Partial sums of hypergeometric series of unit argument

The asymptotic behaviour of partial sums of generalized hypergeometric series of unit argument is investigated.

     

Finite Aomoto-Ito-Macdonald sums

We present finite truncations of the Aomoto-Ito-Macdonald sums associated with root systems through a two-step reduction procedure. The first reduction restricts the sum from the root lattice to a Weyl chamber; the second reduction arises after imposing a truncation condition on the parameters, and gives rise to a finite sum over a Weyl alcove.

     

Supercongruences for truncated hypergeometric series with applications to certain weight three newforms

We prove general results on supercongruences between values of truncated hypergeometric functions and their character analogs. As a consequence of the main results of this paper, we prove Beukers-type supercongruences for certain weight three newforms.

     

Summation formulae for elliptic hypergeometric series

Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.

     

On partial sums of mock theta functions of order three

The object of this paper is to define and study the properties of partial mock theta functions of order three, on the lines Ramanujan had studied partial θ-functions. These new partial functions have been expressed in terms of basic hypergeometric function₂Φ₁. Their continued fractions representations have also been given.     

A reinforcement-depletion urn model: A contiguity case

Summary An urn contains balls ofs different colors. The problem of the reinforcement of a specified color and random depletion of balls has been considered by Bernard (1977,Bull. Math. Biol.,39, 463–470) and Shenton (1981,Bull. Math. Biol.,43, 327–340), (1983,Bull. Math. Biol.,45, 1–9). Here we consider a special relation between a reinforcement and depletion, leading to a hypergeometric distribution. Research sponsored in part by the Applied Mathematical Sciences Research Program, Office of Energy Research, U.S. Department of Energy under contract DE-AC05-840R21400 with the Martin Marietta Energy Systems, Inc.     

A Hypergeometric Hierarchy for the Andrews Evaluations

Several 6F5(1) evaluations are given which generalize Andrews' 5F4(1) evaluations. All such evaluations are shown to be equivalent to transformations for a 4F3(z). The methodology allows for higher evaluations, for example an 8F7(1) is given which specializes to over one hundred 5F4(1) results, including all of Andrews'.     

Evaluation of infinite series involving special products and their algebraic characterization

The aim of the paper is the investigation of special infinite series of the form

A generating function for sums of multiple zeta values and its applications

A generating function for specified sums of multiple zeta values is defined and a differential equation that characterizes this function is given. As applications, some relations for multiple zeta values over the field of rational numbers are discussed.

     

Diophantine Problems for q-Zeta Values

Mathematical Notes -     

Asymptotic formulas for sequence factorial of arithmetic progression

This note provides asymptotic formulas for approximating the sequence factorial of members of a finite arithmetic progression by using Stirling, Burnside and other more accurate asymptotic formulas for large factorials that have appeared in the literature.     

The Well-Poised Thread: An Organized Chronicle of Some Amazing Summations and their Implications

This paper provides an organized history of well-poised hypergeometric series. The object is to reveal the process by which a rather narrow mathematical study blossomed into a topic of widespread importance. In addition, short biographies of the early contributors are included.     

Sequence factorial and its applications

In this note, we introduce sequence factorial and use this to study generalized M-bonomial coefficients. For the sequence of natural numbers, the twin concepts of sequence factorial and generalized M-bonomial coefficients, respectively, extend the corresponding concepts of factorial of an integer and binomial coefficients. Some latent properties of generalized M-bonomial coefficients by which a vast majority of practical problems involving generalized M-bonomial coefficients can be solved are derived.     

A simple demonstration of zero factorial equals one

When asked, a number of students answer zero factorial to be zero as a continuation to the answer of one factorial to be one. Any instructor would then seek a justification of zero factorial to be one from computing _nC_n via the well- known combination formula. This article conveys a simple presentation of zero factorial to be one based on lower and upper bounds of n factorial. We have not seen this explanation covered in any algebra textbook.     

两个q-级数恒等式

本文用部分求和项满足反演关系的方法给出了两个 q -级数恒等式 .证明了这种方法对寻求新的恒等式还是很有效的