On Rational Solutions of Dressing Chains of Even Periodicity

Aratyn, H.; Gomes, J. F.; Lobo, G. V.; Zímerman, A. H.

doi:10.3390/sym15010249

Cited by 2 publications

(29 citation statements)

References 18 publications

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“…The Young diagrams including hook length corresponding to (A) 𝝀 = (4 2 , 2, 1 3 ) and its core (B) 𝝀 = (2, 1), and corresponding abacus diagrams (C) and (D).…”

Section: Partitionsmentioning

confidence: 99%

“…The 2-core 𝝀 is found from the abacus by sliding all beads vertically up as far as possible and reading off the resulting partition. Figure 1 shows the Young diagram and hooklengths of (4 2 , 2, 1 3 ) in (A), an abacus representation in (C), its 2-core 𝝀 = (2, 1) in (B), and the abacus corresponding to 𝝀 that is obtained from (C) by pushing up all beads.…”

Section: Partitionsmentioning

confidence: 99%

“…Only one bead is moved on runner 2, by one space, and so 𝝂 2 = (1). Therefore, the 2-core and 2-quotient of 𝝀 = (4 2 , 2, 1 3 ) are (2,1) and ((3, 1), ( 1)), respectively.…”

Section: Partitionsmentioning

confidence: 99%

“…It follows from ( 7) that the abacus of the partition 𝚲(3, (4, 2, 1)) has beads in places 2,6,12 and 1,3,5,7,9,11. Therefore, 𝐡 𝚲 = (12,11,9,7,6,5,3,2,1) and thus 𝚲(3, (4, 2, 1)) = (4 2 , 3, 2 3 , 1 3 ). In Section 7, we use the first column hook set (7) to determine an explicit formula for the family of partitions with 2-core 𝑘 and 2-quotient (((𝑚 + 1) 𝑛 ), ∅).…”

Section: Partitionsmentioning

confidence: 99%

“…(ii) Some rational solutions of sP V (80) are given in Refs. 3,23,24, where a different normalization of the symmetric system is used.…”

Section: Rational Solutions Of the Symmetric P 𝐕 Systemmentioning

confidence: 99%

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Rational solutions of the fifth Painlevé equation. Generalized Laguerre polynomials

Clarkson,

Dunning

2023

Stud Appl Math

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In this paper, rational solutions of the fifth Painlevé equation are discussed. There are two classes of rational solutions of the fifth Painlevé equation, one expressed in terms of the generalized Laguerre polynomials, which are the main subject of this paper, and the other in terms of the generalized Umemura polynomials. Both the generalized Laguerre polynomials and the generalized Umemura polynomials can be expressed as Wronskians of Laguerre polynomials specified in terms of specific families of partitions. The properties of the generalized Laguerre polynomials are determined and various differential‐difference and discrete equations found. The rational solutions of the fifth Painlevé equation, the associated σ‐equation, and the symmetric fifth Painlevé system are expressed in terms of generalized Laguerre polynomials. Nonuniqueness of the solutions in special cases is established and some applications are considered. In the second part of the paper, the structure of the roots of the polynomials are investigated for all values of the parameters. Interesting transitions between root structures through coalescences at the origin are discovered, with the allowed behaviors controlled by hook data associated with the partition. The discriminants of the generalized Laguerre polynomials are found and also shown to be expressible in terms of partition data. Explicit expressions for the coefficients of a general Wronskian Laguerre polynomial defined in terms of a single partition are given.

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“…The Young diagrams including hook length corresponding to (A) 𝝀 = (4 2 , 2, 1 3 ) and its core (B) 𝝀 = (2, 1), and corresponding abacus diagrams (C) and (D).…”

Section: Partitionsmentioning

confidence: 99%

Section: Partitionsmentioning

confidence: 99%

“…Only one bead is moved on runner 2, by one space, and so 𝝂 2 = (1). Therefore, the 2-core and 2-quotient of 𝝀 = (4 2 , 2, 1 3 ) are (2,1) and ((3, 1), ( 1)), respectively.…”

Section: Partitionsmentioning

confidence: 99%

Section: Partitionsmentioning

confidence: 99%

“…(ii) Some rational solutions of sP V (80) are given in Refs. 3,23,24, where a different normalization of the symmetric system is used.…”

Section: Rational Solutions Of the Symmetric P 𝐕 Systemmentioning

confidence: 99%

See 3 more Smart Citations

Rational solutions of the fifth Painlevé equation. Generalized Laguerre polynomials

Clarkson,

Dunning

2023

Stud Appl Math

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Two-fold degeneracy of a class of rational Painlev\'e V solutions

Aratyn,

Gomes,

Lobo

et al. 2024

Open Communications in Nonlinear Mathematical Physics

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We present a construction of a class of rational solutions of the Painlev\'e V equation that exhibit a two-fold degeneracy, meaning that there exist two distinct solutions that share identical parameters. The fundamental object of our study is the orbit of translation operators of $A^{(1)}_{3}$ affine Weyl group acting on the underlying seed solution that only allows action of some symmetry operations. By linking points on this orbit to rational solutions, we establish conditions for such degeneracy to occur after involving in the construction additional B\"acklund transformations that are inexpressible as translation operators. This approach enables us to derive explicit expressions for these degenerate solutions. An advantage of this formalism is that it easily allows generalization to higher Painlev\'e systems associated with dressing chains of even period $N>4$.

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On Rational Solutions of Dressing Chains of Even Periodicity

Cited by 2 publications

References 18 publications

Rational solutions of the fifth Painlevé equation. Generalized Laguerre polynomials

Rational solutions of the fifth Painlevé equation. Generalized Laguerre polynomials

Two-fold degeneracy of a class of rational Painlev\'e V solutions

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