Bijective proofs of shifted tableau and alternating sign matrix identities

We give a bijective proof of an identity relating primed shifted gl(n)-standard tableaux to the product of a gl(n) character in the form of a Schur function and . This result generalises a number of well-known results due to Robbins and Rumsey, Chapman, Tokuyama, Okada and Macdonald. An analogous result is then obtained in the case of primed shifted sp(2n)-standard tableaux which are bijectively related to the product of a t-deformed sp(2n) character and . All results are also interpreted in terms of alternating sign matrix (ASM) identities, including a result regarding subsets of ASMs specified by conditions on certain restricted column sums.     

On denominators of the Kontsevich integral and the universal perturbative invariant of 3-manifolds

 The integrality of the Kontsevich integral and perturbative invariants is discussed. It is shown that the denominator of the degree n part of the Kontsevich integral of any knot or link is a divisor of (2!3!…n!)⁴(n+1)!. We prove this by establishing the existence of a Drinfeld's associator in the space of chord diagrams with special denominators. We also show that the denominator of the degree n part of the universal perturbative invariant of homology 3-spheres is not divisible by any prime greater than 2n+1. Oblatum 20-VI-1997 & 28-IV-1998 / Published online: 12 November 1998     

Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part I. Maehly type approximants

Laurent Padé-Chebyshev rational approximants,A _m (z,z ⁻¹)/B _n (z, z ⁻¹), whose Laurent series expansions match that of a given functionf(z,z ⁻¹) up to as high a degree inz, z ⁻¹ as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients off up to degreem+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions betweenf(z,z ⁻¹)B _n (z, z ⁻¹)). The derivation was relatively simple but required knowledge of Chebyshev coefficients off up to degreem+2n. In the present paper, Padé-Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé-Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m, n) Padé-Chebyshev approximant, of degreem in the numerator andn in the denominator, is matched to the Chebyshev series up to terms of degreem+n, based on knowledge of the Chebyshev coefficients up to degreem+2n. Numerical tests are carried out on all four Padé-Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent-Chebyshev series on a variety of functions. In part II of this paper [7] Padé-Chebyshev approximants of Clenshaw-Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

The Flagged Double Schur Function

The double Schur function is a natural generalization of the factorial Schur function introduced by Biedenharn and Louck. It also arises as the symmetric double Schubert polynomial corresponding to a class of permutations called Grassmannian permutations introduced by A. Lascoux. We present a lattice path interpretation of the double Schur function based on a flagged determinantal definition, which readily leads to a tableau interpretation similar to the original tableau definition of the factorial Schur function. The main result of this paper is a combinatorial treatment of the flagged double Schur function in terms of the lattice path interpretations of divided difference operators. Finally, we find lattice path representations of formulas for the symplectic and orthogonal characters for sp(2n) and so(2n + 1) based on the tableau representations due to King and El-Shakaway, and Sundaram. Based on the lattice path interpretations, we obtain flagged determinantal formulas for these characters.     

Some asymptotic Bessel function ratios

We derive leading terms in the expansion of ratios of the formB _n+α(nβ)/B_n(nβ) for largen, whereB _n(x) is any one of the Bessel functionsJ _n(x), Y_n(x), H_n(x),I _n(x) andK _n(x).     

Second-order evaluations of orthogonal and symplectic Yangians

Orthogonal or symplectic Yangians are defined by the Yang–Baxter RLL relation involving the fundamental R-matrix with the corresponding so(n) or sp(2m) symmetry. We investigate the second-order solution conditions, where the expansion of L(u) in u ^?1 is truncated at the second power, and we derive the relations for the two nontrivial terms in L(u).     

Technical Note: Reduction of rational integrals related to linear and convex quadrilateral finite elements

We reduce a rational function of bivariate nth degree polynomial numerator with a linear denominator to a simple bivariate polynomial of degree (n ? 1) and a rational function of a single variate nth degree polynomial numerator with the same bivariate linear denominator. This has very greatly contributed to the evaluation of (n + 1)(n + 2)/2 rational integrals in bivariates to mere (n + 1) rational integral of a single variate and an integration of simple polynomial in bivariates. Thus the effort of integration is reduced several times and leads to simple analytical expressions in terms of the nodal coordinates. In order to illustrate the numerical process two examples are considered. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 759–770, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10026.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part I. Maehly Type Approximants

Laurent Padé–Chebyshev rational approximants, A _m (z,z ^–1)/B _n (z,z ^–1), whose Laurent series expansions match that of a given function f(z,z ^–1) up to as high a degree in z,z ^–1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

Alternating Sign Matrices and Some Deformations of Weyl's Denominator Formulas

An alternating sign matrix is a square matrix whose entries are 1, 0, or –1, and which satisfies certain conditions. Permutation matrices are alternating sign matrices. In this paper, we use the (generalized) Littlewood's formulas to expand the products and 2 as sums indexed by sets of alternating sign matrices invariant under a 180° rotation. If we put t = 1, these expansion formulas reduce to the Weyl's denominator formulas for the root systems of type B _n and C _n. A similar deformation of the denominator formula for type D _n is also given.     

A monotonicity property of bessel functions

Some monotonicity results are given for the remainder terms in the asymptotic expansion for x → ∞ of the function J_v(x)J_v+n(x) + Y_v(x Y_v+n(x), v ε R, n ε Z.     

Growth Behavior and Zero Distribution of Rational Approximants

We investigate the growth and the distribution of zeros of rational uniform approximations with numerator degree ≤n and denominator degree ≤m _n for meromorphic functions f on a compact set E of ℂ where m _n =o(n/log n) as n→∞. We obtain a Jentzsch–Szegő type result, i.e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain E _ρ(f) of meromorphy of f if f has a singularity of multivalued character on the boundary of E _ρ(f). The paper extends results for polynomial approximation and rational approximation with fixed degree of the denominator. As applications, Padé approximation and real rational best approximants are considered.     

The partition function and Hecke operators

The theory of congruences for the partition function p(n) depends heavily on the properties of half-integral weight Hecke operators. The subject has been complicated by the absence of closed formulas for the Hecke images P(z)|T(?²), where P(z) is the relevant modular generating function. We obtain such formulas using Euler?s Pentagonal Number Theorem and the denominator formula for the Monster Lie algebra. As a corollary, we obtain congruences for certain powers of Ramanujan?s Delta-function.     

Comparison of germ expansion¶on inner forms of GL(n)

We prove that the germ expansion of a discrete series representation π^′ on GL _n (D) where D is a division algebra over k of index m and the germ expansion of the representation π of GL _mn (k) associated to π^′ by the Deligne–Kazhdan–Vigneras correspondence are closely related, and therefore certain coefficients in the germ expansion of a discrete series representation of GL _mn (k) can be interpreted (and therefore sometimes calculated) in terms of the dimension of a certain space of (degenerate) Whittaker models on GL _n (D). Received: 30 September 1999 / Revised version: 11 February 2000     

An extremal property of Hermite polynomials

Let H_n be the nth Hermite polynomial, i.e., the nth orthogonal on polynomial with respect to the weight w(x)=exp(−x²). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f||H_n| at the zeros of H_n+1, then for k=1,…,n we have f^(k)H_n^(k), where · is the norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the norm, and estimates for the expansion coefficients in the basis of Hermite polynomials.     

On the number of antipodal bicolored necklaces

Let ν(2n) be the number of antipodal bicolored necklaces with 2n pearls. In this note, we find the first two terms of the asymptotic expansion of ν(2n). As a byproduct of this result, we also show that the sequence (ν(2n)) _n≥1 is non-holonomic, i.e., it satisfies no linear recurrence of a fixed finite order k with polynomial coefficients.     

Minimal Bar Tableaux

Motivated by recent results of Stanley, we generalize the rank of a partition λ to the rank of a shifted partition S(λ). We show that the number of bars required in a minimal bar tableau of S(λ) is max(o, e + (ℓ(λ) mod 2)), where o and e are the number of odd and even rows of λ. As a consequence we show that the irreducible projective characters of S_n vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur’s Q_λ symmetric functions in terms of the power sum symmetric functions. Received November 20, 2003     

ADMISSIBLE WAVELETS ASSOCIATED WITH THE TRANSFORM GROUP ON R

Let p be the transform group on R, then P has a natural unitary representation U onL2 (R^n). Decompose L2(R^n) into the direct sum of irreducible invariant closed subspace,s. The re-striction of U on these suhspaces is square-intagrable. In this paper the characterization of admissi-ble condition in tarrns of the Fourier transform is given. The wavelet transform is defined, and theorthogorml direct sum decomposition of function space L2 (P,du1) is obtained.     

An expansion theorem for an eigenvalue problem of an arbitrary multiply connected domain with an eigenparameter in a general type of boundary conditions

The object of this paper is to establish an expansion theorem for a regular right-definite eigenvalue problem with an eigenvalue parameter which is contained in the Schrödinger partial differential equation and in a general type of boundary conditions on the boundary of an arbitrary multiply connected bounded domain inR ⁿ (n2). We associate with this problem an essentially self-adjoint operator in a suitably defined Hilbert space and then we develop an associated eigenfunction expansion theorem.