Analysis of a generalized Lebesgue identity in Ramanujan’s Lost Notebook

We analyze a two-parameter q-series identity in Ramanujan’s Lost Notebook that generalizes the product part of the fundamental one-parameter Lebesgue identity. From reformulations of this two-parameter identity, we deduce new partition theorems including variants of the Gauss triangular number identity and Euler’s pentagonal number theorem. We discuss connections with a partial theta identity of Ramanujan and with several classical results such as those of Sylvester and Göllnitz–Gordon.     

An AZ-style identity and Bollobás deficiency

The powerful AZ identity is a sharpening of the famous LYM-inequality. More generally, Ahlswede and Zhang discovered a generalization in which the Bollobás inequality for two set families can be lifted to an identity.In this paper, we show another generalization of the AZ identity. The new identity implies an identity which characterizes the deficiency of the Bollobás inequality for an intersecting Sperner family. We also give some consequences relating to Helly families and LYM-style inequalities.     

Identity excluding groups

We consider identity excluding groups. We first show that motion groups of totally disconnected nilpotent groups are identity excluding. We prove that certain class of p-adic algebraic groups which includes algebraic groups whose solvable radical is type R have identity excluding property. We also prove the convergence of averages of representations for some solvable groups which are not necessarily identity excluding.     

Bijective proofs of Gould's and Rothe's identities

We first give a bijective proof of Gould's identity in the model of binary words. Then we deduce Rothe's identity from Gould's identity again by a bijection, which also leads to a double-sum extension of the q-Chu-Vandermonde formula.     

Convolution identities for tensors

The concept of a convolution identity for tensors is introduced and it is proved that any convolution identity for tensors on a finite-dimensional space follows from a convolution identity equivalent to the classical Cayley-Hamilton identity.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 211–214, 1982.     

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.     

A modular look at Farkas’ identities

Recently H. Farkas introduced a new simple arithmetic function and found an identity which involves this function. It is immediate to rewrite this identity as an identity between modular forms and reprove it in this way. We discuss natural generalizations of Farkas’ identity. Surprisingly, in a certain sense, there is only one identity which is an exact analogue of that found by Farkas. At the same time, we present a way to produce infinitely many similar identities. As an application, we obtain a result on non-vanishing of the central critical value of L-functions associated to a cusp Hecke eigenform. Supported by NSF grant DMS-0700933.     

A generalization of Clausen’s identity

The paper gives an extension of Clausen’s identity to the square of any Gauss hypergeometric function. Accordingly, solutions of the related third-order linear differential equation are found in terms of certain bivariate series that can reduce to ₃F₂ series similar to those in Clausen’s identity. The general contiguous variation of Clausen’s identity is found as well. The related Chaundy’s identity is generalized without any restriction on the parameters of the Gauss hypergeometric function. The special case of dihedral Gauss hypergeometric functions is underscored.     

Minimal differential identities in prime rings

It is known that a prime ring which satisfies a polynomial identity with derivations applied to the variables must satisfy a generalized polynomial identity, but not necessarily a polynomial identity. In this paper we determine the minimal identity with derivations which can be satisfied by a non-PI prime ringR. The main result shows, essentially, that this identity is the standard identityS ₃ withD applied to each variable, whereD = ad(y) fory inR, y ² = 0, andy of rank one in the central closure ofR.     

On Local LYM Identities

The LYMinequality (Lubell, Yamamoto, Meshalkin) is a generalization of Sperner’s theorem for antichains. Kleitman and Harper independently proved that the LYM inequality and the normalized matching property (or local LYM inequality) are equivalent. Many contributions have been proposed to sharpen the LYM inequality. Noticeably, Ahlswede and Zhang lifted the LYM inequality to an identity, called the AZ identity. Thus, one expects that the same sharpening of the local LYM inequality is equivalent to the AZ identity. In this paper, we introduce a local LYM identity which sharpens the local LYM inequality and prove that it is equivalent to the AZ identity. The local LYM identity shows local relationships between components in the AZ identity.