Number Theory and Discrete Mathematics

Frontmatter

Multiple Polylogarithms: An Introduction

Abstract

Multiple polylogarithms in a single variable are defined by

$$L{i_{\left( {{s_1}, \cdots ,{s_k}} \right)}}\left( z \right) = \sum\limits_{{n_1} > {n_2} > \cdots > {n_k} \geqslant 1} {\frac{{{z^{{n_1}}}}} {{n_1^{{s_1}} \cdots n_k^{{s_k}}}}}$$

, when s₁, … , s _k are positive integers and z a complex number in the unit disk. For k = 1, this is the classical polylogarithm Li _s (z). These multiple polylogarithms can be defined also in terms of iterated Chen integrals and satisfy shuffle relations. Multiple polylogarithms in several variables are defined for s _i ≥ 1 and |z _i | < 1(1 ≤ i ≤ k) by

$$L{i_{\left( {{s_1}, \cdots ,{s_k}} \right)}}\left( {{z_1}, \cdots {z_k}} \right) = \sum\limits_{{n_1} > {n_2} > \cdots > {n_k} \geqslant 1} {\frac{{z_1^{{n_1}} \cdots z_k^{{n_k}}}} {{n_1^{{s_1}} \cdots n_k^{{s_k}}}}}$$

, and they satisfy not only shuffle relations, but also stuffle relations. When one specializes the stuffle relations in one variable at z = 1 and the stuffle relations in several variables at z₁ = ⋯ = z _k = 1, one gets linear or quadratic dependence relations between the Multiple Zeta Values

$$\zeta \left( {{s_1}, \cdots ,{s_k}} \right)\sum\limits_{{n_1} > {n_2} > \cdots > {n_k} \geqslant 1} {\frac{1} {{n_1^{{s_1}} \cdots n_k^{{s_k}}}}}$$

which are defined for k, s₁, … ,s _k positive integers with s₁ ≥ 2. The Main Diophantine Conjecture states that one obtains in this way all algebraic relations between these MZV.

M. Waldschmidt

A (Conjectural) 1/3-phenomenon for the Number of Rhombus Tilings of a Hexagon which Contain a Fixed Rhombus

Abstract

We state, discuss, provide evidence for, and prove in special cases the conjecture that the probability that a random tiling by rhombi of a hexagon with side lengths 2n + a, 2n + b, 2n + c, 2n + a, 2n + b, 2n + c contains the (horizontal) rhombus with coordinates (2n + x, 2n + y) is equal to \(\frac{1}{3} + {g_{a,b,c,x,y}}\left( n \right){\left( {\begin{array}{*{20}{c}} {2n} \\ n \\ \end{array} } \right)^3}/\left( {\begin{array}{*{20}{c}} {6n} \\ {3n} \\ \end{array} } \right)\), where g_a,b,c,x,y(n) is a rational function in n. Several specific instances of this “1/3-phenomenon” are made explicit.

C. Krattenthaler

The Influence of Carr’s Synopsis on Ramanujan

Abstract

Almost all biographers of Ramanujan (e.g., P.V. Seshu Aiyar and R. Ramachandra Rao [13, p. xii]) point to G.S. Carr’s A Synopsis of Elementary Results in Pure Mathematics [10] as the book which kindled the fire of Ramanujan’s devotion to mathematics. How much did Carr’s Synopsis influence Ramanujan? Which published papers and which entries in his notebooks [14] have their seeds in the Synopsis? We cannot provide definitive answers to these questions, because we know very little about the other books Ramanujan might have studied in his formative years. However, upon a close examination of Carr’s book, we can suggest some topics which Ramanujan might have learned from Carr.

Bruce C. Berndt

A Bailey Lemma from the Quintuple Product

Abstract

In a previous paper, the discovery of further Rogers-Ramanujan type identities from new Bailey Lemmas was discussed. In that paper, the starting point was a product of independent Jacobi triple products. In this paper, we start from the quintuple product.

George E. Andrews

Little Flowers to G.H. Hardy (07-02-1877–01-12-1947)

Abstract

Honouring Ramanujan is not complete without honouring G.H. Hardy who collaborated with him in an epoch-making way and brought his contributions to the lime light of the world. In this small article I list a few results of mine and offer it to G.H. Hardy as little flowers.

K. Ramachandra

Rogers-Ramanujan Type Identities for Burge’s Restricted Partition Pairs Via Restricted Frobenius Partitions

Abstract

We obtain generating functions for two sets of infinite families of restricted Frobenius partition functions by giving combinatorial arguments. We also establish a connection between three particular cases of these restricted Frobenius partition functions and Burge’s restricted partition pairs (J. Combin. Theory Ser. A, 63, 1993, 210–222). This connection and Burge’s Theorem 1 give us three new analytic identities. A comparison of these analytic identities with three known identities from Slater’s compendium (Proc. London Math. Soc. (2). 54, 1952, 147–167) leads us to Rogers-Ramanujan type identities for Burge’s restricted partition pairs.

A. K. Agarwal, Padmavathamma

On q-additive and q-multiplicative Functions

I. Kátai

Antimagic Labeling of Complete m-ary Trees

Abstract

A function f is called an antimagic labeling of a graph G with q edges if f is an injection from the edges of G to the set {1, 2, 3,…, q} such that when each vertex v is assigned the sum of all the labels of edges incident with v, the resulting vertex labels are all distinct. It is shown that all complete m-ary trees have antimagic labelings.

P. D. Chawathe, Vijaya Krishna

Some Recent Advances on Symmetric, Quasi-Symmetric and Quasi-Multiple Designs

Abstract

Since the advent of design theory that began with constructive results of R.C. Bose and M. Hall, symmetric designs have occupied a very special status. This is due to several reasons, two most important of which are the structural symmetry and the difficulty in constructions of these designs (compared to other classes of designs). The initial part of this exposition will concentrate on new results on symmetric designs. Quasi-symmetric designs are closely related to symmetric designs. These are designs that have (at the most) two block intersection numbers. One can associate a block graph with a quasi-symmetric design and in many cases of interest, this also turns out to be a graph with special properties, and is called a strongly regular graph. One trivial way of constructing quasi-symmetric designs is to take multiple copies of a symmetric design. Since a symmetric design has exactly one block intersection number, the resulting design will have two block intersection numbers. Such quasi-symmetric designs are called improper. Only one example of a proper quasi-multiple quasi-symmetric design seems to be known so far. The question of the existence of a quasi-multiple of a symmetric design is of some importance particularly when the corresponding symmetric design is known not to exist. In that case, obtaining a quasi-multiple with least multiplicity has attracted attention of combinatorialists in the last thirty years.

Sharad Sane

On T-core Partitions and Quadratic Forms

Abstract

We derive a recurrence for c _t (n), the number of t-core partitions of n, where t ≥ 4. Use of the recurrence requires the solution of a Diophantine equation involving a quadratic form in \(t - 2[\frac{t}{3}]\) variables.

Neville Robbins

Observations on Some Algebraic Equations Associated with Ramanujan’s Work

Abstract

This note collects different observations stimulated by Ramanujan’s results on algebraic equations, namely, the equation with 3 iterated square roots \(x = \sqrt {t + \sqrt {t + \sqrt {t + x} } }\), which he solved by radicals, and the Diophantine equation x³ + y³ + z³ = 1, which appears in The Lost Notebook, along with an astonishing solution. It is shown that, in general, the equation with 5 iterated square roots cannot be solved by radicals and that the Diophantine equation has solutions not previously quoted.

Michele Elia

On Rapidly Convergent Series for Dirichlet L-function Values Via the Modular Relation

Abstract

Special values of various zeta- and L-functions in number theory and related areas at integral (or almost integral) arguments have been the major subject of research over the years.

S. Kanemitsu, Y. Tanigawa, M. Yoshimoto

On a Conjecture of Andrews-II

Abstract

The case k = a of the 1974 conjecture of Andrews on two partition functions A_{λ, k, a}(n) and B_{λ, k, a}(n) was proved by the first author and T.G. Sudha [On a conjecture of Andrews, Internat. J. Math, and Math. Sci. Vol. 16, No. 4 (1993), 763–774]. In this paper we prove the two cases of k = a + 1 and k = a + 2 of the same conjecture.

Padmavathamma, M. Ruby Salestina

Integrity of P2 × P n

Abstract

The vertex Integrity, I(G), of a graph G is defined as

$$I\left( G \right) = \min \left\{ {\left| S \right| + m\left( {G - S} \right)\left| {S\mathop \subset \limits_ \ne } \right.V\left( G \right)} \right\}$$

where m(G − S) is the order of the largest component of G − S. In this paper, we compute I(P₂ × P _n ), the vertex integrity of the Cartesian product of P₂ and P _n .

P. D. Chawathe, S. A. Shende

A Note on Cordial Labelings of Multiple Shells

Abstract

Let G be a graph with vertex set V and edge set E. A vertex labelling f : V → {0, 1} induces an edge labelling \(\bar f:E \to \left\{ {0,1} \right\}\) defined by \(\bar f\left( {uv} \right) = \left| {f\left( u \right) - f\left( v \right)} \right|\). Let v _f (0), v _f (1) denote the number of vertices v with f(v) = 0 and f(v) = 1 respectively. Let e _f (0), e _f (1) be similarly defined. A graph is said to be cordial if there exists a vertex labeling f such that |v _f (0) − v _f (1)| ≤ 1 and |e _f (0) − e _f (1)| ≤ 1. In this paper, we show that every multiple shell \(MS\left\{ {n_1^{{t_1}}, \ldots ,n_r^{{t_r}}} \right\}\) is cordial for all positive integers n₁, …, n _r , t₁, …, t _r .

Mahesh Andar, Samina Boxwala, N. B. Limaye

A Report on Additive Complements of the Squares

Abstract

This article provides an account of some recent investigations into the behaviour of additive complements of the sequence of squares. We begin by defining this notion.

D. S. Ramana

Transcendental Infinite Sums and Some Related Questions

Abstract

Erdős and Chowla put forward some questions regarding non-vanishing of certain infinite sums. In this article, we present an expository account of results obtained in that direction. These include some interesting results of Baker, Birch and Wirsing and some recent work of the present author jointly with Saradha, Shorey and Tijdeman.

Sukumar Das Adhikari

The Lehmer Problem on the Euler Totient: A Pendora’s Box of Unsolvable Problems

Abstract

The celebrated seventy year old, innocent looking problem of D.H. Lehmer [5] asking for composite numbers, if any, satisfying the relation φ(n)|(n − 1), where φ(n) is the Euler totient, is still unsolved. This is easily seen to be equivalent to asking the

M. V. Subbarao

The Problems Solved by Ramanujan in the Journal of the Indian Mathematical Society

Abstract

Between 1912 and 1914, eight solutions by Ramanujan to questions posed in the Journal of the Indian Mathematical Society were published. Since these solutions have not heretofore appeared elsewhere, and since some of these problems evidently motivated certain entries in his notebooks [6], in this paper, we present all eight problems and solutions and provide some commentary on them.

Bruce C. Berndt

On the gcd and lcm of Matrices Over Dedekind Domains

Abstract

In an interesting paper published in 1986, Thompson [8] gave a brief and elegant account of classical theory (traceable to the work of Cahen 1914 and Châtlet 1924, conveniently through the book of McDuffee [3]). Earlier Hua [2] had dealt with the subject in his book. Thompson also answered some questions of Morris Newman regarding relations between matrices A and B and their gcd’s and lcm’s in case the underlying ring R is a principal ideal domain (PID). Some of these are repeated here, albeit with different proofs (but for the last section, we too work over PID’s).

V. C. Nanda

The Billiard Ball Motion Problem I: A Markoff Type Chain for the Octahedron in ℜ3

Abstract

Suppose that a particle moves in a rectilinear, uniform motion inside the unit cube

$$ U:-1/2\leq {{x}_{i}}\leq 1/2,\quad i=1,\ldots ,n $$

.

R. J. Hans-Gill

Hilbert’s Seventeenth Problem and Pfister’s Work on Quadratic Forms

Abstract

On August 8, 1900, David Hilbert [5], in his famous address at the International Congress of Mathematicians in Paris, proposed twenty three problems as sign posts for twentieth century Mathematics; the seventeenth being

A. R. Rajwade

Certain Representations of Mock-Theta Functions

Abstract

Recently, Denis [4], making use of the identity

$${e_q}\left( {xt - xyt} \right){e_q}\left( {t - tx} \right) = {e_q}\left( {t - xyt} \right)$$

(1)

where \({e_q}\left( x \right) = \sum\nolimits_{n = 0}^\infty {\frac{{{x^n}}}{{{{[q]}_n}}}}\) with [q]n = (1 − q)(1 − q²)…(1 − q ⁿ ), for n ≥ 1, [q]₀ = 1, established the following result:

$$\begin{gathered}\sum\limits_{m = 0}^n {\left[ {\begin{array}{*{20}{c}}n \\m \\\end{array} } \right]} \frac{{{x^m}{{[(a)]}_m}}}{{{{[(b)]}_m}}}A + 1{\Phi _B}\left[ {\begin{array}{*{20}{c}}{{q^{ - n + m}},} & {(a){q^m};x{q^{n - m}}} \\{} & {(b){q^m}} \\\end{array} } \right] \hfill \\{\times _{C + 1}}{\Phi _D}\left[ {\begin{array}{*{20}{c}}{{q^{ - m}},} & {(c);y{q^m}} \\{} & {(d)} \\\end{array} } \right] \hfill \\{ = _{A + C + 1}}{\Phi _{B + D}}\left[ {\begin{array}{*{20}{c}}{{q^{ - n}},} & {(a),(c);xy{q^n}} \\{} & {(b),(d)} \\\end{array} } \right]. \hfill \\\end{gathered}$$

(2)

with A = B, C = D and \(\left[ {\begin{array}{*{20}{c}}n \\m \\\end{array} } \right]\) is the q-binomial coefficient, defined by [q] _n /[q] _m [q]_n−m and the ϕ-functions are the usual basic hypergeometric functions (cf. Section 2 for detailed definitions). The parameters of the type (a _m ), m ∈ N in small brackets shall stand for the sequence of m parameters a₁, a₂, …, a _m . If m = A, we shall denote it by (a) instead of (a _A ).

R. Y. Denis, S. N. Singh, D. Sulata

Bi-Graceful Graphs

Abstract

A graph G is said to be bi-graceful if both G and its line graph L(G) are graceful. In this paper we study bi-graceful graphs and prove that C₄ with two disjoint paths of the same length attached at any two adjacent points is bi-graceful.

M. Murugan

Wheels, Cages and Cubes

Abstract

Let G = 〈V, E〉 be a graph of order p ≥ 2 and P = {V₁, V₂, … V _k } be a partition of V of order k. The k-complement G _k ^p of G is obtained as follows: For all V _i and V _j in P, i ≠ j, remove the edges between V _i and V _j , and add the missing edges between them. G is said to be k-self-complementary if for some partition P of V of order k, G _k ^p ≈ G; and it is said to be k-co-self-complementary if \(G_k^p \approx \overline G\). In this paper we characterize the k-self-complementary generalized wheels, cubes and cages.

G. Sudhakara

Relevance of Srinivasa Ramanujan at the Dawn of the New Millennium

Abstract

The work of Ramanujan will be appreciated, as long as people do mathematics, opined the Astrophysicist Nobel Laureate Dr. S. Chandrasekhar, at the time of the birth centenary of Ramanujan. Prof. E.H. Neville began a broadcast in Hindustani, in 1941, as follows: Srinivasa Ramanujan was a mathematician so great that his name transcends jealousies, the one superlatively great mathematician whom India has produced in the last thousand years. Undoubtedly, Srinivasa Ramanujan (Dec. 22, 1887 – April 26, 1920) is one of the greatest Mathematicians of the twentieth century. For his mathematical abilities and natural genius he has been compared, by his contemporaries, Professors G.H. Hardy and J.E. Littlewood, with all-time great mathematicians, Leonhard Euler, Carl Friedrich Gauss and Karl Gustav Jacobi. Marc Kac said: An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we too, could have done it. It is different with the magicians … the working of their minds is for all intents and purposes incomprehensible. Even after we understand what they have done, the process by which they have done it is completely dark. Prof. Bruce C. Berndt who has methodically and thoroughly edited every one of the 3254 entries of Ramanujan in his three Notebooks — in Ramanujan’s Notebooks, Parts I to V, published by Springer-Verlag (1985–1997) — states that though there are a few scattered errors in these notebooks, Ramanujan’s accuracy is amazing and mystery … still surrounds some of his work.

K. Srinivasa Rao

Number of Solutions of Equations over Finite Fields and a Conjecture of Lang and Weil

Abstract

A brief survey of the conjectures of Weil and some classical estimates for the number of points of varieties over finite fields is given. The case of partial flag manifolds is discussed in some details by way of an example. This is followed by a motivated account of some recent results on counting the number of points of varieties over finite fields, and a related conjecture of Lang and Weil. Explicit combinatorial formulae for the Betti numbers and the Euler characteristics of smooth complete intersections are also discussed.

Sudhir R. Ghorpade, Gilles Lachaud

On An Additive Question

Abstract

Let k > 0 be a given integer. Here we obtain some results concerning solvability of A ⊕ B = ℤ ^k , in B, with respect to a finite set A of a given ‘diameter’. And also announce some other results regarding a conjecture from R. Tijdeman in the case k = 1.

S. Srinivasan

n-Colour Partitions

Abstract

In this paper we shall survey the advances that have been made in recent years in dealing with problems on n-colour partitions. An n-colour partition (also called a partition with n copies of n) of a positive integer vis a partition in which a part of size n, can come in n different colours denoted by subscripts: n₁, n₂, …, n _n and the parts satisfy the order.

A. K. Agarwal