Geodesic
Series solution of a~ten-parameter second-order differential equation with three regular singularities and one irregular singularity
Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 1, pp. 20-33

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We consider a ten-parameter second-order ordinary linear differential equation with four singular points. Three of them are finite and regular, while the fourth is irregular at infinity. We use the tridiagonal representation approach to obtain a solution of the equation as a bounded infinite series of square-integrable functions written in terms of Jacobi polynomials. The expansion coefficients of the series satisfy a three-term recurrence relation, which is solved in terms of a modified version of the continuous Hahn orthogonal polynomial. We present a physical application in which we identify the quantum mechanical systems that could be described by the differential equation, give the corresponding class of potential functions and energy in terms of the equation parameters, and write the system wave function.
Keywords: differential equation, tridiagonal representation, $J$-matrix method, recurrence relation, continuous Hahn polynomial. 
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     author = {A. D. Alhaidari},
     title = {Series solution of a~ten-parameter second-order differential equation with three regular singularities and one irregular singularity},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {20--33},
     publisher = {mathdoc},
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     number = {1},
     year = {2020},
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     url = {http://geodesic.mathdoc.fr/item/TMF_2020_202_1_a2/}
}
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A. D. Alhaidari. Series solution of a~ten-parameter second-order differential equation with three regular singularities and one irregular singularity. Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 1, pp. 20-33. http://geodesic.mathdoc.fr/item/TMF_2020_202_1_a2/