A Few Theta-Function Identities and Some of Ramanujan's Modular Equations

We prove three modular equations of Ramanujan using theta-function identities. Proofs via methods known to Ramanujan were not available hitherto. One had previously been proved by classical methods, and two had been proved using the theory of modular forms.     

The Borweins' Cubic Theta Function Identity and Some Cubic Modular Identities of Ramanujan

In this paper the author will give new proofs of the Borweins' cubic theta function identity and a related identity relying on the properties of elliptic functions and the technique of comparing constant terms.     

Asymptotics of Orthogonal Polynomials: Some Old, Some New, Some Identities

We briefly review some asymptotics of orthonormal polynomials. Then we derive the Bernstein–Szeg, the Riemann–Hilbert (or Fokas–Its–Kitaev), and Rakhmanov projection identities for orthogonal polynomials and attempt a comparison of their applications in asymptotics.     

Gauss和与广义Kloosterman和的几个新的恒等式

主要应用Gauss和的性质和解析方法来研究Gauss和与广义Kloosterman和之间的关系,并且给出了几个有趣的恒等式.     

Some Combinatorial and Analytical Identities

We give new proofs and explain the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu. We use the theory of basic hypergeometric functions, and generalize these identities. We also exploit the theory of polynomial expansions in the Wilson and Askey-Wilson bases to derive new identities which are not in the hierarchy of basic hypergeometric series. We demonstrate that a Lagrange interpolation formula always leads to very-well-poised basic hypergeometric series. As applications we prove that the Watson transformation of a balanced ${_{4}\phi_{3}}$ to a very-well-poised ${_{8}\phi_{7}}$ is equivalent to the Rodrigues-type formula for the Askey-Wilson polynomials. By applying the Leibniz formula for the Askey-Wilson operator we also establish the ${_{8}\phi_{7}}$ summation theorem.     

N-graphs,Modular Sidon and Sum-Free Sets,and Partition Identities

Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of a set that is a finite union of arithmetic progressions associated with a modular sum-free Sidon set. Partition identities are also constructed for sets associated with modular sum-free sets.     

Some Algebraic Identities in 3-Prime Near-Rings

Ukrainian Mathematical Journal - We extend the domain of applicability of the concept of (1, ??)-derivations in 3-prime near-rings by analyzing the structure and commutativity of the...     

Graded Identities of Some Simple Lie Superalgebras

We study ?₂-graded identities of Lie superalgebras of the type b(t), t?≥?2, over a field of characteristic zero. Our main result is that the n-th codimension is strictly less than \((\dim b(t))^n\) asymptotically. As a consequence we obtain an upper bound for ordinary (non-graded) PI-exponent for each simple Lie superalgebra b(t), t?≥?3.     

Some Identities for Enumerators of Circulant Graphs

We establish analytically several new identities connecting enumerators of different types of circulant graphs mainly of prime, twice prime and prime-squared orders. In particular, it is shown that the half-sum of the number of undirected circulants and the number of undirected self-complementary circulants of prime order is equal to the number of directed self-complementary circulants of the same order. Several identities hold only for prime orders p such that (p + 1)/2 is also prime. Some conjectured generalizations and interpretations are discussed.     

关于两类广义多项式的一些恒等式(英文)

By means of generating function and partial derivative methods, we investigate and establish several general summation formulas involving two classes of polynomials. The general results would apply to yield some identities for the Pell polynomials and Pell-Lucas polynomials, and other general polynomials can also be recovered in this paper.     

一些包含勒让德多项式的积分恒等式

设P_n(x)表示勒让德多项式.即就是P_0(x)=1,P_1(x)=x,当n≥2时有递推关系,P_(n+1)(x)=(2n+1)/(n+1)·xP_n(x)-n/(n+1)·P_(n-1)(x).主要目的是运用初等方法以及幂级数的性质讨论一类包含P_(n)(x)的卷积的定积分计算问题,并给出一些确切的计算公式.     

Some Finite Abelian Group Theory and Some $${}$$-Series Identities

The purpose of this article is to compute certain weighted sums over various subposets of the poset of isomorphism classes of finite abelian \({\ell}\)-groups, where \({\ell}\) is a fixed odd prime. These sums are similar to previous sums computed by Hall and Cohen-Lenstra. The computation expands upon the previous analysis of Cohen-Lenstra while also using the tools of hypergeometric functions and \({q}\)-series. The identities computed in this article, while aesthetically pleasing in their own right, also turn out to be applicable to the construction of a random matrix version of the Cohen-Lenstra-Martinet heuristics.     

Some Identities for Elements of a Symmetric Matrix

Let Sym (n) be the space of n-dimensional real symmetric matrices and let X ∈ Sym (n). The matrices E, X, X²,... , X^{n − 1} may be considered as vectors of the Euclidean space of dimension n². Denote by V(E, X,... , X^{n − 1}) the volume of the parallelepiped spanned by these vectors. It is proved that

Some Properties of Reduced Modular Polynomials

In this paper we will discuss some properties of reduced modular polynomials used in SEA algorithm for computing the number of rational points of elliptic curves on finite fields.     

Minimal Characteristic Algebras for Some Properties of Identities

We consider algebras of a type , without nullary fundamental operation symbols. A structural property p of an identity of type is hereditary if for every set I of identities of type having the property p every consequence of I (derived identity) has the property p. An algebra of type we call characteristic for a hereditary property p if for every variety V of type we have: V if and only if every identity from Id(V) has the property p. In this paper we show minimal characteristic algebras for several hereditary properties, e.g., to be regular, to be normal, etc.1991 Mathematics Subject Classification 08B05     

Some Skew Linear Groups Satisfying Generalized Group Identities

In this article, we consider a type of generalized group identity and extend some earlier results. For example, we show that, if D is a division ring with infinite center, then every subnormal subgroup of GL_n(D) satisfying a generalized group identity over GL_n(D) is central.     

A New Approach to Polynomial Identities

Using the formal derivative idea, we give a generalization for the Cauchys Theorem relating to the factors of (x + y)ⁿ–xⁿ– yⁿ. We determine the polynomials A(n, a, b) and B(n, a, b) such that the polynomial

Some Identities in Algebras Obtained by the Cayley-Dickson Process

Polynomial identities in algebras are the central objects of Polynomial Identities Theory. They play an important role in learning of algebras properties. In particular, the Hall identity is fulfilled in the quaternion algebra and does not hold in other non-commutative associative algebras. For this reason, the Hall identity is important for the quaternion algebra. The idea of this work is to generalize the Hall identity to algebras obtained by the Cayley-Dickson process. Starting from the above remarks, in this paper, we prove that the Hall identity is true in all algebras obtained by the Cayley-Dickson process and, in some conditions, the converse of this statement is also true for split quaternion algebras. From Hall identity, we will find some new properties and identities in algebras obtained by the Cayley-Dickson process.