Summing Series Using Residues by Anthony Sofo

206 Pages·2010·3.55 MB·English

Summing Series Using Residues
by
Anthony Sofo
A Thesis Submitted at Victoria University of Technology
in fulfilment of the requirements for the
degree of Doctor of Philosophy.
School of Communications and Informatics
Faculty of Engineering and Science
Victoria University of Technology
Melbourne, Australia
July, 1998.
V
I
FTS THESIS
515.243 SOF
30001005475365
Sofo, Anthony
Summing series using
residues
Contents
0.1 Abstract, 5
0.2 Statement 6
0.3 Acknowledgments 7
0.4 Dedication 8
0.5 List of Tables 9
0.6 List of Figures 10
0.7 Summary. 11
1 A review of methods for closed form summation 13
1.1 Some Methods 14
1.1.1 Introduction 14
1.1.2 Contour Integration 16
1.1.3 Cerone's method and extension 17
1.1.4 Wheelon's results 20
1.1.5 Hypergeometric functions 25
1.2 A tree search sura and some relations 28
1.2.1 Binomial summation 28
1.2.2 Riordan 28
1.2.3 Method of Jonassen and Knuth 30
1.2.4 Method of Gessel 30
1.2.5 Method of Rousseau 31
1.2.6 Hypergeometric form 32
1.2.7 Snake oU method 36
1.2.8 Some relations 38
1.2.9 Method of Sister Celine 40
1.2.10 Method of creative telescoping 41
1.2.11 WZ pairs method 42
2 Summing series arising from integro-differential-difference equations 45
2.1 Introduction 46
2.2 Method 47
2.3 Biirmann's theorem and application 51
2.4 Differentiation and Integration 54
2.5 Forcing terms 56
2.6 Multiple delays, mixed and neutral equations 58
2.7 Bruwier series 60
2.8 Teletraffic example 60
2.9 Neutron behaviour example 63
2.10 A renewal example 65
2.11 Ruin problems in compound Poisson processes 66
2.12 A grazing system 67
2.13 Zeros of the transcendental equation 68
2.14 Numerical examples 70
2.15 Euler'swork 71
2.16 Jensen's work 73
2.17 Ramanujan's question 75
2.18 Cohen's modification and extension 75
2.19 Conolly's Problem 78
3 Biirmann's theorem 81
3.1 Introduction 82
3.2 Biirmann's theorem and proof 82
3.2.1 Applying Biirmann's theorem 86
3.2.2 The remainder 88
3.3 Convergence region 89
3.3.1 Extension of the series 90
4 Binomial type sums. 92
4.1 Introduction 93
4.2 Problem statement 93
4.3 A recurrence relation 94
4.4 Relations between Gfc (m) and Fk+i (m) 100
5 Generalization of Euler's identity. 105
5.1 Introduction 106
5.2 1-dorainant zero 106
5.2.1 The system 106
5.2.2 QR,k (0) recurrences and closed forms 109
5.2.3 Lemma and proof of theorem 17 115
5.2.4 Extension of results 118
5.2.5 Renewal processes 121
5.3 The A;-dominant zeros case 123
5.3.1 The A;-system 123
5.3.2 Examples 126
5.3.3 Extension 127
6 Fibonacci and related series. 129
6.1 Introduction 130
6.2 The difference-delay system 130
6.3 The infinite sum 132
6.4 The Lagrange form 134
6.5 Central binomial coefficients 136
6.5.1 Related results 140
6.6 Fibonacci, related polynomials and products 143
6.7 Functional forms 147
7 A convoluted Fibonacci sequence. 153
7.1 Introduction 154
7.2 Technique 154
7.3 Multiple zeros 157
7.4 More sums 161
7.5 Other forcing terms 162
8 Sums of Binomied variation. 164
8.1 Introduction 165
8.2 One dominant zero 165
8.2.1 Recurrences 166
8.2.2 Proof of conjecture 169
8.2.3 Hypergeometric functions 172
8.2.4 Forcing terms 175
8.2.5 Products of central binomial coefficients 176
8.3 Multiple dominant zeros 179
8.3.1 The A; theorem 180
8.3.2 Numerical results and special cases 183
8.3.3 The Hypergeometric connection 184
8.4 Non-zero forcing terms 185
8.5 Appendix A: A recurrence for Q's 187
8.6 Appendix B: Zeros 190
Conclusion and suggestions for further work 195
Bibliography.
0.1 Abstract.
This thesis deals with the problem of representation of series in closed form, mainly by the use
of residue theory. Forced differential-difference equations of arbitrary order are considered from
which infinite sums of the form
Si(R,k,b,a,t) = ^^y . J (r.k^Rk-.y. '
with arbitrary parameters (R, k, b, a, t), are generated. For the most basic case of
(R, k) = (1,1) the infinite sum ^i (1,1,6, a, t), in different form, has been considered by various
mathematicians including Euler, Jensen and P6lya and Szego. Their methods of representing
5"! (1,1, b, a, t) in closed form are different than those developed by the author; moreover the au
thor demonstrates that Si (1,1,6, a, t) has many appUcations in a wide area of study including
teletraffic theory, neutron behaviour, renewal processes and grazing systems. The author proves
that for the general case, ^i (R, k, 6, a, t) may be represented in closed form which depends on
k dominant zeros of an associated transcendental characteristic function.
In a similar vein, arbitrary order forced difference-delay equations are considered, from
which infinite sums of the form
,^ f r-^R-l\ f n-akr \ . „. ^,
S2(R,k,b,a,n) = J2\ U^-akr-Rk+l^
r.=o \ r J \kr-\-Rr-l J
with arbitrary parameters {R,k,b,a,n) are generated. It is shown that the finite form,
S2F(R,k,b,a,n), of S2{R,k,b,a,n) is associated with Fibonacci and other related polyno
mials. Many functional forms of 5*2^ (1,1, b, 1, n) are also proved. For some special cases of the
finite form, S2F {R,k,b,a,n) may be represented as an identity and residue theory, together
with automated techniques and recurrences are employed in their proof. By the use of residue
theory and induction the author proves that the infinite sum 5*2 (R, k, 6, a, n) may be represented
in closed form, which depends on A; zeros of an associated polynomial characteristic function.
Moreover, S2 (R, k, 6, a, n) may be represented in hypergeometric form which in some particular
instances reduce to known identities incorporating, for example, Kummer's form.
0.2 Statement.
To the best of my knowledge and belief this thesis does not contain any material previously
published or written by another author, except where due reference is made in the text. This
thesis has not been submitted for any other degree or diploma, at any tertiary institution.
Anthony Sofo.
0.3 Acknowledgments.
It is indeed a pleasure to acknowledge the assistance received from my supervisors, Associate
Professor Pietro Cerone and Dr. Don Watson. I've been fortunate in being able to spend many
hours of constructive discourse with them.
To the people I love the most: Lucia, Fiona, Jessica, Clara and Matthew, thank you.
I gratefully acknowledge the allocation of an Australian Postgraduate Award, through Vic
toria University of Technology, which has assisted this research.
0.4 Dedication.
A mia madre; lei non sa leggere ne scrivere, ma la sua saggezza e
infinitamente pivi gremde della mia.