Related papers: Painlev\'e equations for semi-classical recurrence…
The Painlev\'e equations are here connected to other classes of equations with the Painlev\'e Property (Ince's equations) by the same degeneracy procedure that connects the Painlev\'e equations (coalescence). These Ince's equations here are…
Special polynomials associated with rational solutions of the second Painlev\'{e} equation and other members of its hierarchy are discussed. New approach, which allows one to construct each polynomial is presented. The structure of the…
We discuss polynomials orthogonal with respect to a semi-classical generalised higher order Freud weight \[\omega(x;t,\lambda)=|x|^{2\lambda+1}\exp\left(tx^2-x^{2m}\right),\qquad x\in\mathbb{R},\] with parameters $\lambda > -1$,…
In this paper, We study the asymptotics of the leading coefficients and the recurrence coefficients for the orthogonal polynomials with repect to the Laguerre weight with singularity of root type and jump type at the soft edge via the…
The standard formula for the multi-section of the general linear three-term recurrence relation is simplified in terms of Chebyshev S-polynomials.
By using quasi-Banach techniques as key ingredient we prove Poincar\'e- and Sobolev- type inequalities for $m$-subharmonic functions with finite $(p,m)$-energy. A consequence of the Sobolev type inequality is a partial confirmation of B\l…
In the present paper we study the structure of the WKB series for the polynomial potential $V(x)=x^N$ ($N$ even). In particular, we obtain relatively simple recurrence formula of the coefficients $\s'_k$ of the semiclassical approximation…
We look at some extensions of the Stieltjes-Wigert weight functions. First we replace the variable x by x^2 in a family of weight functions given by Askey in 1989 and we show that the recurrence coefficients of the corresponding orthogonal…
For an arbitrary homogeneous linear recurrence sequence of order d with constant coefficients, we derive recurrence relations for all subsequences with indices in arithmetic progression. The coefficients of these recurrences are given…
We formulate and prove a general recurrence relation that applies to integrals involving orthogonal polynomials and similar functions. A special case are connection coefficients between two sets of orthonormal polynomials, another example…
Our goal is to find asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with the recurrence coefficients slowly stabilizing as $n\to\infty$. To that end, we develop spectral theory of Jacobi operators with long-range coefficients and…
We show that any graph polynomial from a wide class of graph polynomials yields a recurrence relation on an infinite class of families of graphs. The recurrence relations we obtain have coefficients which themselves satisfy linear…
We investigate the uniform asymptotic of some Sobolev orthogonal polynomials. Three term recurrence relation is given, moreover we give a recurrence relation between the so-called Sobolev orthogonal polynomials and Freud orthogonal…
Polynomials related to rational solutions of Painleve' equations satisfy certain difference equations. Conditions are given to acertain that all solutions really are polynomials.
We show that the Painleve equations P3-P5 can be derived (in a unified way) from a periodic sequence of Darboux transformations for a Schrodinger problem with quadratic eigenvalue dependency. The general problem naturally divides into three…
In this work we show that the $ N\times N $ Toeplitz determinants with the symbols $ z^{\mu}\exp(-{1/2}\sqrt{t}(z+1/z)) $ and $ (1+z)^{\mu}(1+1/z)^{\nu}\exp(tz) $ -- known $\tau$-functions for the \PIIIa and \PV systems -- are characterised…
We consider some bilinear recurrences that have applications in number theory. The explicit solution of a general three-term bilinear recurrence relation of fourth order is given in terms of the Weierstrass sigma function for an associated…
We present a determinant expression for a family of classical transcendental solutions of the Painlev\'e V and the Painlev\'e VI equation. Degeneration of these solutions along the process of coalescence for the Painlev\'e equations is…
We extend the work of Fuchs, Painlev\'e and Manin on a Calogero-like expression of the sixth Painlev\'e equation (the ``Painlev\'e-Calogero correspondence'') to the other five Painlev\'e equations. The Calogero side of the sixth Painlev\'e…
In this article explicit formulas for the recurrence equation p_{n+1}(x) = (A_n x + B_n) p_n(x) - C_n p_{n-1}(x) and the derivative rules sigma(x) p'_n(x) = alpha_n p_{n+1}(x) + beta_n p_n(x) + gamma_n p_{n-1}(x) and sigma(x) p'_n(x) =…