A generalization of Calkin's identity

The purpose of this short note is to give a generalization of Calkin's identity.     

A generalization of the AZ identity

The identity discovered in [1] can be viewed as a sharpening of the LYM inequality ([3], [4], [5]). It was extended in [2] so that it covers also Bollobás' inequality [6]. Here we present a further generalization and demonstrate that it shares with its predecessors the usefullness for uniqueness proofs in extremal set theory.     

A generalization of the Subbarao identity

Arithmetical functionsf andh are said to satisfy the Subbarao identity if

On one generalization of modular subgroups

We study the influence of generalized modular subgroups on the structure of finite groups.     

A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms

Rankin and Swinnerton-Dyer (1970) prove that all zeros of the Eisenstein series in the standard fundamental domain for lie on . In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc . Using this result we prove a speculation of Ono, namely that the zeros of the unique ``gap function" in , the modular form with the maximal number of consecutive zero coefficients in its -expansion following the constant , has zeros only on . In addition, we show that the -invariant maps these zeros to totally real algebraic integers of degree bounded by a simple function of weight .

     

A simple proof of the modular identity for theta series

We characterize the function spanned by theta series. As an application we derive a simple proof of the modular identity of the theta series.

     

A simple proof of the modular identity for theta functions

The modular identity arises in the theory of theta functions in one complex variable. It states a relation between theta functions for parameters and situated in the complex upper half-plane. A standard proof uses Poisson summation and hence builds on results from Fourier theory. This paper presents a simple proof using only a uniqueness property and the heat equation.

     

An elliptic generalization of Schur's Pfaffian identity

We present a Pfaffian identity involving elliptic functions, whose rational limit gives a generalization of Schur's Pfaffian identity for Pf((x_j-x_i)/(x_j+x_i)). This identity can be regarded as the Pfaffian counterpart of Frobenius’ generalization of Cauchy's determinant identity for det(1/(x_i+x_j)).     

A generalization of Kawanaka’s identity for Hall-Littlewood polynomials and applications

An infinite summation formula of Hall-Littlewood polynomials due to Kawanaka is generalized to a finite summation formula, which implies, in particular, twelve more multiple q-identities of Rogers-Ramanujan type than those previously found by Stembridge and the last two authors.     

A generalization of Bessel"s inequality and Parseval"s identity

We present an inequality and an identity involving Gram determinants, which can be viewed as a generalization of Bessel"s and Parseval"s, respectively. This revised version was published online in June 2006 with corrections to the Cover Date.     

Lattices with modular identity and Lie algebras

The classification of Lie algebras over a principal ideal ring, the lattice of whose subalgebras satisfies a modular identity, is given.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 26, pp. 213–257, 1985.     

A generalization of Sylvester's identity on determinants and some applications

A generalization of Sylvester's identity on determinants is proved by elimination techniques. As an application some inequalities for determinants of totally positive or positive definite matrices are derived.     

A generalization of the Ricci identity for a function of a field of arbitrary type given in a space of affine connectedness

For a function of a field of arbitrary type given in a space of affine connectedness a relationship is deduced which is analogous to the well-known Ricci identity for tensor fields.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 409–411, April, 1969.