New series representations for Jacobi?s triple product identity and more via the q-Markov method

We derive new series representations for Jacobi?s triple product identity, the q-binomial theorem, q-analogs of the exponential function, and more with several special cases using the q-Markov-WZ method.     

Bilateral truncated Jacobi’s identity

Recently, Andrews and Merca considered the truncated version of Euler’s pentagonal number theorem and obtained a non-negative result on the coefficients of this truncated series. Guo and Zeng showed the coefficients of two truncated Gauss’ identities are non-negative and they conjectured that truncated Jacobi’s identity also has non-negative coefficients. Mao provided a proof of this conjecture by using an algebraic method. In this paper, we consider bilateral truncated Jacobi’s identity and show that when the upper and lower bounds of the summation satisfy some certain restrictions, then this bilateral truncated identity has non-negative coefficients. As a corollary, we show the conjecture of Guo and Zeng holds. Our proof is purely combinatorial and mainly based on a bijection for Jacobi’s identity.     

A proof of Ewell’s octuple product identity

We give a new proof of Ewell’s octuple product identity in [7] by using a general theorem developed by the first author in [3].     

An algebraic identity and the Jacobi–Trudi formula

The first Jacobi–Trudi identity expresses Schur polynomials as determinants of matrices, the entries of which are complete homogeneous polynomials. The Schur polynomials were defined by Cauchy in 1815 as the quotients of determinants constructed from certain partitions. The Schur polynomials have become very important because of their close relationship with the irreducible characters of the symmetric groups and the general linear groups, as well as due to their numerous applications in combinatorics. The Jacobi–Trudi identity was first formulated by Jacobi in 1841 and proved by Nicola Trudi in 1864. Since then, this identity and its numerous generalizations have been the focus of much attention due to the important role which they play in various areas of mathematics, including mathematical physics, representation theory, and algebraic geometry. Various proofs of the Jacobi–Trudi identity, which are based on different ideas (in particular, a natural combinatorial proof using Young tableaux), have been found. The paper contains a short simple proof of the first Jacobi–Trudi identity and discusses its relationship with other well-known polynomial identities.     

Richter’s local limit theorem and Black–Scholes type formulas

We prove a Black–Scholes type formula when the geometric Brownian motion originates from approximations by multinomial distributions. It is shown that the variance appearing in the Black–Scholes formula for option pricing can be structured according to occurrences of different types of events at each time instance using a local limit theorem for multinomial distributions in Richter (1956). The general approach has first been developed in Kan (2005).     

Jacobi and Kummer’s ideal numbers

In this article we give a modern interpretation of Kummer’s ideal numbers and show how they developed from Jacobi’s work on cyclotomy, in particular the methods for studying “Jacobi sums” which he presented in his lectures on number theory and cyclotomy in the winter semester 1836/37.     

Darboux’s identity and its analogs

Let us choose a positive integern and denoteF(x, y)= , wheref(·) andg(·) are arbitrary sufficiently smooth functions. Three different proofs of the validity of the relation

Generalizations of girstmair’s formulas

In 1994 and 1995 GIRSTMAIR gave (relative) class number formulas for the imaginary quadratic field $\mathbb{Q}(\sqrt { - p} )$ , P an odd prime with p ≡ 3 (mod 4) and p ≥ 7, using the coefficients of the digit expression of 1/p and z/p, respectively, where z is an integer with 1 ≤ z ≤p - 1. We extend the formulas to an imaginary abelian number field.     

A nonlinear Picone’s identity and its applications

The purpose of this note is to show a generalization to Picone’s identity in a nonlinear framework. The classical Picone’s identity turns out to be a particular case of our result. We show, as an application of our results, that the Morse index of the zero solution to a semilinear elliptic boundary value problem is 0 and also establish a linear relationship between the components of the solution of a nonlinear elliptic system.     

Some extensions for Ramanujan’s circular summation formulas and applications

The Ramanujan Journal - In this paper, we give some extensions for Ramanujan’s circular summation formulas with the mixed products of two Jacobi’s theta functions. As applications, we...     

Heat Kernel and Hardy’s Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associated with Jacobi transform, and get some analogues of Hardy's Theorem for Jacobi transform by using the sharp estimate of the heat kernel.     

Heat Kernel and Hardy’s Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associated with Jacobi transform, and get some analogues of Hardy’s Theorem for Jacobi transform by using the sharp estimate for the heat kernel.     

Heat Kernel and Hardy’s Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower hounds for the heat kernel associated with Jacobi transform, and get some analogues of Hardy‘s Theorem for Jacobi transform by using the sharp estimate of the heat kernel.     

Domination in the hierarchical product and Vizing’s conjecture

Given a graph $G$, a set $S?V\left(G\right)$ is a dominating set of $G$ if every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. The domination number of $G$, denoted $\gamma \left(G\right)$, is the minimum cardinality of a dominating set of $G$. Vizing’s conjecture states that $\gamma (G\square H)\ge \gamma \left(G\right)\gamma \left(H\right)$ for any graphs $G$ and $H$ where $G\square H$ denotes the Cartesian product of $G$ and $H$. In this paper, we continue the work by Anderson et al. (2016) by studying the domination number of the hierarchical product. Specifically, we show that partitioning the vertex set of a graph in a particular way shows a trend in the lower bound of the domination number of the product, providing further evidence that the conjecture is true.     

Whitney’s formulas for curves on surfaces

The classical Whitney formula relates the number of times an oriented plane curve cuts itself to its rotation number and the index of a base point. In this paper we generalize Whitney’s formula to curves on an oriented punctured surface Σ _{m, n} , obtaining a family of identities indexed by elements of π ₁(Σ _{m, n} ). To define analogs of the rotation number and the index of a base point of a curve γ, we fix an arbitrary vector field on Σ _{m, n} . Similar formulas are obtained for non-based curves.     

Hartwig’s triple reverse order law revisited

We extend the classical Hartwig’s triple reverse-order law for the Moore–Penrose inverse to closed range bounded linear operators on infinite dimensional Hilbert spaces.