Two matrix inversions associated with the Hagen-Rothe formula, their q-analogues and applications

By means of the Hagen-Rothe formula, we establish two new matrix inversions with four parameters. These new inversions uniformize Riordan's inverse relations of Abel-, Chebyshev-, and Legendre-type as well as Gould's inversions based on Vandermonde-type convolutions. Some related q-series inverse relations using the known q-analogues of the Hagen-Rothe formula are established. A Λ-extension of Gould's g-inverse, a novel expression for all Chebyshev-type inversions, and several new summation and transformation formulas of series are presented as applications.     

ON ABEL-GONTSCHAROFF-GOULD’S POLYNOMIALS

Abstract In this paper a connective study of Gould's annihilation coefficients and Abel-Gontscharoff polynomialsis presented. It is shown that Gould's annihilation coefficients and Abel-Gontscharoff polynomials are actu-ally equivalent to each other under certain linear substitutions for the variables. Moreover, a pair of relatedexpansion formulas involving Gontscharoff's remainder and a new form of it are demonstrated, and also il-lustrated with several examples.     

The method of combinatorial telescoping

We present a method for proving q-series identities by combinatorial telescoping, in the sense that one can transform a bijection or a classification of combinatorial objects into a telescoping relation. We shall illustrate this method by giving a combinatorial derivation of Watson's identity, which implies the Rogers-Ramanujan identities.     

More on the identity of Chaundy and Bullard

In what follows we present a homogeneous identity which implies a more elementary treatment of the Chaundy–Bullard identity with n variables. In a different direction we bring another ramification of the Chaundy–Bullard identity.     

On a partition identity of Lehmer

Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called “Beck-type” companions to other identities.In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of n with an even number of even parts over the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.     

Anti-lecture hall compositions and overpartitions

We show that the number of anti-lecture hall compositions of n with the first entry not exceeding k−2 equals the number of overpartitions of n with non-overlined parts not congruent to 0,±1 modulo k. This identity can be considered as a finite version of the anti-lecture hall theorem of Corteel and Savage. To prove this result, we find two Rogers-Ramanujan type identities for overpartitions which are analogous to the Rogers-Ramanujan type identities due to Andrews. When k is odd, we give another proof by using the bijections of Corteel and Savage for the anti-lecture hall theorem and the generalized Rogers-Ramanujan identity also due to Andrews.     

On the quest for positivity in operator algebras

We show that in every nonzero operator algebra with a contractive approximate identity (or c.a.i.), there is a nonzero operator T such that ‖I−T‖?1. In fact, there is a c.a.i. consisting of operators T with ‖I−2T‖?1. So, the numerical range of the elements of our contractive approximate identity is contained in the closed disk center and radius . This is the necessarily weakened form of the result for C^?-algebras, where there is always a contractive approximate identity consisting of operators with 0?T?1 - the numerical range is contained in the real interval [0,1]. So, if an operator algebra has a c.a.i., it must have operators with a “certain amount” of positivity.     

Extended Bressoud-Wei and Koike skew Schur function identities

The Jacobi-Trudi identity expresses a skew Schur function as a determinant of complete symmetric functions. Bressoud and Wei extend this idea, introducing an integer parameter t?−1 and showing that signed sums of skew Schur functions of a certain shape are expressible once again as a determinant of complete symmetric functions. Koike provides a Jacobi-Trudi-style definition of universal rational characters of the general linear group and gives their expansion as a signed sum of products of Schur functions in two distinct sets of variables. Here we extend Bressoud and Wei's formula by including an additional parameter and extending the result to the case of all integer t. Then we introduce this parameter idea to the Koike formula, extending it in the same way. We prove our results algebraically using Laplace determinantal expansions.     

Characterizing the Hilbert transform by the Bedrosian theorem

It is proved that a bounded linear translation invariant operator on L²(Rd) satisfies the Bedrosian theorem if and only if it is a linear combination of the compositions of the partial Hilbert transforms and the identity operator. This observation justifies a definition of multidimensional analytic signals in the papers [T. Bulow, G. Sommer, Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case, IEEE Trans. Signal Process. 49 (2001) 2844-2852] and [S.L. Hahn, Multidimensional complex signals with single-orthant spectra, Proc. IEEE 80 (1992) 1287-1300].     

The products of three theta functions and the general cubic theta functions

In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan＇s identities, Winquist＇s identity and many other interesting identities.     

Frames and bases of subspaces in Hilbert spaces

In this paper we develop the frame theory of subspaces for separable Hilbert spaces. We will show that for every Parseval frame of subspaces {Wi}_i∈I for a Hilbert space H, there exists a Hilbert space K⊇H and an orthonormal basis of subspaces {Ni}_i∈I for K such that Wi=P(Ni), where P is the orthogonal projection of K onto H. We introduce a new definition of atomic resolution of the identity in Hilbert spaces. In particular, we define an atomic resolution operator for an atomic resolution of the identity, which even yield a reconstruction formula.     

Applications of residues to combinatorial identities

A concrete aspect of Grothendieck Duality is used to give local cohomology proofs of combinatorial identities including MacMahon's master theorem, Grosswald identity, identity of Shoo, Tepper identity, and others.

     

Eigentime identity for transient Markov chains

An eigentime identity is proved for transient symmetrizable Markov chains. For general Markov chains, if the trace of Green matrix is finite, then the expectation of first leap time is uniformly bounded, both of which are proved to be equivalent for single birth processes. For birth-death processes, the explicit formulas are presented. As an application, we give the bounds of exponential convergence rates of (sub-) Markov semigroup Pt from l_∞ to l_∞.     

Several identities for certain products of theta functions

By means of a technique used by Carlitz and Subbarao to prove the quintuple product identity (Proc. Am. Math. Soc. 32(1):42–44, 1972), we recover a general identity (Chu and Yan, Electron. J. Comb. 14:#N7, 2007) for expanding the product of two Jacobi triple products. For applications, we briefly explore identities for certain products of theta functions φ(q), ψ(q) and modular relations for the Göllnitz-Gordon functions.     

A general class of fuzzy operators,the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators

In this paper we examine operators which can be derived from the general solution of functional equations on associativity. We define the characteristics of those functions f(x) which are necessary for the production of operators. We shall show, that with the help of the negation operator for every such function f(x) a function g(x) can be given, from which a disjunctive operator can be derived, and for the three operators the DeMorgan identity is fulfilled. For the fulfillment of the DeMorgan identity the necessary and sufficient conditions are given.We shall also show that an f_λ(x) can be constructed for every f(x), so that for the derived k_λ(x,y) and d_λ(x,y) lim_λ→∞k_λ(x,y) and lim_λ→∞d_λ(x,y) = max(x,y).As Yager's operator is not reducible, for every λ there exists an α, for which, in case x < α and y<α, k_λ(x,y) = 0.We shall give an f(x) which has the characteristics of Yager's operator, and which is strictly monotone.Finally we shall show, that with the help of all those f(x), which are necessary when constructing a k(x,y), an F(x) can be constructed which has the properties of the measures of fuzziness introduced by A. De Luca and S. Termini. Some classical fuzziness measures are obtained as special cases of our system.     

On totally positive sequences and functions

Let a>0 be a fixed number. A function f:R→R is said to be a-shift-generating (a-SG) if for every x∈R, is a totally positive sequence and it does not coincide with a sequence of the form , where A?0 and λ>0. In this paper, we describe all a-SG functions and obtain a new characterization of totally positive functions in the terms of a-SG functions. In addition, using characteristic properties of a-SG functions, we generalize the famous Jacobian identity in theory of elliptic functions.     

ON POLYNOMIAL AUTOMORPHISM IDENTITY SETS AND IDENTITY POLYNOMIALS

In this paper, we propose a new condition for hypersurfaces to be polynomial automorphism identity sets. This new condition can be used to give a new proof of Mckay-Wang's problem. Moreover, we also study the concepts of identity polynomials, and give a criterion for a polynomial to be identity polynomial.     

Fq-linear codes over Fq-algebras

Huffman (2013) [12] studied ${\mathbb{F}}_q$-linear codes over ${\mathbb{F}}_{q^m}$ and he proved the MacWilliams identity for these codes with respect to ordinary and Hermitian trace inner products. Let S be a finite commutative ${\mathbb{F}}_q$-algebra. An ${\mathbb{F}}_q$-linear code over S of length n is an ${\mathbb{F}}_q$-submodule of $S^n$. In this paper, we study ${\mathbb{F}}_q$-linear codes over S. We obtain some bounds on minimum distance of these codes, and some large classes of MDR codes are introduced. We generalize the ordinary and Hermitian trace products over ${\mathbb{F}}_q$-algebras and we prove the MacWilliams identity with respect to the generalized form. In particular, we obtain Huffman's results on the MacWilliams identity. Among other results, we give a theory to construct a class of quantum codes and the structure of ${\mathbb{F}}_q$-linear codes over finite commutative graded ${\mathbb{F}}_q$-algebras.     

A new proof of the septuple product identity

By means of Liouville’s theorem and the log-derivative method, we give a new proof of the septuple product identity.     

Another homogeneous q-difference operator

In this paper we show the equivalence between Goldman-Rota q-binomial identity and its inverse. We may specialize the value of the parameters in the generating functions of Rogers-Szegö polynomials to obtain some classical results such as Euler identities and the relation between classical and homogeneous Rogers-Szegö polynomials. We give a new formula for the homogeneous Rogers-Szegö polynomials h_n(x,y|q). We introduce a q-difference operator θ_xy on functions in two variables which turn out to be suitable for dealing with the homogeneous form of the q-binomial identity. By using this operator, we got the identity obtained by Chen et al. [W.Y.C. Chen, A.M. Fu, B. Zhang, The homogeneous q-difference operator, Advances in Applied Mathematics 31 (2003) 659-668, Eq. (2.10)] which they used it to derive many important identities. We also obtain the q-Leibniz formula for this operator. Finally, we introduce a new polynomials s_n(x,y;b|q) and derive their generating function by using the new homogeneous q-shift operator L(bθ_xy).