Two transformations of series that commute with compositional inversion

It is shown that the compositional inverse of either of two transformations of a given series can be determined from the compositional inverse of the series. Specifically, if t · f(t) and t · g(t) are compositional inverses, then so are t · f_k(t) and $\text{t · g}{}_{\mathrm{k}}{}^1\text{(t)}$, where f_k(t) is the kth Euler transformation of f(t) and $\text{g}{}_{\mathrm{k}}{}^1\text{(t) =}\text{g(t)}\text{(1 ? kt · g(t))}$.     

Nonsymmetric Toeplitz matrices that commute with tridiagonal matrices

Translated from Matematicheskie Zametki, Vol. 55, No. 5, pp. 69–79, May, 1994.     

A unified interpretation of several combinatorial dualities

Several combinatorial structures exhibit a duality relation that yields interesting theorems, and, sometimes, useful explanations or interpretations of results that do not concern duality explicitly. We present a common characterization of the duality relations associated with matroids, clutters (Sperner families), oriented matroids, and weakly oriented matroids. The same conditions characterize the orthogonality relation on certain families of vector spaces. This leads to a notion of abstract duality.     

Canonical forms for normal matrices that commute with their complex conjugate

We consider the class of normal complex matrices that commute with their complex conjugate. We show that such matrices are real orthogonally similar to a canonical direct sum of 1-by-1 and certain 2-by-2 matrices. A canonical form for quasi-real normal matrices is obtained as a special case. We also exhibit a special form of the spectral theorem for normal matrices that commute with their conjugate.     

Hankel operators that commute with second-order differential operators

Suppose that Γ is a continuous and self-adjoint Hankel operator on L²(0,∞) with kernel ?(x+y) and that with a(0)=0. If a and b are both quadratic, hyperbolic or trigonometric functions, and ? satisfies a suitable form of Gauss's hypergeometric differential equation, or the confluent hypergeometric equation, then ΓL=LΓ. The paper catalogues the commuting pairs Γ and L, including important cases in random matrix theory. There are also results proving rapid decay of the singular numbers of Hankel integral operators with kernels that are analytic and of exponential decay in the right half-plane.     

A combinatorial interpretation of the inverse kostka matrix

The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K_λ,μenumerates the number of column strict tableaux of shape λ and type μ. We make use of the Jacobi -Trudi identity to give a combinatorial interpretation for the inverse of the Kostka matrix in terms of certain types of signed rim hook tabloids. Using this interpretation, the matrix identity KK^-1=Iis given a purely combinatorial proof. The generalized Jacobi-Trudi identity itself is also shown to admit a combinatorial proof via these rim hook tabloids. A further application of our combinatorial interpretation is a simple rule for the evaluation of a specialization of skew Schur functions that arises in the computation of plethysms.     

A combinatorial interpretation of the inverse kostka matrix

The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K _λ,μenumerates the number of column strict tableaux of shape λ and type μ. We make use of the Jacobi -Trudi identity to give a combinatorial interpretation for the inverse of the Kostka matrix in terms of certain types of signed rim hook tabloids. Using this interpretation, the matrix identity KK ^?1=Iis given a purely combinatorial proof. The generalized Jacobi-Trudi identity itself is also shown to admit a combinatorial proof via these rim hook tabloids. A further application of our combinatorial interpretation is a simple rule for the evaluation of a specialization of skew Schur functions that arises in the computation of plethysms.     

A combinatorial proof of the multivariable lagrange inversion formula

Part I contains a combinatorial proof of a multivariable Lagrange inversion formula. Part II discusses the various multivariable Lagrange inversion formulas of Jacobi, Stieltjes, Good, Joni, and Abhyankar and shows how they can be derived from each other.     

Bounded linear operators that commute with shifts are scaled identity

We prove that every bounded linear operator on L^p?L^p(R'),s≥1 and 1≤p<∞, that commutes with both the spatial shift operator and the phase shift operator must be a constant multiple of the identity. We also apply this result to identify analysis-synthesis dual L^q-L^p paris and to characterize bi-orthogonal unconditional bases of L^q-L^p, where p^?1+q^?1=1.     

A combinatorial interpretation of the Seidel generation ofq-derangement numbers

In [8] Dumont and Randrianarivony have given several combinatorial interpretations for the coefficients of the Euler-Seidel matrix associated withn!. In this paper we consider aq-analogue of their results, which leads to the discovery of a new Mahonian statistic “maf” on the symmetric group. We then give new proofs and generalizations of some results of Gessel and Reutenauer [12] and Wachs [17].     

Tripotency of a linear combination of two involutory matrices and a tripotent matrix that mutually commute

This paper concerns with the tripotency of a linear combination of three matrices, which has a background in statistical theory. We demonstrate all the possible cases that lead to the tripotency of a linear combination of two involutory matrices and a tripotent matrix that mutually commute. By utilizing block technique and returning the partitioned matrices to their original forms, we derive the sufficient and necessary conditions such that a linear combination of three mutually commuting involutory matrices is tripotent.     

Convergence for the regularized inversion of Fourier series

The problems of smoothing data through a transform in the Fourier domain and of retrieving a function from its Fourier coefficients are analyzed in the present paper. For both of them a solution, based on regularization tools, is known. Aim of the paper is to prove strong results of convergence of the regularized solution and optimality of the Generalized Cross Validation criterion for choosing the regularization parameter.     

Some transformations and reduction formulas for multipleq-hypergeometric series

Summary Some simple ideas of G. E. Andrews ([2], [3]) are used here to derive a transformation formula for a general multiple q-series with essentially arbitrary terms. As applications of (or motivated by) this q-series transformation, several transformation and reduction formulas for q-hypergeometric series in two and more variables are presented. Relevant connections of the various q-identities considered here with a number of known results are also indicated.Supported, in part, by N.S.E.R.C. (Canada) Grant A-7353.     

New transformations for elliptic hypergeometric series on the root system A n

Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system A _n , with different dimensions n. We give, with a new, elementary proof, an elliptic extension of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series. 2000 Mathematics Subject Classification Primary—33D67; Secondary—11F50     

Continued fractions for linear fractional transformations of power series

We present an algorithm to produce the continued fraction expansion of a linear fractional transformation of a power series. Giving an application, we demonstrate that the behavior of the algorithm is intimately related with the continued fraction expansions of certain algebraic power series over finite fields.