Generic differential operators on Siegel modular forms and special polynomials

Garrett's pullback formula for vector valued Siegel modular forms

An analogue of the irreducibility is developed for linear differential operators, or, more generally, noncommutative polynomials, and is applied to a few simple examples. Introduction. The question of the irreducibility of an ordinary linear differential operator is of interest in the Picard-Vessiot theory (see Kolchin [1, ?22]). Indeed, the operator is irreducible if and only if the Galois group is linearly irreducible. In this note we develop an Eisenstein criterion to help answer this question. Because a linear differential operator is a special type of noncommutative polynomial, it is more natural to phrase the result in this more general context. The result, and indeed the proof with only slight changes in notation, is valid for partial differential operators, or more generally, noncommutative polynomials in several variables. Because the hypothesis that the ring be principal is unnatural in this setting, and because the reader may make the necessary changes if he so desires, we confine our attention to the ordinary case. 1. The irreducibility criterion. By a differential ring M we shall mean a ring (associative, commutative, with unit) together with two commuting operators a and 3 such that a is an automorphism of R, 3 is additive, and 6(ab)=aa3b+6a.b for every a, b in S. 1[6] will denote the noncommutative ring of differential operators with coefficients in R, so that 1[6] is the set of 0 ai6i, with ai E $, n E N, and right multiplication is defined by 6 * a-=oa * + 3a. Evidently if or=id, then W is an ordinary differential ring with derivation operator 6. By a differential ideal of R we mean an ideal which is closed under the two operators a and 6. The smallest differential ideal which contains the set Ec$S is denoted by [2]. Received by the editors October 7, 1971. AMS 1970 subject classifications. Primary 12H05, 12E05; Secondary 13B25, 16A02.

In this paper we explicitly construct vector valued Siegel modular forms of degree two and the automorphic factor detk| for even k where St denotes the standard representation of GL(2,C). As an application, we prove some congruences between eigenvalues of Hecke operators. For a positive integer n, let F, be the full Siegel modular group of degree n and H, the Siegel upper half plane of degree n. For M ~ F, and Z ~ H,, we put

This paper is motivated by a 2001 paper of Choie and Kim and a 2006 paper of Bump and Choie. The paper of Choie and Kim extends an earlier result of Bol for elliptic modular forms to the setting of Siegel and Jacobi forms. The paper of Bump and Choie provides a representation theoretic interpretation of the phenomenon, and shows how a natural generalization of Choie and Kim's result on Siegel modular forms follows from a natural conjecture regarding (<TEX>$g$</TEX>, K)-modules. In this paper, it is shown that the conjecture of Bump and Choie follows from work of Boe. A second proof which is along the lines of the proof given by Bump and Choie in the genus 2 case is also included, as is a similar treatment of the result of Choie and Kim on Jacobi forms.

Matrix elements of irreducible representations of SU(n + 1)×SU(n + 1) and multivariable matrix-valued orthogonal polynomials

This is the first of two papers on the ALPAK system for nonnumerical algebra on a digital computer. This paper is concerned with polynomials in several variables and truncated power series with polynomial coefficients. The second paper will discuss rational functions of several variables, truncated power series with rational-function coefficients, and systems of linear equations with rational-function coefficients. The ALPAK system has been programmed within the BE-SYS-4 monitor system on the IBM 7090 computer, but the language and concepts are machine independent. The available polynomial arithmetic operations are add, subtract, multiply, divide (if divisible), substitute, differentiate, zero test, nonzero test, and equality test. The speed of the system is indicated by the rule of thumb that one man-hour equals one 7090-second. The available space in core is usually sufficient for approximately 8000 polynomial terms. Section I of this paper consists of a nontechnical description of the system and a brief glimpse into the future. Section II discusses several specific problems to which the ALPAK system has been applied. These two parts do not presuppose any knowledge of computers or computer programming. Section III describes the use and the implementation of the algebraic operations relating to polynomials in several variables and truncated power series with polynomial coefficients. The reader of Section III is assumed to be acquainted with the elements of FAP (FORTRAN Assembly Program) programming, including the use of macros, as described in a series of IBM publications, the latest of which is IBM 7090–7094 Programming Systems MAP (Macro Assembly Program) Language (Form Number C28–6311).

The D1–D5–KK–p system naturally provides an infinite-dimensional module graded by the dyonic charges whose dimensions are counted by the Igusa cusp form, Φ10(Z). We show that the Mathieu group, M24, acts on this module by recovering the Siegel modular forms that count twisted dyons as a trace over this module. This is done by recovering Borcherds product formulae for these modular forms using the M24 action. This establishes the correspondence (‘moonshine’) proposed in arXiv:0907.1410 that relates conjugacy classes of M24 to Siegel modular forms. This also, in a sense that we make precise, subsumes existing moonshines for M24 that relates its conjugacy classes to eta-products and Jacobi forms.

An Analogy of Bol's Result on Jacobi Forms and Siegel Modular Forms

In this paper, we show that the space of vector valued Siegel modular forms of Γ_0 (N)⊂Sp(2,Z) with respect to the symmetric tensor of degree 2 has a simple unified structure for N=2,3,4 . Each structure is similar to the structure of the full modular group.

Some Families of Generating Functions Associated with the Stirling Numbers of the Second Kind

Markov–Bernstein and Landau–Kolmogorov type inequalities in several variables for the Hermite and closely connected measures

(1971). A Representation for Polynomials of Several Variables. The American Mathematical Monthly: Vol. 78, No. 1, pp. 45-47.

In the present paper we shall consider certain problems of best mean approximation for functions of several variables. It is wellknown that there is an essential difficulty in the extension of the theory of best Chebyshev approximation to the multivariate case. The basic problem here is the absence of Haar subspaces of functions of more than one variable. However, this fact seems not to be an insuperable obstacle in developing the theory of best Lt-approximation of functions of several variables. Some first steps in this direction have been made in [1]. In particular, it was shown in [1] that the L~-approximation of continuous functions of two variables by the tensor product of a two-dimensional Haar subspaces and an arbitrary Haar subspace is unique. This result gives some hope that the general problem of unicity of best Ll-approximation of continuous functions of several variables by tensor products ofHaar subspaces can be solved positively. Nevertheless, this problem is still unsolved even in the case of ordinary polynomials. In the present paper we continue the study of uniqueness of multivariate L~-approximation presenting another approach to this problem. We shall consider approximation by arbitrary multivariate polynomials but restrict ourselves only to approximating rational functions. The main advantage of this approach is that it includes the study of the unicity of polynomials of several variables with fixed leading coefficients which have minimal Ll-deviation from zero. In case of Chebyshev approximation this problem was considered in [2], [6] and [7]. The results of our paper imply the unicity of the polynomials of least Ll-deviation having fixed leading coefficients. Thus we show that the solution of the Korkin--Zolotarjov type extremal problem is unique in the multivariate case, too. For rectangular regions this unique solution is given by the product of Chebyshev polynomials of second kind.

Generalized Hermite polynomials

We consider polynomials in several variables orthogonal with respect to a Sobolev-type inner product, obtained from adding a higher order gradient evaluated in a fixed point to a standard inner product. An expression for these polynomials in terms of the orthogonal family associated with the standard inner product is obtained. A particular case using polynomials in the unit ball is analyzed, and some asymptotic results are derived.