Casorati Determinant Solutions for the Discrete Painlevé III Equation

Abstract

The discrete Painlevé III equation is investigated based on the bilinear formalism. It is shown that it admits the solutions expressed by the Casorati determinant whose entries are given by the discrete Bessel function. Moreover, based on the observation that these discrete Bessel functions are transformed to the $q$-Bessel functions by a simple variable transformation, we present a $q$-difference analogue of the Painlevé III equation.

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References in corpus (4)

• Linearization And Solutions Of The Discrete Painlevé-III Equation
• Casorati Determinant Solution for the Relativistic Toda Lattice Equation
• Casorati Determinant Solutions for the Discrete Painlevé-II Equation
• q-Discrete Toda Molecule Equation

Cited by in corpus (9)

• Determinant Structure of the Rational Solutions for the Painlevé II Equation
• Rational Solutions for the Discrete Painlevé II Equation
• Hypergeometric solutions to the q-Painlevé equation of type $A_4^{(1)}$
• On a q-difference Painlevé III equation: I. Derivation, symmetry and Riccati type solutions
• Projective reduction of the discrete Painlevé system of type $(A_2+A_1{)}^{(1)}$
• Hypergeometric $\tau$ Function of the $q$-Painlevé Systems of Type $(A_2+A_1{)}^{(1)}$
• Hypergeometric $\tau$ Functions of the $q$-Painlevé Systems of Types $A_4^{(1)}$ and $(A_1+A_1^{\prime }{)}^{(1)}$
• Hypergeometric Solutions of the $A_4^{(1)}$-Surface $q$-Painlevé IV Equation
• $q$-deformation of random partitions, determinantal structure, and Riemann-Hilbert problem