
- Numerical evaluations
- Others
- Online resources

### Textbooks on the Heun functions

Heun, Karl (1889), "Zur Theorie der Riemann'schen Functionen zweiter Ordnung mit vier Verzweigungspunkten", Mathematische Annalen 33: 161

A. Erdélyi, F. Oberhettinger, W. Magnus and F. Tricomi, "Higher Transcendental functions vol. 3" (McGraw Hill, NY, 1953).

Forsyth, Andrew Russell (1959) [1906], "Theory of differential equations. 4. Ordinary linear equations", New York: Dover Publications, pp. 158, MR 0123757

Ronveaux, A., ed. (1995), "Heun's differential equations", Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-859695-0, MR 1392976

S. Y. Slavyanov and W. Lay, "Special Functions, A Unified Theory Based on Singularities," Oxford Mathematical Monographs, Oxford, 2000.

Sleeman, B. D.; Kuznetzov, V. B. (2010), "Heun functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., "NIST Handbook of Mathematical Functions", Cambridge University Press, ISBN 978-0521192255, MR2723248

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, "NIST Handbook of Mathematical Functions," Cambridge University Press, Cambridge, 2010.

Maplesoft's online documentation: The five Second Order Linear Heun equations and the corresponding Heun function solutions

Return to Contents### Articles on the theory of the Heun functions:

E.W. Leaver (1986), "Solutions to a generalized spheroidal wave equation: Teukolsky's equations in general relativity, and the two-center problem in molecular quantum mechanics", Journal of mathematical physics 27 (1986): 1238

Maier, Robert S. (2007), "The 192 solutions of the Heun equation", Mathematics of Computation 76 (258): 811-843, arXiv:math/0408317, doi:10.1090/S0025-5718-06-01939-9, MR 2291838

Valent, Galliano (2007), "Heun functions versus elliptic functions", Difference equations, special functions and orthogonal polynomials, World Sci. Publ., Hackensack, NJ, pp. 664-686, arXiv:math-ph/0512006, doi:10.1142/9789812770752_0057, MR 2451210,

Plamen P. Fiziev, "Novel relations and new properties of confluent Heun's functions and their derivatives of arbitrary order", .J Phys. A: Math. Theor. 43 (2010) 035203

Raimundas Vidunas, Galina Filipuk, "Parametric transformations between the Heun and Gauss hypergeometric functions", arXiv:0910.3087 [math.CA]

Artur Ishkhanyan, "Incomplete beta-function expansions of the solutions to the confluent Heun equation" arXiv:0909.1686, JPA 38, L491 (2005)

Raimundas Vidunas, Galina Filipuk, "A classification of coverings yielding Heun-to-hypergeometric reductions" , arXiv:1204.2730 [math.CA]

Kouichi Takemura, "Heun's equation, generalized hypergeometric function and exceptional Jacobi polynomial", arXiv:1106.1543 [math.CA]

Peter A. Becker, "Normalization integrals of orthogonal Heun functions", arXiv:funct-an/9707004, JMP (1997), volume 38, page 3692

Lea Jaccoud El-Jaick and Bartolomeu D. B. Figueired, |Transformations of Heun's equation and its integral relations", (To appear in J. Phys. A), arXiv:1002.4559

Lea Jaccoud El-Jaick, Bartolomeu D. B. Figueiredo,|On Certain Solutions for Confluent and Double-Confluent Heun Equations", arXiv:0807.2219 [math-ph], J. Math. Phys., 49, 083508 (2008)

Lea Jaccoud El-Jaick, Bartolomeu D. B. Figueiredo, "Confluent and Double-Confluent Heun Equations: Convergence of Solutions in Series of Coulomb Wavefunctions", arXiv:1209.4673 [math-ph]

B. D. Bonorino Figueiredo, "Ince's limits for confluent and double-confluent Heun equations", arXiv:math-ph/0509013, JMP, DOI: 10.1063/1.2104267

R. Sokhoyan, D. Melikdzanian, A. Ishkhanyan, "New hypergeometric series solutions to the general Heun equation", J. Contemp. Phys. (Armenian Academy of Sciences) Vol. 40, No. 6, pp. 1-7 (2005), arXiv:0909.1286 [math-ph]

Hossein Movasati, Stefan Reiter, "Heun equations coming from geometry", arXiv:0902.0760 [math.AG]

Artur Ishkhanyan, Kalle-Antti Suominen, "New solutions of Heun general equation", J. Phys. A: Math. Gen. 36, L81 (2003), arXiv:0909.1684 [math-ph]

Galliano Valent, "Associated Stieltjes-Carlitz polynomials and a generalization of Heun's differential equation"., arXiv:math/9307204 [math.CA]

E.S. Cheb-Terrab, "Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations", Journal of Physics A: Mathematical and General 37 9923-9949 (2004)., arXiv:math-ph/0404014

Kouichi Takemura, "Integral Transformation And Darboux Transformation Of Heun's Differential Equation", 0911.2162 [math.CA]

Kouichi Takemura, "Towards Finite-Gap Integration of the Inozemtsev Model", SIGMA 3 (2007), 038, 17 pages, arXiv:0703256 [math.CA]

Lea Jaccoud El-Jaick, Bartolomeu D. B. Figueiredo, "A limit of the confluent Heun equation and the Schroedinger equation for an inverted potential and for an electric dipole", arXiv:0902.3202 [math-ph]

Robert S. Maier, P-symbols, Heun Identities, and 3F2 Identities, arXiv:0712.4299 [math.CA]

E. Ovsiyuk and O. Veko, M. Amirfachrian. On Schrodinger equation with potential U and the bi-confluent Heun functions theory, arXiv:1110.5121 [math-ph], NONLINEAR PHENOMENA IN COMPLEX SYSTEMS 2012, Vol.15, No.2, pp.163-170

Yves Gaspar,New transformation law for Heun and Hypergeometric Equations, arXiv:math-ph/0305050

Kouichi Takemura, "Heun equation and Painlevé equation", arXiv:math/0503288 [math.CA]

J.Abad, F. J. Gomez, J. Sesma, "An algorithm to obtain global solutions of the double confluent Heun equation", Numerical Algorithms 49 (2008) 33-51, arXiv:0811.1172 [math.CA]

B. D. Bonorino Figueiredo, "Generalized spheroidal wave equation and limiting cases", J.Math.Phys. 48 (2007) 013503, arXiv:math-ph/0611048

A.O. Smirnov, "Elliptic solitons and Heun's equation", CRM Proceedings and Lecture Notes, 2002, v.32, p.287-305. arXiv:math/0109149 [math.CA]

Kouichi Takemura, "Integral representation of solutions to Fuchsian system and Heun's equation", arXiv:0705.3358 [math.CA]

David Garcia-Alvarez, "A comparison of a few numerical schemes for the integration of stochastic differential equations in the Stratonovich interpretation", arXiv:1102.4401 [physics.comp-ph]

Anthony J Guttmann, "Lattice Green functions in all dimensions", arXiv:1004.1435 [math-ph]

Mahouton Norbert Hounkonnou, André Ronveaux, Komi Sodoga, "Factorization of some confluent Heun's differential equations.", Applied Mathematics and Computation 189(1): 816-820 (2007),

Mahouton Norbert Hounkonnou, André Ronveaux: About derivatives of Heun's functions from polynomial transformations of hypergeometric equations. Applied Mathematics and Computation 209(2): 421-424 (2009), arXiv:0902.2991 [math-ph],

Kouichi Takemura, "Integral transformation of Heun's equation and some applications", arXiv:1008.4007 [math.CA]

Kouichi Takemura, "The Heun equation and the Calogero-Moser-Sutherland system I: the Bethe Ansatz method", Commun. Math. Phys. 235 (2003) 467-494, arXiv:math/0103077 [math.CA]

S.I. Tertychniy, "The modelling of a Josephson junction and Heun polynomials", arXiv:math-ph/0601064

Robert S. Maier, "On Rationally Parametrized Modular Equations", J. Ramanujan Math. Soc. 24 (2009), 1-73, arXiv:math/0611041 [math.NT]

Edwin Langmann, Kouichi Takemura, "Source identity and kernel functions for Inozemtsev-type systems", arXiv:1202.3544 [math-ph]

Yu. V. Brezhnev, "A note on Chudnovsky's Fuchsian equations", Journal of Differential Equations (2012) v.253, 3727-3751, arXiv:1011.0925 [math.CA]

Avinash Khare, Uday Sukhatme, "Quasi-Periodic Solutions of Heun's Equation", arXiv:math-ph/0505077

Bartolomeu D. B. Figueiredo, "Integral relations and new solutions to the double-confluent Heun equation", arXiv:math-ph/0402071

Kouichi Takemura, "Middle Convolution and Heun's Equation", SIGMA 5 (2009), 040, arXiv:0810.3112 [math.CA]

Davide Batic, Runako Williams, Marek Nowakowski, "Potentials of the Heun class", arXiv:1212.0448 [math-ph]

S. Oblezin, "Discrete symmetries of isomonodromic deformations of order two Fuchsian differential equations", arXiv:math-ph/0306009

Hiroyuki Ochiai, "A special value of the spectral zeta function of the non-commutative harmonic osciallators", arXiv:math/0507111 [math.RT]

A. Bostan, S. Boukraa, S. Hassani, J.-M. Maillard, J.-A. Weil, N. Zenine, "Globally nilpotent differential operators and the square Ising model", 10.1088/1751-8113/42/12/125206, arXiv:0812.4931 [math-ph]

Yurii V. Brezhnev, "The sixth Painleve transcendent and uniformization of algebraic curves", arXiv:1011.1645 [math.CA]

S.Yu. Slavyanov, A.Ya. Kazakov, F. R. Vukajlovic, "Relations Between Heun Equations And Painleve Equations", Albanian Journal Of Mathematics Volume 4, Number 4, Pages 195-201 ISSN 1930-1235: (2010)

V. A. Shahnazaryana,c, T. A. Ishkhanyanc, T. A. Shahverdyanc, A. M. Ishkhanyan, "New relations for the derivative of the confluent heun function", Armenian Journal of Physics, 2012, vol. 5, issue 3, pp. 146-155

C.-L. Ho , R. Sasaki and K. Takemura, "Confluence of apparent singularities in multi-indexed

orthogonal polynomials: the Jacobi case", arXiv:1210.0207 [math.CA]

Yuan, Qua, "Finding all Bessel type solutions for linear differential equations with rational function coefficients", THE FLORIDA STATE UNIVERSITY, 2012, 114 pages; 3519384, http:// gradworks.umi.com/35/19/3519384.html

Wolfgang Lay and Sergei Yu Slavyanov, "The central two-point connection problem for the Heun class of ODEs", J. Phys. A: Math. Gen. 31 4249

W. Lay, A. Ya. Kazakovb S. Yu. Slavyanov, "Eigenvalue problems for equations of the Heun class", Algebra i Analiz, 1996, Volume 8, Issue 2, Pages 129-141

S Yu Slavyanov, "Painlevé equations as classical analogues of Heun equations", 1996 J. Phys. A: Math. Gen. 29 7329

N. Gurappa, Prasanta K. Panigrahi, "On polynomial solutions of Heun equation", J.Phys. A37 (2004) L605-L608, arXiv:math-ph/0410015

Vincent X. Genest, Mourad E.H. Ismail, Luc Vinet, Alexei Zhedanov, "The Dunkl oscillator in the plane II : representations of the symmetry algebra", arXiv:1302.6142 [math-ph]

Francois Batola, Jomo Batola, "Integral Representation Of A Solution Of Heun's General Equation", Electronic Journal of Differential Equations, Vol. 2007(2007), No. 94, pp. 1-11. ISSN: 1072-6691

R. Farnoosh, H. R. Rezazadeh, J. Damirchi,"Numerical solution of Heun equation via linear stochastic differential equation" 3rd Conference on Financial Mathematics and Applications. 3031 January 2013, Semnan University, Semnan, Iran

Léa Jaccoud El-Jaick and Bartolomeu D. B. Figueiredo, "Confluent Heun equations: convergence of solutions in series of coulomb wavefunctions", 2013 J. Phys. A: Math. Theor. 46 085203

Dieter Schmidt and Gerhard Wolf, "A Method Of Generating Integral Relations By The Simultaneous Separability Of Generalized Schrodinger Equaitons", SIAM J. Math. Anal. vol. 10 no.4, July 1979

Mark van Hoeij, Raimundas Vidunas, "Belyi functions for hyperbolic hypergeometric-to-Heun transformation", arXiv:1212.3803 [math.AG]

G. Felder, Th. Willwacher, "Jointly orthogonal polynomials", arXiv:1304.4482 [math.QA]

D. Caruntu, " Factorization of self-adjoint ordinary differential equations", AMC 17937, February 2013

Q. S. A. Al-Zamil, "Hankel Determinant structure of the Rational Solutions for Fifth Painlevé Equation", lied Mathematical Sciences, Vol. 7, 2013, no. 9, 445 - 452

Masato Wakayama, "Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun?s differential equation", Proc. Japan Acad. Ser. A Math. Sci. Volume 89, Number 6 (2013), 69-73.

Mark van Hoeij, Raimundas Vidunas, "Algorithms and differential relations for Belyi functions", arXiv:1305.7218 [math.AG]

Mahouton Norbert Hounkonnou, Andre Ronveaux, "On a polynomial transformation of hypergeometric equations, Heun's differential equation and exceptional Jacobi polynomials", arXiv:1306.4889 [math-ph]

Arunesh Roy, Abhijit Sen, Prasanta K. Panigrahi, "Quasi-Exact Solvability and Deformations of Sl(2) Algebra", arXiv:1304.2225 [math-ph]

M. Wakayama, "Equivalence between the eigenvalue problem of non-commutative harmonic oscillators and existence of holomorphic solutions of Heun's differential equations, eigenstates degeneration, and Rabi's mode", MI Preprint Series

Alexey Litvinov, Sergei Lukyanov, Nikita Nekrasov, Alexander Zamolodchikov, "Classical Conformal Blocks and Painleve VI", arXiv:1309.4700 [hep-th]

Runako Williams, Davide Batic, "The Two-Point Connection Problem for a Sub-Class of the Heun Equation", arXiv:1310.4587 [math.CA]

Yik-Man Chiang, Guo-Fu Yu, "On complex oscillation theory, quasi-exact solvability and Fredholm Integral Equations", arXiv:1310.5507 [math.CA]

Kazuo Kaneko, "Special solutions and linear monodromy for the two dimensional degenerate Garnier system G(1112)", arXiv:1310.2006 [math.CA]

Jeroen Sijsling And John Voight,"On Computing Belyi Maps", dartmouth.edu

Raimundas Vidunas,"Degenerate and dihedral Heun functions with parameters", arXiv:1309.7656 [math.CA]

S. Y. Slavyanov,"Relations Between Linear Equations and Painlevé’s Equations", Constructive Approximation , DOI:10.1007/s00365-013-9216-0

L. J. El-Jaick, B. D. B. Figueiredo,"Integral relations for solutions of confluent Heun equations", arXiv:1311.3703 [math-ph]

Lea Jaccoud El-Jaick, Bartolomeu D. B. Figueiredo, "New solutions to the confluent Heun equation and quasiexact solvability", arXiv:1311.7677 [math-ph]

Tolga Birkandan, "Physical Examples of the Heun-to-Hypergeometric Reduction", arXiv:1401.0449 [math-ph]

Pietro Menotti, "On the monodromy problem for the four-punctured sphere", arXiv:1401.2409 [hep-th]

T.A. Ishkhanyan, A.M. Ishkhanyan, "Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions", arXiv:1403.1833 [math.CA]

T.A. Ishkhanyan, A.M. Ishkhanyan, "Confluent hypergeometric expansions of the solutions of the double-confluent Heun equation", arXiv:1403.1884 [math.CA]

A.M. Ishkhanyan, A.E. Grigoryan, "Fifteen classes of solutions of the quantum two-state problem in terms of the confluent Heun function", arXiv:1402.1326 [physics.atom-ph]

A.M. Ishkhanyan, A.E. Grigoryan, "Time-dependent level crossing models solvable in terms of the confluent Heun functions", arXiv:1402.1330 [physics.atom-ph]

W. Robin, "Frobenius Series Solution of Fuchs Second-Order Ordinary Differential Equations via Complex Integration ", http://dx.doi.or g/10.12988/imf.2014.4491

A.M. Ishkhanyan , "The Appell hypergeometric expansions of the solutions of the general Heun equation", arXiv:1405.2871 [math-ph]

P. P Fiziev , "Novel representation of the general Heun's functions", arXiv:1405.6837 [math-ph]

A.M. Manukyan, M.V. Hakobyan , "A series solution of the general Heun equation in terms of incomplete Beta functions", International Journal of Differential Equations and Applications 13(4), 231-239 (2014)

K. Aydemir , "Boundary value problems with eigenvalue-dependent boundary and transmission conditions", Boundary Value Problems 2014, 2014:131

M.A. Gonzalez Leon, J. Mateos Guilarte, A. Moreno Mosquera, M. de la Torre Mayado , "On the Quasi-Exact Solvability of the Confluent Heun Equation", arXiv:1406.2643 [math-ph]

H. Karayer, D. Demirhan and F. Buyukkilic , "A particular solution of Heun equation for Hulthen and Woods-Saxon potential", Annalen der Physik, Vol. 526, Issue 11-12, pages 527–532, December 2014

arXiv:1406.2643 [math-ph]T.A. Ishkhanyan, T.A. Shahverdyan, A.M. Ishkhanyan , "Hypergeometric expansions of the solutions of the general Heun equation governed by two-term recurrence relations for expansion coefficients", arXiv:1403.7863 [math.CA]

A. Intissar, T.A. Shahverdyan, A.M. Ishkhanyan , "On the zeros of some families of polynomials satisfying a three-term recurrence associated to Gribov operator", arXiv:1404.3499 [math-ph]

Y. Alavi, P-F. Hsieh, "Trends and Developments in Ordinary Differential Equations ", Trends and Developments in Ordinary Differential Equations Proceedings of the International Symposium. Trends and Developments in Ordinary Differential Equations: pp. 1-426.

W. He, "A new treatment for some periodic Schrödinger operators ", arXiv:1412.6776 [math-ph]

D. Batic, D. Mills-Howell, M. Nowakowski, "Potentials of the Heun class: the triconfluent case", arXiv:1412.3847 [math-ph]

T.A. Shahverdyan, T.A. Ishkhanyan, A.E. Grigoryan, A.M. Ishkhanyan, "Analytic solutions of the quantum two-state problem in terms of the double-, bi- and tri-confluent Heun functions ", arXiv:1412.1378 [quant-ph]

A.M. Ishkhanyan, "Series solutions of confluent Heun equations in terms of incomplete Gamma-functions", arXiv:1412.0654 [math-ph]

C. Leroy, Y. Pashayan-Leroy, A.M. Ishkhanyan, "Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions", arXiv:1410.8575 [math-ph]

P. P Fiziev , "Novel representation of the general Heun's functions. Back to the beginning", Proceedings of the International Conference DAYS on DIFFRACTION 2014, pp. 76-79

A. Eremenko, A. Gabrielov, V. Tarasov , "Metrics with four conic singularities and spherical quadrilaterals", arXiv:1409.1529 [math.CV]

I. Rasa , "Special functions associated with positive linear operators", arXiv:1409.1015 [math.CA]

W. Miller and A. V. Turbine , "(Quasi)-exact-solvability on the sphere Sn", pdf notes

B. Shapiro And Milos Tater , "On Spectral Asymptotics Of Quasi-Exactly Solvable Quartic And Yablonskii-Vorob'Ev Polynomials", pdf notes

M. Dariescu, D. Mihu, C. Dariescu , "Stationary Solutions To Kompaneets Equation For Relativistic Processes In Astrophysical Objects ", Rom. Journ. Phys., Vol. 59, N os. 3–4, P. 224–232, Bucharest, 2014

S.Yu.Slavyanov, "Antiquantization and corresponding symmetries", pdf only

A. Eremenko, A. Gabrielov and V. Tarasov, "Spherical polygons and differential equations", pdf only

P. B. Acosta-Humanez, J. T. Lazaro-Ochoa, J. J. Morales-Ruiz, C. Pantazi, "On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory", doi:10.3934/dcds.2015.35.1767

L. Jaccoud El-Jaick and B. D. B. Figueiredo, C. Pantazi, "Integral relations for solutions of the confluent Heun equation", Applied Mathematics and Computation Vol. 256, 1 April 2015, Pages 885–904

A. Martinez-Finkelshtein, E. A. Rakhmanov, and S. P. Suetin, "Asymptotics of type I Hermite(Pade polynomials for semiclassical functions)", arXiv:1502.01202v2 [math.CA]

C. Leroy & A.M. Ishkhanyan, "Expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta and the Appell generalized hypergeometric functions", Integral Transforms and Special Functions Vol. 26, Issue 6, 2015

G. Natanson, "Single-Source Nets of Algebraically-Quantized Reflective Liouville Potentials on the Line I. Almost-Everywhere Holomorphic Solutions of Rational Canonical Sturm-Liouville Equations with Second-Order Poles", arXiv:1503.04798 [math-ph]

C.Lauter and M. Mezzarobba, "Semi-Automatic Floating-Point Implementation of Special Functions", author's website

A. Eremenko, A. Gabrielov and V. Tarasov, "Spherical quadrilaterals with three non-integer angles", author's website

M. N. Hounkonnou and A. Ronveaux, "On some differential transformations of hypergeometric equations", 2015 J. Phys.: Conf. Ser. 597 012044

H. Karayer, D. Demirhan, F. Buyukkilic , "Extension of Nikiforov-Uvarov Method for the Solution of Heun Equation", arXiv:1504.03518 [math-ph]

A.V. Shanin, R.V. Craster, "Removing false singular points as a method of solving ordinary differential equations", Euro. Jnl of Applied Mathematics 2002, vol. 13, pp. 617-639.

L.J. El-Jaick, B.D.B. Figueiredo , "Integral relations for solutions of the confluent Heun equation [CBPF-NF--001/2015]", [pdf]

Y.M. Chiang, C.Y. Tsang, A. Ching "Symmetries of the Darboux equation", arXiv:1509.03995

V.M. Buchstaber, A.A. Glutsyuk "On determinants of modified Bessel functions and entire solutions of double confluent Heun equations", arXiv:1509.01725

N. Saad, S.B. Medina "On the solvability of confluent Heun equation and associated orthogonal polynomials" , arXiv:1509.00090

R. Mendez-Fragoso, E. Ley-Koo "Angular Momentum Theory in Bases of Lamé Spheroconal Harmonics" , Advances in Quantum Chemistry, 2015

F. Novaes, B.C. da Cunha "Isomonodromic Approach to Scattering by Black Holes" , [pdf]

O.V. Motygin "On evaluation of the Heun functions", arXiv:1506.03848

D. BATIC, R. WILLIAMS " THE TWO-POINT CONNECTION PROBLEM FOR A SUB-CLASS OF THE HEUN EQUATION", Journal of Inequalities & Special Functions, 2015

L. Jaccoud El-Jaick and B. D. B. Figueiredo "Convergence and applications of some solutions of the confluent Heun equation", Notas de Fisica (2016)

V. M. Bukhshtaber , S. I. Tertychnyi "On a remarkable sequence of Bessel matrices", Mathematical Notes (2016)

H. Karayer et al. "Some Special Solutions of Biconfluent and Triconfluent Heun Equations in Elementary Functions by Extended Nikiforov–Uvarov Method", Reports on Mathematical Physics (2015)

F.ALBERTO GRUNBAUM et al. "TRIDIAGONALIZATION AND THE HEUN EQUATION", arXiv:1602.04840

C. FERREIRA et al. "Second order differential equations with an irregular singularity at the origin and a large parameter: convergent and asymptotic expansions", CARPATHIAN J. (2016)

FP Petkov - Rn, 2016, "Novel representation of the general Fuchsian and Heun equations and their solutions" https://www.scipress.com/IFSL.7.11.pdf

S. Yu. Slavyanov1, DA Shat'ko1, AM Ishkhanyan, "Generation and removing apparent singularities in linear ordinary differential equations with polynomial coefficients "

P. P. Fiziev, "A new approach to the connection problem for local solutions to the general Heun" arXiv:1606.08539 [math-ph]

Yoon Seok Choun "The violation of uniqueness theorems for power series solutions by applying the Poincare-Perron theorem" arXiv:1605.08960 [math.CA]

Farrokh Atai, Edwin Langmann "Series solutions of the non-stationary Heun equation" arXiv:1609.02525 [math-ph]

Victor M. Buchstaber, Alexey A. Glutsyuk "On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect" Proc. Steklov Inst. of Math

Kouichi Takemura "Degenerations of Ruijsenaars-van Diejen operator and q-Painleve equations" arXiv:1608.07265 [math-ph]

A. Ya. Kazakov "Confluent Heun equation with single added apparent singularity" IEEE

Y. Abdelaziz, J.-M. Maillard "Modular forms, Schwarzian conditions, and symmetries of differential equations in physics" arXiv:1611.08493 [math-ph]

Tomas Andrade, Alejandra Castro, Diego Cohen-Maldonado "The Spectrum of Static Subtracted Geometries" arXiv:1611.09330 [hep-th]

Alexandre Eremenko, Andrei Gabrielov "Circular pentagons and real solutions of Painleve VI equations" arXiv:1611.01356 [math.CV]

T. A. Shahverdyan, V. M. Red’kov, and A. M. Ishkhanyan, "Expansions of the Solutions of the General Heun Equation in Terms of the Incomplete Beta Functions " NONLINEAR PHENOMENA IN COMPLEX SYSTEMS

S. Yu. Slavyanov, O. L. Stesik, "Antiquantization of deformed Heun-class equations" Theoretical and Mathematical Physics > 2016 > 186 > 1 > 118-125

### Articles with applications of the Heun functions

M. Hortacsu, "Heun Functions and their uses in Physics", arXiv:1101.0471 [math-ph]

D. Momeni, Koblandy Yerzhanov, Ratbay Myrzakulov, "Quantized Black Hole and Heun function", Can. J. Phys. 90: 877-881 (2012), arXiv:1009.0130 [physics.gen-ph]

Plamen P. Fiziev, Denitsa R. Staicova, "Solving systems of transcendental equations involving the Heun functions", AJCM. 02 : 02, pp.95 (2012) DOI:10.4236/ajcm.2012.22013, arXiv:1201.0017 [cs.NA]

Plamen Fiziev, Denitsa Staicova, "Application of the confluent Heun functions for finding the quasinormal modes of nonrotating black holes", Phys. Rev. D 84, 127502 (2011), arXiv:1109.1532 [gr-qc]

T.Birkandan and M. Hortacsu, "Examples of Heun and Mathieu Functions as Solutions of Wave Equations in Curved Spaces", arxiv: 0607108 [gr-qc]

M. S. Cunha, H. R. Christiansen, "Confluent Heun functions in gauge theories on thick braneworlds", Phys.Rev.D84:085002,2011, arXiv:1109.3486 [hep-th]

A.Ralko, T.T.Truong, "Heun Functions and the energy spectrum of a charged particle on a sphere under magnetic field and Coulomb force", JOPA, DOI:10.1088/0305-4470/35/45/306, arXiv:quant-ph/0209152

Yongjoon Kwon, Soonkeon Nam, Jong-Dae Park, Sang-Heon Yi, "Quasi Normal Modes for New Type Black Holes in New Massive Gravity", arXiv:1102:0138

T.Birkandan and M. Hortacsu, "Heun and Mathieu functions as solutions of the Dirac equation", arXiv:0710.4243

Norio Konno, Takuya Machida, Tohru Wakasa, "The Heun differential equation and the Gauss differential equation related to quantum walks", arXiv:1109.2662 [quant-ph], YMJ (2011)

C. A. Downing, "On a solution of the Schrodinger equation with a hyperbolic double-well potential", arXiv:1211.0913, J. Math. Phys. 54, 072101 (2013)

Richard L. Hall, Nasser Saad, and K. D. Sen, "Soft-core Coulomb potentials and Heun's differential equation", arXiv:0912.3445

Patrick Dorey, Junji Suzuki, Roberto Tateo, "Finite lattice Bethe ansatz systems and the Heun equation", J.Phys.A37:2047-2062,2004,arXiv:hep-th/0308053

Roumen S. Borissov, Plamen P. Fiziev, "Exact Solutions of Teukolsky Master Equation with

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M.-A. Dariescu, C. Dariescu, . Cretu, O. Buhucianu, "?nalytic Study Of Fermions In Graphene; Heun Functions And Beyond", Rom. Journ. Phys., Vol. 58, Nos. 7-8, P. ppi?ppf, Bucharest, 2013

M. Renardy, "On the eigenfunctions for Hookean and FENE dumbbell models ", J. Rheol. 57, 1311 (2013)

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Marina-Aura Dariescu And Ciprian Dariescu, "Polynomial Solutions Of Heun Equation Describing Fermions In Graphene", Int. J. Mod. Phys. B DOI: 10.1142/S0217979213501907

C. S. Lopez-Monsalvo, F. Nettel, H. Quevedo , "Conformally invariant thermodynamics of a Maxwell-Dilaton black hole", General Relativity and Gravitation, 10.1007/s10714-013-1604-z

Marina-Aura Dariescu And Ciprian Dariescu, "On The Mathieu Equation Describing Ultra-Relativistic Fermions Evolving In A Time-Harmonic Electric Field Parallel To A Static Magnetic Induction", Mod. Phys. Lett. A DOI: 10.1142/S0217732313501575

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E.M. Ovsiyuk, O.V. Veko, K.V. Kazmerchuk, V.V. Kisel, V.M. Red’kov, "Quantum Mechanics Of A Spin 1 Particle In The Magnetic Monopole Potential, In Spaces Of Euclid And Lobachevsky: Non-Relativistic Approximation", ISSN 2071-0194. Ukr. J. Phys. 2013. Vol. 58, No. 11

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M. T. Arredondo, "Mathematical model for solar drying of potato cylinders with thermal conductivity radially modulated", Proc. SPIE 9108, Sensing for Agriculture and Food Quality and Safety VI, 91080N (May 28, 2014);

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O.V. Veko, K.V. Kazmerchuk, E.M. Ovsiyuk, V.M. Red'kov, A.M. Ishkhanyan, "Hydrogen atom in de Sitter spaces", arXiv:1410.8344 [math-ph]

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G. Natanson, "Heun-Polynomial Representation of Regular-at-Infinity Solutions for the Basic SUSY Ladder of Hyperbolic Poschl-Teller Potentials Starting from the Reflectionless Symmetric Potential Well", arXiv:1410.1515 [math-ph]

A. Schulze-Halberg, J. Wang, "Bound State Solutions of the Klein-Gordon Equation for the Mathews-Lakshmanan Oscillator", Few-Body Systems December 2014, Volume 55, Issue 12, pp 1223-1232

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M. Wakayama, "Remarks on Quantum Interaction Models by Lie Theory and Modular Forms via Non-commutative Harmonic Oscillators", Springer Japan : 17-34, ISBN: 978-4-431-55060-0

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M. Bednarik, M. Cervenka, R. Balek, "Acoustic plane waves in a gas-filled duct with an axial temperature gradient", Forum Acusticum, 7-12.09.2014, Krakow

O. L. Petersen, "The wave equation and redshift in Bianchi type I spacetimes", Master of Science Thesis, Stockholm, Sweden 2014

J. J. Cadavid Munoz, "Generalized Electroosmosis transport with a spatially modulated electric permittivity inside a cylindrical micro channel ", Proc. SPIE 9107, Smart Biomedical and Physiological Sensor Technology XI, 91070K (May 22, 2014); doi:10.1117/12.2048847

D. Bykov, "Ricci-flat metrics on cones over toric complex surfaces", Quarks-2014 Proceedings

P. Valtancoli, "Hawking radiation for a scalar field conformally coupled to an AdS black hole", arXiv:1502.01716v1 [gr-qc]

D. Bouaziz , "Kratzer’s molecular potential in quantum mechanics with a generalized uncertainty principle", Annals of Physics, Vol. 355, April 2015, Pages 269–281

H. S. Vieira, V. B. Bezerra , "Quantum Newtonian cosmology and the biconfluent Heun functions", arXiv:1502.03071 [gr-qc]

V. M. Buchstaber, S. I. Tertychnyi , "Holomorphic solutions of the double confluent Heun equation associated with the RSJ model of the Josephson junction ", Teoret. Mat. Fiz., 2015, Volume 182, Number 3, Pages 373–404

O. L. Petersen, "The mode solution of the wave equation in Kasner spacetimes and redshift", arXiv:1503.02411 [math-ph]

J. R. F. Lima, Fernando Moraes , "The combined effect of inertial and electromagnetic fields in a fullerene molecule", The European Physical Journal B, March 2015, 88:63

O. Christov, G. Georgiev, "On the integrability of a system describing the stationary solutions in Bose--Fermi mixtures", arXiv:1503.08171 [math-ph]

D. Philipp, V. Perlick, "Schwarzschild radial perturbations in Eddington-Finkelstein and Painlevé-Gullstrand coordinates", arXiv:1503.08361 [gr-qc]

D. Philipp, V. Perlick, " On analytic solutions of wave equations in regular coordinate systems on Schwarzschild background", arXiv:1503.08101 [gr-qc]

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H. S. Vieira, V. B. Bezerra and Andre A. Costa, "Scalar fields in the Lense-Thirring background with a cosmic string and Hawking radiation ", 2015 EPL 109 60006

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A. Chichurin, E. Ovsiyuk, A.Red’ko, V. Red’kov, "Spin 1 particle in the Coulomb field on the background of Lobachevsky geometry: general Heun functions, analytical and numerical study", Computer Algebra Systems in Teaching and Research vol. 5

H.S. Vieira , "Quasinormal modes of the hydrodynamic vortex", arXiv:1510.08298

M. Bednarik, M. Cervenka, P. Lotton, G. Penelet, "Behavior of plane waves propagating through a temperature-inhomogeneous region", Journal of Sound and Vibration

A.M. Ishkhanyan, "Exact solution of the Schrodinger equation for the inverse square root potential", Europhysics Letters, 2015

C. R. Muniz, H. S. Vieira, M. O. Tahim, M. S. Cunha, "Global Casimir Effect in the Schwarzschild Spacetime", arXiv:1510.06655 [gr-qc]

A.S. Tarloyan, T.A. Ishkhanyan, A.M. Ishkhanyan, "Four five-parametric and five four-parametric independent confluent Heun potentials for the stationary Klein-Gordon equation", arXiv:1510.03700 [quant-ph]

T. Kereselidze, G. Chkadua, P. Defrance, J.F. Ogilvie, "Derivation, properties and application of Coulomb Sturmians defined in spheroidal coordinates", Molecular Physics, 2015

M.T. Batchelor, Z.M. Li, H.Q. Zhou, "Energy landscape and conical intersection points of the driven Rabi model", arXiv:1509.06572

D. Braak "Analytical solutions of basic models in quantum optics", arXiv:1509.05748

D. Bini, E. Bittencourt, A. Geralico "Massless Dirac particles in the vacuum C-metric", arXiv:1509.04878

A.E. Sitnitsky "Probability distribution function for reorientations in Maier-Saupe potential", arXiv:1509.03439

L. Xin, S. Yang, D. Zhou, G. Dui "An approximate analytical solution based on the Mori-Tanaka method for functionally graded thick-walled tube subjected to internal pressure", Composite Structures, Volume 135, 2016

A.M. Ishkhanyan "The Lambert W-barrier-an exactly solvable confluent hypergeometric potential", arXiv:1509.00846

T.A. Ventimiglia "Heun Polynomials in the Construction of Vector Valued Slepian Functions on a Spherical Cap", thesys

A. Frolova, A. Vasil'ev "Combinatorial description of jumps in spectral networks", arXiv:1509.00674

A.M. Ishkhanyan "Exact solution of the Schr\" odinger equation for the potential $ V_0/{\ sqrt {x}} $", arXiv:1509.00019

H.S. Vieira, V.B. Bezerra, G.V. Silva "Analytic solutions in the dyon black hole with a cosmic string: Scalar fields, Hawking radiation and energy flux", Annals of Physics, 2015

A. Ishkhanyan, V. Krainov "Discretization of Natanzon potentials", arXiv:1508.06989

H. Neyazi, A.A. Rajabi, H. Hassanabadi "Exactly separable Bohr Hamiltonian with the Killingbeck potential for triaxial nuclei", Nuclear Physics A, 2015

D. Batic, K. Morgan, M. Nowakowski, S.B. Medina " The Dirac equation in the Kerr-de Sitter metric", arXiv:1509.00452

A. Eremenko, A. Gabrielov "On metrics of curvature 1 with four conic singularities on tori and on the sphere", arXiv:1508.06510

C. Kallendorf, A. Fath, M. Oberlack, Y. Wang "Exact solutions to the interfacial surfactant transport equation on a droplet in a Stokes flow regime", Physics of Fluids (1994-present), 2015

G. Natanson "Dutt-Khare-Varshni Potential and Its Quantized-by-Heun-Polynomials SUSY Partners as Nontrivial Examples of Solvable Potentials Explicitly Expressible in Terms of Elementary Functions", arXiv:1508.04738

O.V. Veko et al. "Spin 1 Particle in the Magnetic Monopole Potential for Minkowski and Lobachevsky Spaces: Nonrelativistic Approximation", NONLINEAR PHENOMENA IN COMPLEX SYSTEMS, An Interdisciplinary Journal, 2015, Vol.18, No.2, pp.243-258

L. Mao, S. Huai, Y. Zhang "The two-qubit quantum Rabi model: inhomogeneous coupling", Journal of Physics A: Mathematical and Theoretical, 2015

M. Bednarik, M. Cervenka "DESCRIPTION AND ANALYSIS OF PLANE WAVES IN A TEMPERATURE-INHOMOGENEOUS REGION", [pdf]

B.C. da Cunha, M.C. de Almeida, A.R. de Queiroz "On the Existence of Monodromies for the Rabi model", arXiv:1508.01342

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B.C. da Cunha, F. Novaes "Scattering Amplitudes for Near-Extremal Kerr Geometries", arXiv:1506.06588

Z.M. Li, M.T. Batchelor "Algebraic equations for the exceptional eigenspectrum of the generalised Rabi model", arXiv:1506.04038

S. Hod "The large-mass limit of cloudy black holes", Classical and Quantum Gravity, 2015

J. Karwowski, H.A. Witek "Analytical Solutions of the Schrödinger Equation with Power Potentials", Journal of the Chinese Chemical Society, 2015

S.R. Roy, D. Sarkar "Hologram of a pure state black hole", arXiv:1505.03895

M. Jiang, X. Wang, Z. Li "Spin field equations and Heun's equations", Astrophysics and Space Science, 2015

M.T. Batchelor, H.Q. Zhou "Integrability versus exact solvability in the quantum Rabi and Dicke models", Physical Review A, 2015

G.H. Sun, S.H. Dong, K.D. Launey, T Dytrych, JP Draayer "Shannon information entropy for a hyperbolic double‐well potential", [pdf]

P. Fiziev "A realistic model of neutron star in minimal dilatonic gravity ", arXiv:1506.08585, 2015

D. Staicova, P. Fiziev "Numerical stability of the electromagnetic quasinormal and quasibound modes of Kerr black holes", Bulg. Astron. Journal vol 23. 2015

L.S. Ridao, M. Bellini "Towards relativistic quantum geometry ", arXiv:1506.09141

M.A. Fiset, J. Walcher " Bounding the Heat Trace of a Calabi-Yau Manifold", arXiv:1506.08407

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Return to ContentsM. A. Gonzalez Leon, J. Mateos Guilarte, M. de la Torre Mayado "Elementary solutions of the quantum planar two center problem", arXiv: 1603.05454

M. R. A. Arcodía, M. Bellini "Geometric back-reaction in pre-inflation from relativistic quantum geometry", arXiv: 1603.03821

H. S. Vieira, V. B. Bezerra "Solution of the Wheeler-DeWitt equation and the triconfluent Heun functions", arXiv: 1603.02236

H. S. Vieira, V. B. Bezerra "Application of the confluent Heun functions to study the physics of black holes: resonant frequencies and scattering of scalar waves", arXiv: 1603.02233

H. S. Vieira, V. B. Bezerra "Quantum effects in Reissner-Nordström black hole surrounded by magnetic field: the scalar wave case", arXiv: 1603.02240

Wen-Du Li, Wu-Sheng Dai "Exact solution of inverse-square-root potential V(r)=−alpha/sqrt(r)", arXiv: 1603.00792

G. V. Kraniotis "The Klein-Gordon-Fock equation in the curved spacetime of the Kerr-Newman (anti) de Sitter black hole", arXiv: 1602.04830

Q. Xie et al. "Analytical results for non-Hermitian parity–time-symmetric and Hermitian asymmetric volcano potentials", PRAMANA (2016)

A.M. Ishkhanyan "The third exactly solvable hypergeometric quantum-mechanical potential ", arXiv: 1602.07685

L. Duan et al. "Two-photon Rabi model: Analytic solutions and spectral collapse", arXiv: 1603.04503

V.G. Marikhin "Two new integrable cases of two-dimensional quantum mechanics with a magnetic field", arXiv: 1602.03375

E. Renzi , P. Sammarco "The hydrodynamics of landslide tsunamis: current analytical models and future research directions", Landslides (2016)

P. M. T. Barboza, K. Bakke "On the Aharonov-Casher system subject to a Coulomb-type potential", EPJ Plus (2016)

A. Lopez-Ortega "New conditionally exactly solvable potentials of exponential type", arXiv:1602.00405

M. Assis, M. van Hoeij and J-M. Maillard "The perimeter generating functions of three-choice, imperfect, and 1-punctured staircase polygons", arXiv:1602.00868

F. Ferrari, M.R. Piątek, A.R. Pietrykowski "2d CFT/Gauge/Bethe correspondence and solvable quantum-mechanical systems", Journal of Physics (2016)

A. Eremenko and A. Gabrielov "Spherical rectangles", arXiv:1601.04060

M.A. Escobar-Ruiz, Willard Miller Jr "Conformal Laplace superintegrable systems in 2D: polynomial invariant subspaces", arXiv:1602.03149

A. B. Oliveira, K. Bakke "On the effects on a Landau-type system for an atom with no permanent electric dipole moment due to a Coulomb-type potential", Annals of Physics (2016)

P. Fiziev "The Heun’s functions as a modern powerful tool for research in different scientific domains", arXiv:1512.04025

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B. C. da Cunha and F. Novaes "Kerr-de Sitter Greybody Factors via Isomonodromy", arXiv:1508.04046

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Return to Contents### Numerical evaluations and codes

E.M. Ferreira, J. Sesma "Global solutions of the biconfluent Heun equation" , Numerical Algorithms, 2015

Oleg V. Motygin, "On evaluation of the Heun functions", arXiv:1506.03848 [math.NA] , source-code Return to Contents### Others (we are in process of reorganizing the categories of the bibliography)

1) V.I.Vlasov, D.B.Volkov. "Inversion problem for an equation of Fuchsian type" // Diff. Equations, 1987. (Translated from Differentsialye Uravneniya, 1986, vol. 22, #11, p. 1854-1865.

2) V.I.Vlasov. "Boundary value problems in domains with curvilinear boundary". Moscow, Computing Centre of Academy of Sciences of the USSSR, 1987.(A monograph in Russian)

3) D.B.Volkov-Bogorodsky. "On integration of Fuch-class equations with four singular points in connection with a conformal mapping problem" // Pitman Research Notes in Mathematics Series, vol. 374: Integral methods in science and engineering. Volume one: analytic methods. 1997, p. 215-219. <

4) V.I.Vlasov, S.L.Skorokhodov. "An efficient analytic-numerical method of conformal mapping of circular arc polygons" // International Conference “Computation Methods and Function Theory”, 2001, Aveiro, Portugal. Book of Abstracts, 2001.

5) S.I.Bezrodnykh. "The Jacoby-type relation for generalized hypergeometric function" // III Conference “P.L.Chebyshev mathematical ideas and their applications to modern problems of natural science”. 2006, Obninsk, Russia. Book of Abstracts, 2006. P.18-19. (In Russian)

6) S.I.Bezrodnykh. "Singular Riemann – Hilbert problem and its application". PhD thesis. Moscow, Dorodnicyn Computing Centre of RAS, 2006.

7) S.I.Bezrodnykh, V.I.Vlasov. "An approach to finding unknown parameters of the Schwarz --- Christoffel integral" // IV Conference “P.L.Chebyshev mathematical ideas and their applications to modern problems of natural science”. 2008, Obninsk, Russia. Book of Abstracts, 2008. P. 9-10. (In Russian)

8)S.I.Bezrodnykh, V.I.Vlasov "To the parameters problem of the Schwarz --- Christoffel integral" // XVII Conference “Theoretical base and construction of numerical algorithms and solution of mathematical physics problems with application to multiprocessor systems”, dedicated to memory of K.I.Babenko. 2008, Durso, Russia. Book of Abstracts, 2008. P. 12-13. (In Russian)

9) V.I.Vlasov. "To the inversion problem for the Fuch-class equation" // International Conference on Differential Equations and Dynamic Systems. 2008, Suzdal, Russia. Book of Abstracts. 2008. P. 59-61. (In Russian)

10) S.I.Bezrodnykh. "On analytic continuation of the Lauricella function" // International Conference on Differential Equations and Dynamic Systems. 2008, Suzdal, Russia. Book of Abstracts. 2008. P. 34-36. (In Russian)

11) V.I.Vlasov. "Inversion of the Schwarz – Christoffel integral" // V International Conference “Differential and functional Differential Equations”. 2008, Moscow, Russia. Book of Abstracts. 2008. P. 91. (In Russian)

12) S.I.Bezrodnykh. "On analytic continuation of some generalized hypergeometric functions" // V International Conference “Differential and functional Differential Equations”. 2008, Moscow, Russia. Book of Abstracts. 2008. P. 88-89. (In Russian)

13) S.I.Bezrodnykh. "Certain formulae of analytic continuation for the Lauricella function" // International Conference “Differential equations and their application”. 2013, Belgorod, Russia. Book of Abstracts. 2013.(In Russian)

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