Eisenstein series associated with Γ<Subscript>0</Subscript>(2)

In this paper, we define the normalized Eisenstein series ℘, e, and associated with Γ₀(2), and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas, we define a differential equation depending on the weights of modular forms on Γ₀(2) and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from a formula involving the triangular numbers.      

Special polynomials associated with rational solutions of some hierarchies

New special polynomials associated with rational solutions of the Painlevé hierarchies are introduced. The Hirota relations for these special polynomials are found. Differential-difference hierarchies to find special polynomials are presented. These formulae allow us to search special polynomials associated with the hierarchies. It is shown that rational solutions of the Caudrey–Dodd–Gibbon, the Kaup–Kupershmidt and the modified hierarchy for these ones can be obtained using new special polynomials.     

Special polynomials and rational solutions of the hierarchy of the second Painlevé equation

We study special polynomials used to represent rational solutions of the hierarchy of the second Painlevé equation. We find several recursion relations satisfied by these polynomials. In particular, we obtain a differential-difference relation that allows finding any polynomial recursively. This relation is an analogue of the Toda chain equations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 58–67, October, 2007.     

Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomialsP _n ^α _n ^β _n are studied, assuming that

A Siegel modular threefold and Saito-Kurokawa type lift to <Emphasis Type="Italic">S</Emphasis><Subscript>3</Subscript>(Γ<Subscript>1,3</Subscript>(2))

Hulek and others conjectured that the unique differential three-form F (up to scalar) on the Siegel threefold associated to the group Γ_1,3(2) comes from the Saito-Kurokawa lift of the elliptic newform h of weight 4 for Γ₀(6). This F have been already constructed as a Borcherds product (cf. Gritsenko and Hulek in Int Math Res Notices 17:915–937, 1999). In this paper, we prove this conjecture by using the Yoshida lift and we settle a conjecture which relates our theorem. A remarkable fact is that the Yoshida lift using the usual test function cannot give the Saito-Kurokawa type lift of weight 3 associated to the group Γ_1,3(2). So important task is to find special test functions for the Yoshida lift at the bad primes 2 and 3. Dedicated to Professor Tomoyoshi Ibukiyama on his 60th birthday.     

Biorthogonal polynomials for two-matrix models with semiclassical potentials

We consider the biorthogonal polynomials associated to the two-matrix model where the eigenvalue distribution has potentials V₁,V₂ with arbitrary rational derivative and whose supports are constrained on an arbitrary union of intervals (hard-edges). We show that these polynomials satisfy certain recurrence relations with a number of terms d_i depending on the number of hard-edges and on the degree of the rational functions . Using these relations we derive Christoffel–Darboux identities satisfied by the biorthogonal polynomials: this enables us to give explicit formulæ for the differential equation satisfied by d_i+1 consecutive polynomials, We also define certain integral transforms of the polynomials and use them to formulate a Riemann–Hilbert problem for (d_i+1)×(d_i+1) matrices constructed out of the polynomials and these transforms. Moreover, we prove that the Christoffel–Darboux pairing can be interpreted as a pairing between two dual Riemann–Hilbert problems.     

Faber polynomials corresponding to rational exterior mapping functions

Faber polynomials corresponding to rational exterior mapping functions of degree (m, m − 1) are studied. It is shown that these polynomials always satisfy an (m + 1)-term recurrence. For the special case m = 2, it is shown that the Faber polynomials can be expressed in terms of the classical Chebyshev polynomials of the first kind. In this case, explicit formulas for the Faber polynomials are derived.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-rigidity in von Neumann algebras

We introduce the notion of L ²-rigidity for von Neumann algebras, a generalization of property (T) which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group. We show that L ²-rigidity passes to normalizers and is satisfied by nonamenable II₁ factors which are non-prime, have property Γ, or are weakly rigid. As a consequence we obtain that if M is a free product of diffuse von Neumann algebras, or if M=LΓ where Γ is a finitely generated group with β₁ ⁽²⁾(Γ)>0, then any nonamenable regular subfactor of M is prime and does not have properties Γ or (T). In particular this gives a new approach for showing solidity for a free group factor thus recovering a well known recent result of N. Ozawa.     

String Equations of the q-KP Hierarchy

Based on the Lax operator L and Orlov-Shulman’s M operator, the string equations of the q-KP hierarchy are established from the special additional symmetry flows, and the negative Virasoro constraint generators {L _−n , n ≥ 1} of the 2-reduced q-KP hierarchy are also obtained.     

A lattice in more than two Kac-Moody groups is arithmetic

Let Γ < G ₁ × … × G _n be an irreducible lattice in a product of infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n ≥ 3, then each G _i is a simple algebraic group over a local field and Γ is an S-arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n ≥ 2: either Γ is an S-arithmetic (hence linear) group, or Γ is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout.     

Polynomial Approximation of Conformal Maps

Let G be a finite domain, bounded by a Jordan curve Γ , and let f ₀ be a conformal map of G onto the unit disk. We are interested in the best rate of uniform convergence of polynomial approximation to f ₀ , in the case that Γ is piecewise-analytic without cusps. In particular, we consider the problem of approximating f ₀ by the Bieberbach polynomials π _n and derive results better than those in [5] and [6] for the case that the corners of Γ have interior angles of the form π/N . In the proof, the Lehman formulas for the asymptotic expansion of mapping functions near analytic corners are used. We study the question when these expansions contain logarithmic terms. December 6, 1995. Date revised: August 5, 1996.     

Characterization of algebraic curves by Chebyshev quadrature

Complex potential theory is used to show that Chebyshev-type quadrature works particularly well on algebraic Jordan curves Γ in ℝ ^d , supplied with normalized arc length or a similar probability measure μ. Evaluating the integral ∫_Γ fdμ by the arithmetic mean of the value off on any cycle ofN equally spaced nodes on Γ (relative to μ), the quadrature error will, be bounded byAe ^−bN sup_Γ|f| for allN and all polynomialsf(x) of degree ≤cN. It is plausible that small shifts of the nodes would give quadrature error zero for such polynomials. There are related results for algebraic Jordan arcs and certain algebraic surfaces. The situation is completely different for nonalgebraic curves and surfaces, where corresponding quadrature remainders are at least of order 1/N.     

Multiple Orthogonal Polynomials Associated with the Modified Bessel Functions of the First Kind

   Abstract. We consider polynomials which are orthogonal with respect to weight functions, which are defined in terms of the modified Bessel function I _ν and which are related to the noncentral χ ² -distribution. It turns out that it is the most convenient to use two weight functions with indices ν and ν+1 and to study orthogonality with respect to these two weights simultaneously. We show that the corresponding multiple orthogonal polynomials of type I and type II exist and give several properties of these polynomials (differential properties, Rodrigues formula, explicit formulas, recurrence relation, differential equation, and generating functions).     

Amalgamations of the Painlevé Equations

We present new hierarchies of nonlinear ordinary differential equations (ODEs) that are generalizations of the Painlevé equations. These hierarchies contain the Painlevé equations as special cases. We emphasize the sixth-order ODEs. Special solutions for one of them are expressed via the general solutions of the P ₁ and P ₂ equations and special cases of the P ₃ and P ₅ equations. Four of the six Painlevé equations can be considered special cases of these sixth-order ODEs. We give linear representations for solving the Cauchy problems for the hierarchy equations using the inverse monodromy transform.     

Stochastic Integration of Operator-Valued Functions with Respect to Banach Space-Valued Brownian Motion

Let E be a real Banach space with property (α) and let W _Γ be an E-valued Brownian motion with distribution Γ. We show that a function is stochastically integrable with respect to W _Γ if and only if Γ-almost all orbits Ψx are stochastically integrable with respect to a real Brownian motion. This result is derived from an abstract result on existence of Γ-measurable linear extensions of γ-radonifying operators with values in spaces of γ-radonifying operators. As an application we obtain a necessary and sufficient condition for solvability of stochastic evolution equations driven by an E-valued Brownian motion. The first named author gratefully acknowledges the support by a ‘VIDI subsidie’ in the ‘Vernieuwingsimpuls’ programme of The Netherlands Organization for Scientific Research (NWO) and the Research Training Network HPRN-CT-2002–00281. The second named author was supported by grants from the Volkswagenstiftung (I/78593) and the Deutsche Forschungsgemeinschaft (We 2847/1–1).     

The structure of a kind of connected components with oriented cycles and semi-stable vertices

LetF be an algebraically closed field. A be a finite-dimensional algebra overF,Γ _A be the Auslander-Reiten quiver ofA,Γ be a connected component of Γ _A with oriented cycles and semi-stable vertices and each non-stable vertex In Γ be a projective-injective vertex. The structure of Γ is studied. Projcct supported by Chinese Postdoctor Fund.     

Differential-Operator Representations of S n and Singular Vectors in Verma Modules

Given a weight of sl(n, \mathbb C{\mathbb C}), we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module, and a differential-operator representation of the symmetric group S _n on the related space of truncated power series. We prove that the solution space of the system of partial differential equations is exactly spanned by {σ(1)|σ ∈ S _n }. Moreover, the singular vectors of sl(n, \mathbb C{\mathbb C}) in the Verma module are given by those σ(1) that are polynomials. The well-known results of Verma, Bernstein–Gel’fand–Gel’fand and Jantzen for the case of sl(n, \mathbb C{\mathbb C}) are naturally included in our almost elementary approach of partial differential equations.     

Existence of a stationary symmetric solution of the two-dimensional Navier-Stokes equations with a given head and with nonzero fluxes

We consider a stationary boundary value problem for the Navier-Stokes equations of a homogeneous incompressible fluid in a two-dimensional bounded domain with boundary consisting of connected components Γ _i . On each part Γ _i , we specify the tangent component of the flow velocity vector, the total flow head (up to an additive constant), and the fluid flux through Γ _i . For the case in which the domain and the original data are symmetric around some line, we prove the existence of a solution of the problem with such a symmetry. We also present some results on the solvability in the nonsymmetric case.     

A class of polynomials defined by generalized Rodrigues’ formula

Summary The present paper incorporates a systematic study of linear, bilinear and bilateral generating functions, pure as well as mixed recurrence relations, and the differential equations associated with the class of polynomials {G _n ^α (x, r, p, k)|n=0, 1, 2, ... }, defined by (1.3) below, which evidently provides an elegant generalization of the various recent extensions of the classical Hermite and Laguerre polynomials. This work was supported in part by the National Research Council of Canada under Grant A7353. See also Abstract 71T-B15, Notices Amer. Math. Soc.18 (1971), p. 252. Entrata in Redazione il 16 novembre 1970.     

Differential operators associated with zonal polynomials. I

Summary Associated with each zonal polynomial,C _k(S), of a symmetric matrixS, we define a differential operator ∂_k, having the basic property that ∂_kC_λδ_kλ, δ being Kronecker's delta, whenever κ and λ are partitions of the non-negative integerk. Using these operators, we solve the problems of determining the coefficients in the expansion of (i) the product of two zonal polynomials as a series of zonal polynomials, and (ii) the zonal polynomial of the direct sum,S⊕T, of two symmetric matricesS andT, in terms of the zonal polynomials ofS andT. We also consider the problem of expanding an arbitrary homogeneous symmetric polynomial,P(S) in a series of zonal polynomials. Further, these operators are used to derive identities expressing the doubly generalised binomial coefficients ( _P ^λ ),P(S) being a monomial in the power sums of the latent roots ofS, in terms of the coefficients of the zonal polynomials, and from these, various results are obtained.