source stringlengths 2 152 | text stringlengths 1 13.4k |
|---|---|
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | Series ISSN 1939-5221
Generating Functions in Engineering and the Applied Sciences
Rajan Chattamvelli, VIT University, Vellore, Tamil Nadu
Ramalingam Shanmugam, Texas State University
This is an introductory book on generating functions (GFs) and their applications. It discusses
commonly encountered ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | apply the lessons learned in
their respective disciplines. The purpose is to give a broad exposure to commonly used techniques of
combinatorial mathematics, highlighting applications in a variety of disciplines.
ABOUT SYNTHESIS
This volume |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | apply the lessons learned in
their respective disciplines. The purpose is to give a broad exposure to commonly used techniques of
combinatorial mathematics, highlighting applications in a variety of disciplines.
ABOUT SYNTHESIS
This volume is a printed version of a work that appears in the Synthesi... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | velli and Ramalingam Shanmugam
2019
Transformative Teaching: A Collection of Stories of Engineering Faculty’s Pedagogical
Journeys
Nadia Kellam, Brooke Coley, and Audrey Bok |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | velli and Ramalingam Shanmugam
2019
Transformative Teaching: A Collection of Stories of Engineering Faculty’s Pedagogical
Journeys
Nadia Kellam, Brooke Coley, and Audrey Boklage
2019
Ancient Hindu Science: Its Transmission and Impact of World Cultures
Alok Kumar
2019
Value Relational Engineering
Shuichi Fukuda
2018
... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 13
The Making of Green Engineers: Sustainable Development and the Hybrid Imagination
Andrew Jamison
2013
Crafting Your Research Future: A Guide to Successful Master’s and Ph.D. Degrees in
Science & Engineering
Charles |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 13
The Making of Green Engineers: Sustainable Development and the Hybrid Imagination
Andrew Jamison
2013
Crafting Your Research Future: A Guide to Successful Master’s and Ph.D. Degrees in
Science & Engineering
Charles X. Ling and Qiang Yang
2012
iv
Fundamentals of Engineering Economics and Decision Analysis
David ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | ids, Part Two: Transport Properties of Solids
Richard F. Tinder
2007
Tensor Properties of Solids, Part One: Equilibrium Tensor Properties of Solids
Richard F. Tinder
2007
Essentials |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | ids, Part Two: Transport Properties of Solids
Richard F. Tinder
2007
Tensor Properties of Solids, Part One: Equilibrium Tensor Properties of Solids
Richard F. Tinder
2007
Essentials of Applied Mathematics for Scientists and Engineers
Robert G. Watts
2007
Project Management for Engineering Design
Charles Lessard and ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | sciences, such as or-
dinary generating functions (OGF), exponential generating functions (EGF), probability gen-
erating functions (PGF), etc. Some new GFs like Pochhammer generating functions for both
rising and falling factor |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | sciences, such as or-
dinary generating functions (OGF), exponential generating functions (EGF), probability gen-
erating functions (PGF), etc. Some new GFs like Pochhammer generating functions for both
rising and falling factorials are introduced in Chapter 2. Two novel GFs called “mean deviation
generating function”... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | , Pochhammer generating functions, poly-
mer chemistry, power series, recurrence relations, reliability engineering, special
numbers, statistics, strided sequences, survival function, truncated distributions.
Contents
ix
List of Tables . . . . . |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | , Pochhammer generating functions, poly-
mer chemistry, power series, recurrence relations, reliability engineering, special
numbers, statistics, strided sequences, survival function, truncated distributions.
Contents
ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Ordinary Generating Functions . . . . . . . . . . . . |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Ordinary Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.2 Types of Sequences . . . . . . . .... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | .7.1 Auto-Covariance Generating Function . . . . . . . . . . . . . . . . . . . . . . . . 16
Information Generating Function (IGF) . . . . . |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | .7.1 Auto-Covariance Generating Function . . . . . . . . . . . . . . . . . . . . . . . . 16
Information Generating Function (IGF) . . . . . . . . . . . . . . . . . . . . . . . 16
1.7.2
1.7.3 Generating Functions in Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7.4 Generating Functions in Number Theo... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 1.2 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
x
2 |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 1.2 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
x
2.1.3 Multiplication by Non-Zero Constant . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.4 Linear Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Shifting . .... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | of Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Summary . . . . . . . . . . . |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | of Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4
3
Generating Functions in Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book |
3.6 MD of Some Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book |
3.6 MD of Some Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6.1 MD of Geometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6.2 MD of Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6.3 MD o... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | . . . 54
3.10 Factorial Moment Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3 |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | . . . 54
3.10 Factorial Moment Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.11 Conditional Moment Generating Functions (CMGF) . . . . . . . . . . . . . . . . . . 58
3.12 Generating Functions of Truncated Distributions . . . . . . . . . . . . . . . . . . . . . . 58
3.13 Convergenc... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | . . . . . . . . . . . . . . . . . . . . . 64
Applications in Computing . . . . . . . . . . . . . . . . . . . . . . |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | . . . . . . . . . . . . . . . . . . . . . 64
Applications in Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.1 Merge-Sort Algorithm Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.2 Quick-Sort Algorithm Analysis . . . . . . . . . . . . . . . ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.4 Towers of Hanoi Puzzle . . . . . . . . . . . |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.4 Towers of Hanoi Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Applications in Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.1 Graph Enumeration . . . . . . . . . . . . . . . ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | Applications in Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
S |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | Applications in Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Sums of IID Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.8.1
4.8.2
Infinite Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | . . . . . . . . . . . . . . . . 86
4.12 Applications in Management . . . . . . . . . . . . . . . . . . . . . . . |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | . . . . . . . . . . . . . . . . 86
4.12 Applications in Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.12.1 Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.6
4.7
4.8
4.9.1
4.5
xii
4.13 Applications in Ec... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | .1
Some standard generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Convolution as |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | .1
Some standard generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Convolution as diagonal sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2
2.3
Summary of convolutions and powers . . . . . . . . . . . . . . . . . . . . . . . . . .... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | . . . . . . . 53
Summary table of zero-truncated generating functions . . . . . . . . . . . . . . . . . . 59
C H A P T E R |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | . . . . . . . 53
Summary table of zero-truncated generating functions . . . . . . . . . . . . . . . . . . 59
C H A P T E R 1
1
Types of Generating Functions
After finishing the chapter, readers will be able to
(cid:1) (cid:1) (cid:1)
• explain the meaning and uses of generating functions.
• describe the ordin... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | GENERATING FUNCTIONS
Definition 1.1 (Definition of Generating Function) A GF is a simple and concise expression
in one or more dummy variables that captures the coefficients of a finite or |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | GENERATING FUNCTIONS
Definition 1.1 (Definition of Generating Function) A GF is a simple and concise expression
in one or more dummy variables that captures the coefficients of a finite or infinite series, and
generates a quantity of interest using calculus or algebraic operations, or simple substitutions.
GFs are used in... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | f .0/
C
D
C
1.1.2 EXISTENCE OF GENERATING FUNCTIONS
GFs are much easier to work with than a whole lot of numbers or symbols. This does not mean
that any sequence can be |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | f .0/
C
D
C
1.1.2 EXISTENCE OF GENERATING FUNCTIONS
GFs are much easier to work with than a whole lot of numbers or symbols. This does not mean
that any sequence can be converted into a GF. For instance, a random sequence may not have a
nice and compact GF. Nevertheless, if the sequence follows a mathematical rule... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | range is (cid:140)
t
1(cid:141). As the input can be complex numbers in some signal processing applications, the
GF can also be complex. A formal power series implies that nothing is assumed on the |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | range is (cid:140)
t
1(cid:141). As the input can be complex numbers in some signal processing applications, the
GF can also be complex. A formal power series implies that nothing is assumed on the conver-
gence. As shown below, the sequence 1; x; x2; x3; : : : has GF F(x) = 1=.1
x/ which converges
x2/,
in the region ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book |
1.2 NOTATIONS AND NOMENCLATURES
A finite sequence will be denoted by placing square-brackets around them, assuming that each
element is distinguishable. An English alphabet will be used to identify it when necessary |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book |
1.2 NOTATIONS AND NOMENCLATURES
A finite sequence will be denoted by placing square-brackets around them, assuming that each
element is distinguishable. An English alphabet will be used to identify it when necessary. An
infinite series will be denoted by simple brackets, and named using Greek alphabet. Thus, S
D
.a0; ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | C
1/
D
(cid:3) (cid:1) (cid:1) (cid:1) (cid:3)
.x
j
C
(cid:0)
1/
D
j
1
(cid: |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | C
1/
D
(cid:3) (cid:1) (cid:1) (cid:1) (cid:3)
.x
j
C
(cid:0)
1/
D
j
1
(cid:0)
Y
.x
0
k
D
k/
C
D
(cid:128).x
j /
C
(cid:128).x/
:
(1.1)
2. Falling Factorial Notation
In the falling factorial, a variable is decremented successively at each iteration. This is
denoted as
x.j /
.x
x
(cid:3)
(cid:0... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | the dummy variable. For example,
G.x; t/ denotes an OGF of a random variable X. Some authors use x as a subscript as Gx.t/.
These dummy variables assume special values (like 0 or 1) to generate |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | the dummy variable. For example,
G.x; t/ denotes an OGF of a random variable X. Some authors use x as a subscript as Gx.t/.
These dummy variables assume special values (like 0 or 1) to generate a quantity of interest.
Thus, uniform and absolute convergence at these points are assumed. The GFs used in statistics
can be... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | derived easily. This reasoning holds for other GFs too.
C
X
(cid:3)
1.4 ORDINARY GENERATING FUNCTIONS
Let (cid:27)
D
.a0; a1; a2; a3; |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | derived easily. This reasoning holds for other GFs too.
C
X
(cid:3)
1.4 ORDINARY GENERATING FUNCTIONS
Let (cid:27)
D
.a0; a1; a2; a3; : : : an; : : :/ be a sequence of bounded numbers. Then the power series
f .x/
D
anxn
1
X
0
n
D
(1.3)
is called the Ordinary Generating Function (OGF) of (cid:27). Here x ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book |
1
x/(cid:0)
1:
x/(cid:0)
1; 1;
.1;
(cid:0)
(cid:0)
6
(cid:0)
1/=.x
(cid:0)
1 |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book |
1
x/(cid:0)
1:
x/(cid:0)
1; 1;
.1;
(cid:0)
(cid:0)
6
(cid:0)
1/=.x
(cid:0)
1, and odd coefficients a2n
1. TYPES OF GENERATING FUNCTIONS
The OGF is .xn
1/=.x
1/ when there are a finite number of 1’s (say n of them). Thus,
(cid:0)
.1; 1; 1; 1; 1/ has OGF .x5
1 when even coefficients
x2/(cid:0)
0. The OGF is called ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book |
C (cid:1) (cid:1) (cid:1)
.1; 2; 3; 4; 5; : : :/. This has OGF 1
2x
.1; |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book |
C (cid:1) (cid:1) (cid:1)
.1; 2; 3; 4; 5; : : :/. This has OGF 1
2x
.1; 2; 3; 4; : : : ; n; : : :/
(cid:148)
.1
.1
2
(cid:0)
x/(cid:0)
2
x/(cid:0)
1=.1
1=.1
(cid:0)
x/2
x/2:
C
D
D
(cid:148)
C
2; 3;
.1;
(cid:0)
(cid:0)
4; : : : ; n;
.n
(cid:0)
C
1/; : : :/
See Table 1.1 for more examples.
... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | =0k=0k=0k=0k=0k=0k=0 1.4. ORDINARY GENERATING FUNCTIONS 7
1.4.1 RECURRENCE RELATIONS
There are in general two |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | =0k=0k=0k=0k=0k=0k=0 1.4. ORDINARY GENERATING FUNCTIONS 7
1.4.1 RECURRENCE RELATIONS
There are in general two ways to represent the nth term of a sequence: (i) as a function of n, and
(ii) as a recurrence relation between nth term and one or more of the previous terms. For example,
an
.1
and Jones, 1996].
1=n gives a... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | :0)
n
k
n
1.4.2 TYPES OF SEQUENCES
(cid:0)
There are many types of sequences encountered in engineering. Examples are arithmetic se-
quence, geometric sequence, exponential sequence, Harmonic sequence |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | :0)
n
k
n
1.4.2 TYPES OF SEQUENCES
(cid:0)
There are many types of sequences encountered in engineering. Examples are arithmetic se-
quence, geometric sequence, exponential sequence, Harmonic sequence, etc. Alternating se-
quence is one in which the sign of the coefficients alter between plus and minus. This is of
a... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 141). It is called a geometric progres-
sion in mathematics and geometric sequence in engineering. It also appears in various physical,
natural, and life sciences. As examples, the equations of state of gaseous mixtures (like |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 141). It is called a geometric progres-
sion in mathematics and geometric sequence in engineering. It also appears in various physical,
natural, and life sciences. As examples, the equations of state of gaseous mixtures (like van der
Vaals equation, Virial equation), mean-energy modeling of oscillators used in crystall... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 1 Consider the infinite series expansion of .1
C
, which is the OGF of the given sequence. Here “a” is any nonzero constant.
D
C
(cid:0)
1
ax
1
ax/( |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 1 Consider the infinite series expansion of .1
C
, which is the OGF of the given sequence. Here “a” is any nonzero constant.
D
C
(cid:0)
1
ax
1
ax/(cid:0)
a2x2
a3x3
(cid:0)
C
(cid:1) (cid:1) (cid:1)
Example 1.2 (Find the OGF of an) Find the OGF of (i) an
1=3n.
D
2n
C
3n and (ii) an
2n
C
D
Solution ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 0)
t/2 as the required OGF.
(cid:0)
0 t n. The first sum is 2t=.1
t/2 and second one is 1=.1
1/. Write G.t /
t |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 0)
t/2 as the required OGF.
(cid:0)
0 t n. The first sum is 2t=.1
t/2 and second one is 1=.1
1/. Write G.t /
t/. Take .1
P1n
D
C
D
C
(cid:0)
(cid:0)
(cid:0)
C
D
D
D
Example 1.4 (OGF of perfect squares) Find the OGF for all positive perfect
.1; 4; 9; 16; 25; : : :/.
squares
Solution 1.4 Consider the expres... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | =x
2=.1
x/
(cid:0)
D
.1
C
C
x/=(cid:140)x.1
x/(cid:141)
(cid:0)
(1.7)
|
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | =x
2=.1
x/
(cid:0)
D
.1
C
C
x/=(cid:140)x.1
x/(cid:141)
(cid:0)
(1.7)
from which @f
C
perfect squares of all positive integers as x.1
as follows:
@x D
x/=.1
.1
(cid:0)
x/3. Multiply this expression by x to get the desired GF of
x/3. This can be represented symbolically
x/=.1
C
(cid:0)
.1; 1; 1; 1; ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | , 28, 82,: : :).
Solution 1.5 Suppose we subtract 1 from each number in the sequence. Then we get (1, 3, 9,
27, 81, 2 |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | , 28, 82,: : :).
Solution 1.5 Suppose we subtract 1 from each number in the sequence. Then we get (1, 3, 9,
27, 81, 243, : : :), which are powers of 3. Hence, above sequence is the sum of .1; 1; 1; 1; : : :/
.1; 3; 32; 33; 34; : : :/ with respective OGFs 1=.1
3x/. Add them to get 1=.1
x/
3x/ as the answer.
x/ and 1=... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | a1x
x2
x
x/(cid:0)
C
C
2Note that the nth derivative of 1=.1
C (cid:1) (cid:1) (cid:1) C
C |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | a1x
x2
x
x/(cid:0)
C
C
2Note that the nth derivative of 1=.1
C (cid:1) (cid:1) (cid:1) C
C (cid:1) (cid:1) (cid:1)
(cid:0)
x/n
C
x/ is n(cid:138)=.1
(cid:0)
(cid:0)
1 and that of 1=.1
ax/b is .n
b
(cid:0)
C
(cid:0)
C
C
D
,
C (cid:1) (cid:1) (cid:1)
1/(cid:138)an=(cid:140).b
(cid:0)
1/(cid:138).1
... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 1
A
aj
xk
a0
C
D
.a0
C
a1/x
.a0
a1
C
C
C
a2/x2
:
C (cid:1) ( |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 1
A
aj
xk
a0
C
D
.a0
C
a1/x
.a0
a1
C
C
C
a2/x2
:
C (cid:1) (cid:1) (cid:1)
(1.10)
(cid:3)
This result has great implications. Computing the sum of several terms of a sequence
appears in several fields of applied sciences. If the GF of such a sequence has closed form, it
is a simple matter of multiplyi... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | C
(cid:0)
(cid:0)
1 k2.
x/,
D
C
(cid:0)
Example 1.6 (OGF of harmonic series) Find the OGF of 1
1=2
1 |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | C
(cid:0)
(cid:0)
1 k2.
x/,
D
C
(cid:0)
Example 1.6 (OGF of harmonic series) Find the OGF of 1
1=2
1=3
C
C
C (cid:1) (cid:1) (cid:1) C
1=n.
Solution 1.6 We know that
x2=2
x/
(cid:0)
1 xk=k. We also know that when an OGF is multiplied by 1=.1
x3=3
C
C
x/, we get a new
(cid:1) (cid:1) (cid:1) D
OGF in whi... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | is the OGF of .1=2; 1
3 .1
C
2 /; 1
4 .1
1
2 C
1
3 /;
(cid:1) (cid:1) (cid |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | is the OGF of .1=2; 1
3 .1
C
2 /; 1
4 .1
1
2 C
1
3 /;
(cid:1) (cid:1) (cid:1)
/.
C
C (cid:1) (cid:1) (cid:1)
(cid:0)
1.5. EXPONENTIAL GENERATING FUNCTIONS (EGF)
11
m
k
1/k(cid:0)
A direct application of the above result is in proving the combinatorial
0.
P
According to the above theorem, the sum of th... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book |
t/ has coefficients with negative sign
0 a2kt 2k, and G.t/
denotes the greatest integer
1. If F .t/
1t 2k
C
m=2
|
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book |
t/ has coefficients with negative sign
0 a2kt 2k, and G.t/
denotes the greatest integer
1. If F .t/
1t 2k
C
m=2
t// is the GF of Pb
c
k
t//=2 is the GF of P1k
D
k(cid:1)F .t/k is also a GF.
x
b
c
F .t//n
0 a2kt 2k.
1. Here
P1k
D
(cid:0)
1t 2k
m
P
k
D
(cid:20)
0 a2k
t//=2
D
F .
F .
0 (cid:0)
P
D
D
(ci... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | D
gives an OGF. But it is the usual practice to write it either as (1.12) or as
H.x/
D
1
X
0
n
D
anenxn where en
1 |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | D
gives an OGF. But it is the usual practice to write it either as (1.12) or as
H.x/
D
1
X
0
n
D
anenxn where en
1=n(cid:138)
D
(1.13)
(1.14)
where we have two independent multipliers. As the coefficients of exponential function .exp.x//
are reciprocals of factorials of natural numbers, they are most suited wh... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | GF may exist for a sequence. As an example, consider the sequence
1; 2; 4; 8; 16; 32; : : : with nth term an
2x/, whereas the EGF is |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | GF may exist for a sequence. As an example, consider the sequence
1; 2; 4; 8; 16; 32; : : : with nth term an
2x/, whereas the EGF is e2x.
Thus, the EGF may converge in lot many problems where the OGF either is not convergent or
k(cid:1)xk. We could write this as an EGF by expanding
does not exist. Consider .1
n
(cid:0)... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | x/ so that P1n
D
xn=n(cid:138)
D
(cid:2)
1=.1
1=.1
0 n(cid:138)
0 1
xn
(cid:0)
|
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | x/ so that P1n
D
xn=n(cid:138)
D
(cid:2)
1=.1
1=.1
0 n(cid:138)
0 1
xn
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(cid:2)
Now consider
(cid:16)ebt
(cid:0)
eat (cid:17) =.b
a/
(cid:0)
D
t=1(cid:138)
.b2
a2/=.b
a/t 2=2(cid:138)
C
.bn
(cid:0)
an/=.b
(cid:0)
a/t n=n(cid:138)
.b3
(cid:0)
a3/=.b
(cid:0)
a/t... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 2)
(cid:11)/
(cid:0)
D
F0
C
F1t=1(cid:138)
C
F2t 2=2(cid:138)
C
F3t |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | 2)
(cid:11)/
(cid:0)
D
F0
C
F1t=1(cid:138)
C
F2t 2=2(cid:138)
C
F3t 3=3(cid:138)
C (cid:1) (cid:1) (cid:1) C
Fnt n=n(cid:138)
C (cid:1) (cid:1) (cid:1)
;
(1.19)
where Fn is the nth Fibonacci number.
Another reason for the popularity of EGF is that they have interesting product and com-
position formula... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | .t. x of both sides. As a0 is a constant, its derivative is zero. Use
xn
derivative of xn
(cid:0)
(cid:3)
C (cid:1) (cid:1) (cid: |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | .t. x of both sides. As a0 is a constant, its derivative is zero. Use
xn
derivative of xn
(cid:0)
(cid:3)
C (cid:1) (cid:1) (cid:1) C
akxk
1=.k
. Differentiate again (this time a1 being a constant vanishes) to
1/(cid:138)
(cid:0)
(cid:0)
get H 00.x/
a3x=1(cid:138)
pro-
cess k times. All terms whose coefficients are below ... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | uishable balls into m distinguishable urns (or boxes) in such a way that none of the urns is
empty.
Solution 1.8 Let un;m denote the total number of ways. Assume that i th urn has ni balls |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | uishable balls into m distinguishable urns (or boxes) in such a way that none of the urns is
empty.
Solution 1.8 Let un;m denote the total number of ways. Assume that i th urn has ni balls so
n where each ni >
that all urn contents should add up to n. That gives n1
C (cid:1) (cid:1) (cid:1) C
x3=3(cid:138)
0. Let x=1(... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | )
/m;
(1.20)
where ak
each urn gets one ball), and upper limit is n. The RHS is easily seen to be .ex
using binomial theorem to get
uk;m and the lower limit is obviously m |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | )
/m;
(1.20)
where ak
each urn gets one ball), and upper limit is n. The RHS is easily seen to be .ex
using binomial theorem to get
uk;m and the lower limit is obviously m (because this corresponds to the case where
1/m. Expand it
D
(cid:0)
.ex
1/m
(cid:0)
D
emx
(cid:0)
!
m
1
e.m
(cid:0)
1/x
C
!
m
2... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | )
n
k(cid:1) is the classical Laguerre
(cid:0)
1.6
POCHHAMMER GENERATING FUNCTIONS
Analogous to the formal power series expansions given above, we could also define Poch |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | )
n
k(cid:1) is the classical Laguerre
(cid:0)
1.6
POCHHAMMER GENERATING FUNCTIONS
Analogous to the formal power series expansions given above, we could also define Pochhammer
GF as follows. Here the series remains the same but the dummy variable is either rising factorial
or falling factorial type.
1.6.1 RISING P... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | ; k/xn=n(cid:138)
ond kind is P1n
D
constants to get falling EGF for Pochhammer numbers b.k/.
xn. The
0 S.n; k/x.k/
x.n/. The |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | ; k/xn=n(cid:138)
ond kind is P1n
D
constants to get falling EGF for Pochhammer numbers b.k/.
xn. The
0 S.n; k/x.k/
x.n/. The EGF of Stirling number
x//k and that of Stirling number of sec-
1/k=k(cid:138). The falling factorial can also be applied to the
n
k
D
.1=k(cid:138)/.log.1
k s.n; k/xn=n(cid:138)
0 s.n; k/xk... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | /.x
x.x
1 neighbors to it so that
1 choices. Continue arguing like this until a single node is left. Hence, the GF is
2/ : : : .x
1/ or x.n/ as the required |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | /.x
x.x
1 neighbors to it so that
1 choices. Continue arguing like this until a single node is left. Hence, the GF is
2/ : : : .x
1/ or x.n/ as the required GF.
(cid:0)
n
(cid:0)
(cid:0)
(cid:0)
(cid:0)
C
16
1. TYPES OF GENERATING FUNCTIONS
1.7 OTHER GENERATING FUNCTIONS
There are many other GFs used in va... |
https://ieeexplore.ieee.org/xpl/ebooks/bookPdfWithBanner.jsp?fileName=8826062.pdf&bkn=8826061&pdfType=book | GF.
n
aj
j where
D
a
j
j
Solution 1.11 Split
n
as P(cid:0)
n
1=.1
a/
nt (cid:0)
a(cid: |
End of preview. Expand in Data Studio
No dataset card yet
- Downloads last month
- 3