Table Of ContentNumber Theory and
Discrete Mathematics
Edited by
A. K. Agarwal
Bruce C. Berndt
Christian F. Krattenthaler
Gary L. Mullen
K. Ramachandra
Michel Waldschmidt
I1.JOla. HINDUSTAN
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ISBN 978-81-85931-32-6 ISBN 978-93-86279-10-1 (eBook)
DOl 10.1007/978-93-86279-10-1
Contents
List of Participants.. ..... ..... ..... ..... ....... ............ ........ .... ..... ......... ...... .............. .... vii
Preface. ......... .... ... ............ ............... ... ..... ......... ...... ............ ..................... ...... .... XUI
Multiple Polylogarithms: An Introduction
M. Waldschmidt ............................................................................................... .
A (Conjectural) 1I3-phenomenon for the Number of Rhombus Tilings
of a Hexagon which contain a Fixed Rhombus
C. Krattenthaler................................................................................................ 13
The Influence of Carr's Synopsis on Ramanujan
Bruce C. Berndt................................................................................................ 31
A Bailey Lemma from the Quintuple Product
George E. Andrews........................................................................................... 37
Little Flowers to G.H. Hardy (07-02-1877-01-12-1947)
K. Ramachandra .............................................................................................. 47
Rogers-Ramanujan Type Identities for Burge's Restricted Partition
Pairs Via Restricted Frobenius Partitions
A.K. Agarwal and Padmavathamma........................... ... ................... ............... 53
On q-additive and q-multiplicative Functions
I. Katai.............................................................................................................. 61
Antimagic Labeling of Complete m-ary Trees
P.D. Chawathe and Vljaya Krishna.................................................................. 77
Some Recent Advances on Symmetric, Quasi-Symmetric
and Quasi-Multiple Designs
Sharad Sane...................................................................................................... 81
On T-core Partitions and Quadratic Forms
Neville Robbins............. ...... ............... ........ ....... ..... ....... ..... ............. ............ ...... 89
Observations on Some Algebraic Equations Associated with
Ramanujan's Work
Michele Elia ...................................................................................................... 101
iv Contents
On Rapidly Convergent Series for Dirichlet L-function Values
Via the Modular Relation
S. Kanemitsu, Y. Tanigawa and M Yoshimoto.. ......... .................. ............. ...... 113
On a Conjecture of Andrews-II
Padmavathamma and M Ruby Salestina ........................................................ 135
Integrity of P x p.
2
P.D. Chawathe and S.A. Shende ...................................................................... 149
A Note on Cordial Labelings of Multiple Shells
Mahesh Andar; Samina Boxwala and N.B. Limaye ........................................ 157
A Report on Additive Complements of the Squares
D.S. Ramana .................................................................................................... 161
Transcendental Infinite Sums and Some Related Questions
Sukumar Das Adhikari .................................... ................... ............................. 169
The Lehmer Problem on the Euler Totient: A Pendora's Box
of Unsolvable Problems
M. V. Subbarao ......................... ............. ........... ...... ......... ................................. 179
The Problems Solved by Ramanujan in the Journal of the Indian
Mathematical Society
Bruce C. Berndt............... ........ ........ ................................................................. 189
On the gcd and Icm of Matrices Over Dedekind Domains
V.C. Nanda ....................................................................................................... 201
The Billiard Ball Motion Problem I: A MarkofTType Chain
for the Octahedron in ryt3
R.J. Hans-Gill ................................................................................................... 213
Hilbert's Seventeenth Problem and Pfister's Work on Quadratic Forms
A.R. Rajwade.................................................. .................................................. 225
Certain Representations of Mock-Theta Functions
R. Y. Denis, s.N. Singh and D. Sulata.. ............................................................. 231
Bi-Graceful Graphs
M. Murugan ...................................................................................................... 243
Wheels, Cages and Cubes
G. Sudhakara .................................................................................................... 251
Contents v
Relevance ofSrinivasa Ramanujan at the Dawn of the New Millennium
K. Srinivasa Rao. ................................. , ............................................................ 261
Number of Solutions of Equations over Finite Fields
and a Conjecture of Lang and Weil
Sudhir R. Ghorpade and Gilles Lachaud ........ '" ................. '" ...... .... ......... ... 269
On An Additive Question
S. Srinivasan .................................................................................................... 293
n-Colour Partitions
A.K. Agarwal ................................................................................................... 301
1st Row: Sushma, Vandana, Rachna, Mrs. R.I. Hans-Gill, Mrs. Madhu Raka,
Mr. Manohar Madan
2nd Row: Mr. AK. Agarwal. Mr. Imre Katai. Mr. R.P. Bambah,
Mr. K. Ramachandara, Mr. Michele Elia,
Mr. Michele Waldschmidt, Mr. K. Srinivasa Rao, N. Saradha,
G. Kaur, A Prasad
3rd Row: Mrs. Padmavathamma, Ms. Ruby Salestina. M,
Mr. Pranjal Rajkhowa, Mr. N. Sankaran,
Mr. S. Ismail Mohideen, Mr. D.V. Shah, Mr. Marcus
D. Bhoominathan, P.J.A Alphonse, Mr. T.N. Janaki Raman,
Mr. Sudhir R. Ghorpade, Mr. S.D. Adhikari
4th Row: Mrs. Nirmala Limaye, Mrs. P.D. Chawathe, Ms. Aditi Bapat,
Tarkeshwar Singh, Mr. Sampat Kumar, Mr. Christian
F. Krattenthaler, Mr. Remy Y. Denis, Mr. R.N. Maggo,
Mr. V.V. Rane, Mr. AK. Bhandari
5th Row: Mr. Anirban Mukhopadhyay, Mr. D.S. Khassa, Mr. P.K. Deb,
Mr. Neville Robbins, Mr. S.S. Sane, Mr. S.A Katre,
Mr. V.K. Grover
6th Row: Mr. Santosh Shende, Mr. T.N. Shorey, Mr. Surya Ramana,
Mr. Sinnou David
7th Row: Amrit Pal Singh, Amarinder Singh
List of Participants
Dr. Adhikari, S.D. Prof. Bambah, R.P.
The Harish-Chandra Research Institute Department of Mathematics
Chhatnag Road Punjab University
Jhusi Chandigarh 160 014
Allahabad 2111019
Ms. Bapat, Aditi
Prof. Agarwal, A.K.
Department of Mathematics
Department of Mathematics
University of Pune
Panjab University
Pune 411007
Chandigarh
Prof. Berndt, Bruce C.
Prof. Alladi, K.
Department of Mathematics
Department of Mathematics
Univ. of Illinois at UC Urbana
Univ.ofFlorida
IL 61801
Florida-32611
USA
USA (Gainsville)
Mr. Alphonse, P.J.A. Prof. Bhandari, A.K.
Department of Mathematics Department of Mathematics
and Computer Applications Punjab University
Regional Engineering College Chandigarh 160014
Trichy-15
Tamilnadu Mr. Bhatia, Saurabh
Department of Mathematics
Ms. Archana Punjab University
Department of Mathematics Chandigarh 160 014
Panjab University
Chandigarh Dr. Bhatnagar, Savita
Department of Mathematics
Dr. Bajaj, Renu
Panjab University
Department of Mathematics
Chandigarh
Panjab University
Chandigarh
Prof. Bhowmik, Gautami
UFR de Mathematiques
Dr. Bakshi, Gurmeet Kaur
Universite Lille I, 59655
Department of Mathematics
Villeneuve d'adscq Cedex
Panjab University
France
Chandigarh
Prof. Balasubramaniam, R. Dr. Bist, Vikas
Institute of Mathematical Sciences Department of Mathematics
CIT Campus Punjab University
Chennai 600 013 Chandigarh 160 014
viii List of Participants
Mr. Boominathan, M.D. Prof. Elia, Michele
Department of Mathematics Dip. Di Elettronica
Bishop Heber College Politechnico di
Trichy-17 Torino, 1-100129
Tomio
Dr. Chan, Heng Huat Italy
Department of Mathematics
National Univ. of Singapore
Dr. Garg, K.C.
King Ridge
Department of Correspondence
Singapore 119260
Studies
Republic of Singapore
Panjab University
Chandigarh
Prof. Chawathe, P.D.
Department of Mathematics
Dr. Ghorpade, S.R.
University of Mumbai
Department of Mathematics
Mumbai 400 098
Indian Institute of
Technology
Dr. Choi, Youn-Seo
Powai
Korea Institute for Advanced Study
Mumbai 400 076
School of Mathematics
207-43 Cheongryangri dong
Dongdaemun-gu, Seoul, 130-012 Prof. Grover, V.K.
Korea Department of Mathematics
Punjab University
Chandigarh 160 014
Prof. David, S.
Institute de Mathematiques de
Jussieu Theroie des-Nombres Prof. Hans-Gill, R.J.
Case 247 F75252 Paris, Cedex 05 Department of Mathematics
France Punjab University
Chandigarh 160 014
Mr. Deiawat, C.S.
The Harish-Chandra
Dr. Hedge, S.M.
Research Institute
Department of Mathematics
Chhatnag Road, Jhusi
and Computer Sciences
Allahabad 2111019
Kamataka Regional
Engineering College
Prof. Denis, R. Y.
Surthkal
Department of Mathematics
Srinivasnagar 574157
University of Gorakhpur
(Kamataka)
Gorakhpur
Mr. Dev, P.K. Mr. Kamal, Aghigh
Senior Lecturer in Mathematics Department of Mathematics
Cotton College Panjab University
Guwahati 78100 I Chandigarh
List of Participants IX
Prof. Kanemitsu, S. Prof. Luthar, I.S.
Deptt. of Electrical Engineering H.No. 256 Sector 6
Univ. of Kinki lizuka Panchkula
Fukooka 820 001 1 Haryana
Japan
Prof. Madan, M.L.
Prof. Katai, Imre Mathematics Department
Department of Computer Algebra Ohio State University
1117 Budapest Columbus
PAZMA NYPETR SET ANY Ohio 43210
Hungary USA
Prof. Katre, S.A.
Mr. Maggo, R.N.
Department of Mathematics
874, Sector 43A
University of Pune
Chandigarh 160022
Pune 411 007
Prof. Khanduja, Sudesh Kaur Mr. Mohideen, Ismail S.
Department of Mathematics Department of Mathematics
Panjab University Jamal Mohamed College
Chandigarh Trichy-20
Tamilnadu
Prof. Khassa, D.S.
Department of Mathematics Mr. Mukhopadhyay,A.
Panjab University The Harish-Chandra
Chandigarh Research Institute
Chhatnag Road, Jhusi
Prof. Krattenthaler, Christian F. Allahabad 2111019
Institute of Mathematik
Strudlhof gasse 4 (Strudlhof) Dr. Mukku, C.S.
A-I090 WIEN Department of Mathematics
Austria Panjab University
Chandigarh
Prof. Lachaud, Gilles
Directeur de Recherches au CNRS Dr. Murugan, M.
Institute de Math. Ematiques Department of Mathematics
de luminy, UPR 9016 The American College
163, Avenue De Luminy Madurai 625002 (Tamilnadu)
Case 930, 13288
Marseille Cedex 9 Prof. Nanda, V.C.
France 600, Sector l6-D
Chandigarh
Prof. Limaye, N.B.
Department of Mathematics Ms. Noha, Eftekhri
University of Mumbai Department of Mathematics
Vidya Nagri Panjab University
Mumbai 400 098 Chandigarh
