Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity

Kouichi Takemura; Ryu Sasaki

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Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity

Ryu Sasaki, Kouichi Takemura

Classical Analysis and ODEs · Submitted 23 Jul 2012

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The authors' main claim Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent -2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.

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arXiv ID1207.5302 (version 2)

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CitationKouichi Takemura, Ryu Sasaki. "Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity." arXiv:1207.5302 (2012). https://pith.science/paper/D64MB3KC

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Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent -2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.

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s0	Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent -2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.	contribution 0.61	0.92	0.87	asserted 1.00

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Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent -2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.

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5 arXiv papers cite this in 15 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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The third type $$ is now employed to produce a wider class of polynomials, to be called tentatively non-generic multi-indexed Jacobi polynomials (for the non-generic multi-indexed Laguerre polynomials, see [ref] ).

Confluence of apparent singularities in multi-indexed orthogonal polynomials: the Jacobi case · Pöschl-Teller potential · 0.88

Cites it as background 2 papers

During the last few years, a lot of research activity has been devoted to the construction of new exactly solvable rational extensions of well-known quantum potentials, some of which are connected with the novel field of exceptional orthogonal polynomials (EOP) and with the appearance of a so far unsuspected class of (translationally) shape-invariant (SI) potentials [ref] .

Novel Enlarged Shape Invariance Property and Exactly Solvable Rational Extensions of the Rosen-Morse II and Eckart Potentials · Introduction · 0.98
Soon this exciting hot topic attracted many authors and the rich structures of the subjects [ref] – [ref] , including for example, identities satisfied by orthogonal polynomials [ref] , Fuchsian differential equations aspects [ref] , Darboux transformations [ref] , alternatives of the three term recurrence relations [ref] , etc were revealed.

Exactly Solvable Quantum Mechanics · Bochner's Theorem · 0.85

What it cites · 25 sentences without a confident label

Uses the method or tool of 2 papers

To obtain the deformed Hamiltonian which admits an apparent singularity of exponent $-2$ or higher, we use Darboux–Crum transformations [ref] .

Associated Sturm-Liouville systems · Darboux transformations and induced singularities · 1.00
They are obtained by Darboux–Crum transformations [ref] applied to the exactly solvable Schrödinger equations [ref] of the radial oscillator potential and the Pöschl–Teller potential.

Supersymmetry and Quantum Mechanics · Introduction · 0.94

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This is in good contrast with the multi-indexed Laguerre and Jacobi polynomials defined in [ref] , in which the parameters are restricted (equations (23) and (24) of [ref] ) so that the seed solutions are of definite sign and their inverse are not square integrable [equation] These two additional properties guarantee the regularity of the solutions and the positivity of the weight function.

Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials · “Non-generic" multi-indexed Laguerre polynomials · 0.95
Those polynomials belonging to (A) and (B) were derived in a different way in [ref] .

A conjecture on Exceptional Orthogonal Polynomials · Summary and comments · 0.90

Cites as background 7 papers

The role of Darboux transformations in iso-monodromic deformations has been known for some time [ref] and some explicit examples of monodromy free rational potentials were given in [ref] .

Modular Galois covers associated to symplectic resolutions of singularities · Introduction · 0.92
In recent years, global solutions of Fuchsian differential equations with more than three regular singularities and those of the differential equations of confluent types, attracted attention [ref] .

Classification of Fuchsian systems and their connection problem · Introduction · 0.98
The multi-indexed orthogonal polynomials are also established in the framework of difference Schrödinger equations, the multi-indexed ($q$-)Racah, Wilson and Askey–Wilson polynomials [ref] .

Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations · Summary and comments · 0.81

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