Approximate identities from Laguerre functions and singular integrals on the Heisenberg group

Suitably scaled Laguerre functions are an approximate identity for multiplicative convolution with test functions on the half line. As an application, we derive a precise connection between the Mikhlin-type expansion of a singular integral operator on a Heisenberg groupH _n and its natural restriction toH _n modulo the center. Research supported by NSF Grant DMS-9800605 and by NSERC Grant OGP0003017.     

Automorphic integrals with log-polynomial period functions and arithmetical identities

Building on the works of S. Bochner on equivalence of modular relation with functional equation associated to the Dirichlet series, K. Chandrasekharan and R. Narasimhan obtained new equivalences between the functional equation and some arithmetical identities. Sister Ann M. Heath considered the functional equation in the Hawkins and Knopp context and showed its equivalence to two arithmetical identities associated with entire modular cusp integrals involving rational period functions for the full modular group. In this paper we use techniques of Chandrasekharan and Narasimhan to prove results analogous to those of Sister Ann M. Heath. Specifically, we establish equivalence of two arithmetical identities with a functional equation associated with automorphic integrals involving log-polynomial-period functions on the discrete Hecke groups.     

Resultants and contour integrals

Resultants are important special functions used to describe nonlinear phenomena. The resultant R_{r1 ?rn} R_{r_1 \ldots r_n } determines a consistency condition for a system of n homogeneous polynomials of degrees r ₁, ..., r _n in n variables in precisely the same way as the determinant does for a system of linear equations. Unfortunately, there is a lack of convenient formulas for resultants in the case of a large number of variables. In this paper we use Cauchy contour integrals to obtain a polynomial formula for resultants, which is expected to be useful in applications.     

From integrals to multi-sum identities

Ramanujan's Master Theorem and its extension in the method of brackets are a powerful technique for evaluation of many definite integrals, often producing series solutions. In the case that the series are multi-sums, simplification is nontrivial. Recurrence-finding algorithms assist in proving multi-sum identities to verify the correctness of such multi-sum series solutions.     

Combinatorial identities and Polya walks

Two combinatorial identities, which generalize previously known ones, are proved on the basis of the random walks scheme.Translated from Staticheskie Metody, pp. 154–159, 1978.     

The plane wave expansion, infinite integrals and identities involving spherical Bessel functions

This paper shows that the plane wave expansion can be a useful tool in obtaining analytical solutions to infinite integrals over spherical Bessel functions and the derivation of identities for these functions. The integrals are often used in nuclear scattering calculations, where an analytical result can provide an insight into the reaction mechanism. A technique is developed whereby an integral over several special functions which cannot be found in any standard integral table can be broken down into integrals that have existing analytical solutions.     

The commutativity of integrals of motion for quantum spin chains and elliptic functions identities

We prove the commutativity of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. We also show their liner independence for the number of spins larger than 4. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments.      

Combinatorial proofs of some determinantal identities

In this paper, we give combinatorial proofs of some determinantal identities. In fact, we give a combinatorial proof of a theorem of R. P. Stanley regarding the enumeration of paths in acyclic digraphs along with some interesting applications. We also give an almost visual proof of a recent result of Oliver Knill, namely ‘The generalized Cauchy–Binet Theorem.’     

Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions

Integration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the independent variable u is the argument of Jacobian elliptic function theory. This allows a limitless number of indefinite integrals with respect to the amplitude to be derived for these three elliptic integrals. Sample results are given, including the integrals derived from powers of the 12 Glaisher elliptic functions. New recurrence relations and integrals are also given for the 12 Glaisher elliptic functions.     

Asymptotics of zeros of polynomials arising from rational integrals

We prove that the zeros of the polynomials Pm(a) of degree m, defined by Boros and Moll via

     

Some identities involving Inverse Gamma integrals

The Ramanujan Journal - In this paper we will give some identities related with the Fransén–Robinson constant and the Inverse Gamma function. The main result is to use Riemann...     

New identities of differential operators from orbital integrals on

We derive identities of differential operators on complex general linear groups which appear in the differential equations satisfied by weighted orbital integrals. These identities stem from and have applications to comparisons of metaplectic coverings.

     

Tribasic integrals and identities of Rogers-Ramanujan type

Some integrals involving three bases are evaluated as infinite products using complex analysis. Many special cases of these integrals may be evaluated in another way to find infinite sum representations for these infinite products. The resulting identities are identities of Rogers-Ramanujan type. Some integer partition interpretations of these identities are given. Generalizations of the Rogers-Ramanujan type identities involving polynomials are given again as corollaries of integral evaluations.

     

Two identities for integrals of hypergeometric type

Two classic identities for the Gaussian hypergeometric function are extended to the case of hypergeometric type integrals. Some other identities for these integrals are derived.     

Combinatorial generators of the multilinear polynomial identities

A Gröbner-Shirshov basis (a combinatorial system of generators) is defined in the set of multilinear elements of a T-ideal of the free associative algebra with a countable set of indeterminates. A combinatorial version of the well-known Specht problem about the finite basedness of polynomial identities of an arbitrary associative algebra is formulated. A “combinatorial Spechtness” property of the multilinear product of commutators of degree 2 and the same property for the three-linear commutator are established.