Polynomial Solutions of Brioschi-Halphen Equation

Unanaowo Nyong Bassey, Ubong Sam Idiong


In this paper, we give point canonical transformations (PCT) which map the Brioschi Halphen equation (BHE) into differential equations of complex classical orthogonal polynomials. We also construct exactly solvable potentials, without supersymmetry or factorization and the concept of shape invariance, which give rise to the solutions of the BHE which may be expressed in terms of complex Jacobi polynomials, Laguerre polynomials and Hermite polynomials.

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U.N. Bassey and U.S. Idiong : A Transformation of Lam´e Equation. Jour. of Mathematical Science, Vol. 27, No.1 (2016) 21-34.

U.N. Bassey and U.S. Idiong : Algebraization of Brioschi-Halphen Equation. International Journal of Differential Equations and Applications Volume 15 No. 2(2016)69-76.

L. Gendenshtein: Derivation of Exact Spectra of the Schr¨odinger Equation by means of Supersymmetry. JETP. Lett., Vol.38, No.6 (1983) 356-359.

M.N. Hounkonnou, K. Sodoga and E.S. Azatassou: Factorization of Sturm-Louiville Operators: Solvable Potentials and Underlying algebraic structure. J.Phys. A: Math. Gen. 38(2005) 371-390.

G. L´evai: A search for shape-invariant solvable potentials. J.Phys. A: Math. Gen. 22(1989) 689-702.

G. L´evai: A class of exactly solvable potentials related to the Jacobi polynomials. J.Phys. A: Math. Gen. 24(1991)131-146.

G. L´evai : On some exactly solvable potentials derived from supersymmetric quantum mechanics, Journal of Physics A 25 (1992) L521-L524.

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

C. Quesne: Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry. arXiv:0807.4087v3 [quant-ph] 1 Sep 2008, 1-10.

C. Quesne: Exceptional orthogonal polynomials and new exactly solvable potentials in quantum mechanics. arxiv: 1111.6467v2 [math-ph] 25 Sep 2012, 1-21.

A. Ronveaux (ed.): Heun differential equations. Oxford, UK: Oxford Science Publications; Oxford University Press (1995).

P. Rusev : Classical Orthogonal Polynomials and their Associated Functions in Complex Domain. Sofia (2005).

Z.X. Wang and D.R. Guo: Special Functions. World Scientific Publishing Company Pte Ltd., Singapore (1989).


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