Modulated Bi-Orthogonal Polynomials on the Unit Circle: The $$2j-k$$ and $$j-2k$$ Systems

Gharakhloo, Roozbeh; Witte, N. S.

doi:10.1007/s00365-022-09604-2

Cited by 3 publications

(2 citation statements)

References 59 publications

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“…Similar "2j − k" determinants have appeared in various works in the literature, see, e.g., [15]. Denoting…”

Section: Connection With 2j − K Determinantssupporting

confidence: 57%

Painlevé-III Monodromy Maps Under the $D_6\to D_8$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions

Barhoumi,

Lisovyy,

Miller

et al. 2024

SIGMA

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The third Painlevé equation in its generic form, often referred to as Painlevé-III($D_6$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x} \right)^2-\frac{1}{x} \frac{{\rm d}u}{{\rm d}x} + \frac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \qquad \alpha,\beta \in \mathbb C. $$ Starting from a generic initial solution $u_0(x)$ corresponding to parameters $\alpha$, $\beta$, denoted as the triple $(u_0(x),\alpha,\beta)$, we apply an explicit Bäcklund transformation to generate a family of solutions $(u_n(x),\alpha + 4n,\beta + 4n)$ indexed by $n \in \mathbb N$. We study the large $n$ behavior of the solutions $(u_n(x), \alpha + 4n, \beta + 4n)$ under the scaling $x = z/n$ in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution $u_n(z/n)$. Our main result is a proof that the limit of solutions $u_n(z/n)$ exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III($D_8$), $$ \frac{{\rm d}^2U}{{\rm d}z^2} =\frac{1}{U}\left(\frac{{\rm d}U}{{\rm d}z}\right)^2-\frac{1}{z} \frac{{\rm d}U}{{\rm d}z} + \frac{4U^2 + 4}{z}.$$ A notable application of our result is to rational solutions of Painlevé-III($D_6$), which are constructed using the seed solution $(1,4m,-4m)$ where $m \in \mathbb C \setminus \big(\mathbb Z + \frac{1}{2}\big)$ and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at $z = 0$ when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both $D_6$ and $D_8$ at $z = 0$. We also deduce the large $n$ behavior of the Umemura polynomials in a neighborhood of $z = 0$.

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“…Similar "2j − k" determinants have appeared in various works in the literature, see, e.g., [15]. Denoting…”

Section: Connection With 2j − K Determinantssupporting

confidence: 57%

Painlevé-III Monodromy Maps Under the $D_6\to D_8$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions

Barhoumi,

Lisovyy,

Miller

et al. 2024

SIGMA

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“…An interesting question put forward in [1] is whether there is a differential equation in the symplectic and orthogonal cases which plays a part analogous to Painlevé III in the unitary setting. Gharakhloo and Witte [17] have made promising progress in this direction in their study of 2j − k and j − 2k bi-orthogonal polynomial systems on the unit circle.…”

Section: Introductionmentioning

confidence: 99%

Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices

Andrade,

Best

2024

J. Phys. A: Math. Theor.

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We investigate the joint moments of derivatives of characteristic polynomials over the unitary symplectic group $Sp(2N)$ and the orthogonal ensembles $SO(2N)$ and $O^-(2N)$. We prove asymptotic formulae for the joint moments of the $n_1$-th and $n_2$-th derivatives of the characteristic polynomials for all three matrix ensembles. Our results give two explicit formulae for each of the leading order coefficients, one in terms of determinants of hypergeometric functions and the other as combinatorial sums over partitions. We use our results to put forward conjectures on the joint moments of derivatives of $L$-functions with symplectic and orthogonal symmetry.

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Non-intersecting path explanation for block Pfaffians and applications into skew-orthogonal polynomials

Yao,

2025

Advances in Applied Mathematics

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Modulated Bi-Orthogonal Polynomials on the Unit Circle: The $$2j-k$$ and $$j-2k$$ Systems

Abstract: We construct the systems of bi-orthogonal polynomials on the unit circle where the Toeplitz structure of the moment determinants is replaced by $$\det (w_{2j-k})_{0\le j,k \le N-1} $$ det ( w 2 j - k … Show more

Cited by 3 publications

References 59 publications

Painlevé-III Monodromy Maps Under the $D_6\to D_8$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions

Painlevé-III Monodromy Maps Under the $D_6\to D_8$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions

Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices

Non-intersecting path explanation for block Pfaffians and applications into skew-orthogonal polynomials

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