Difference Equations With Public Health Applications - Moye Kapadia

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EQUATIONS WITHPUBLIC HEALTH

APPLICATIONS

EQUATIONS WITHPUBLIC HEALTH

APPLICATIONS

LEMUEL A. MOVEASHA SETH KAPADIA

The University of Texas-Houston School of Public HealthHouston, Texas

MARCEL DEKKER, INC. NEW YORK BASEL

Library of Congress Cataloging-in-Publication Data

Moye, Lemuel A.Difference equations with public health applications / Lemuel A. Moye, Asha SethKapadia.

p. cm. (Biostatistics ; vol 6)Includes index.ISBN 0-8247-0447-9 (alk. paper)1. Public HealthResearchMethodology. 2. Difference equations. I. Kapadia, AshaSeth. II. Title. III. Biostatistics (New York, N.Y.); 6.

RA440.85 .M693 2000614.4'07'27dc21 00-040473

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Biostatistics: A Series of References and Textbooks

Series EditorShein-Chung Chow

President, U.S. OperationsStatPlus, Inc.

Yardley, PennsylvaniaAdjunct ProfessorTemple University

Philadelphia, Pennsylvania

Design and Analysis of Animal Studies in Pharmaceutical Devel-opment, edited by Shein-Chung Chow and Jen-pei LiuBasic Statistics and Pharmaceutical Statistical Applications,James E. De MuthDesign and Analysis of Bioavailability and Bioequivalence Studies,Second Edition, Revised and Expanded, Shein-Chung Chow andJen-pei LiuMeta-Analysis in Medicine and Health Policy, edited by Dalene K.Stangl and Donald A. BerryGeneralized Linear Models: A Bayesian Perspective, edited byDipak K. Dey, Sujit K. Ghosh, and Ban! K. MallickDifference Equations with Public Health Applications, Lemuel A.Moye and Asha Seth Kapadia

ADDITIONAL VOLUMES IN PREPARATION

Medical Biostatistics, Abhaya Indrayan and Sanjeev B. Sarmu-kaddam

Statistical Methods for Clinical Trials, Mark X. Norleans

ToMy mother Florence Moye, my wife Dixie, my daughter Flora, my brother

Eric, and the DELTSLAM

ToMy mother Sushila Seth, my sisters Mira, Gita, Shobha, and Sweety, my son

Dev, and my friend, JiirgenASK

Series Introduction

The primary objectives of the Biostatistics Series are to provide useful referencebooks for researchers and scientists in academia, industry, and government, andalso to offer textbooks for undergraduate and/or graduate courses in the area ofbiostatistics. This series provides a comprehensive and unified presentation ofstatistical designs and analyses of important applications in biostatistics, such asthose in biopharmaceuticals. A well-balanced summary is given of current andrecently developed statistical methods and interpretations for both statisticiansand researchers/scientists with minimal statistical knowledge who are engagedin the field of applied biostatistics. The series is committed to providing easy-to-understand, state-of-the-art references and textbooks. In each volume,statistical concepts and methodologies are illustrated through real worldexamples.

The difference equation is a powerful tool for providing analytical solutionsto probabilistic models of dynamic systems in health related research. The healthrelated applications include, but are not limited to, issues commonlyencountered in stochastic processes, clinical research, and epidemiology. In

Series Introduction

practice, many important applications, such as the occurrence of a clinical eventand patterns of missed clinic visits in randomized clinical trials, can bedescribed in terms of recursive elements. As a result, difference equations aremotivated and are solved using the generating function approach to addressissues of interest. This volume not only introduces the important concepts andmethodology of difference equations, but also provides applications in publichealth research through practical examples. This volume serves as a bridgeamong biostatisticians, public health related researchers/scientists, andregulatory agents by providing a comprehensive understanding of theapplications of difference equations regarding design, analyses, andinterpretations in public health research.

Shein-Chung Chow

Preface

The authors have worked together since 1984 on the development andapplication of difference equations. Sometimes our work was episodic, at othertimes it was continuous. Sometimes we published in the peer reviewed literaturewhile at other times we wrote only for ourselves. Sometimes we taught, and atother times, we merely talked. Difference equations have provided a consistentcadence for us, the background beat to which our careers unconsciously moved.

For each of us, difference equations have held a fascination in and ofthemselves, and it has always been natural for the two of us to immerseourselves in their solution to the exclusion of all else. However, working in aschool of public health means involvement in public health issues. We havecommitted to continually explore the possibility of linking difference equationsto public health topics. Issues in public health, whether environmental,economic, or patient-care oriented, involve a host of interdependencies whichwe suspected might be clearly reflected in families of difference equations.

A familiar medical school adage is that physicians cannot diagnose adisease they know nothing about. Similarly, quantitative students in public

viii Preface

health cannot consider utilizing difference equations to formulate a public healthproblem unless they are comfortable with their formulations and solutions.Difference equations have been the focus of attention by several authorsincluding Samuel Goldberg's classic text, Introduction to Difference Equations,published in 1958. However, none of these have focused on the use of differenceequations in public health. Although generating functions cannot be used tosolve every difference equation, we have chosen to elevate them to the primarytool. Mastering these tools provides a path to the solution of complicateddifference equations.

We would like to express our appreciation and thanks to Ms. KathleenBaldonado for her valuable suggestions, extraordinary editing efforts, andkeeping these turbulent authors calm.

Lemuel A. MoyeAsha Seth Kapadia

Introduction

Difference equations are an integral part of the undergraduate and graduate levelcourses in stochastic processes. Instruction in difference equations providessolid grounding for strengthening the mathematical skills of the students byproviding them with the ability to develop analytical solutions to mathematicaland probabilistic prediction models of dynamic systems. Difference Equationswith Public Health Applications is a new contribution to didactic textbooks inmathematics, offering a contemporary introduction and exploration of differenceequations in public health. The approach we have chosen for this book is anincremental one-we start with the very basic definitions and operations fordifference equations and build up sequentially to more advanced topics. Thetext begins with an elementary discussion of difference equations withdefinitions of the basic types of equations. Since intuition is best builtincrementally, development of the solution of these equations begins with thesimplest equations, moving on to the more complicated solutions of morecomplex equations.

In addition, many examples for the complete solutions of families ofdifference equations in their complete detail are offered. The solution methods

Introduction

of induction and of partial fractions are discussed and provided. Solutions frompartial fractions are contrasted to other forms of the solution in terms of theeffort in obtaining the solution (identification and the relative comprehensiveease of the solutions). Expansive exercises are provided at the end of eachchapter.

The major contribution of Difference Equations with Public HealthApplications is its scope of coverage. We begin with a simple, elementarydiscussion of the foundations of difference equations and move carefully fromthere through an examination of their structures and solutions. Solutions forhomogeneous and nonhomogeneous equations are provided. A major emphasisof the text is obtaining solutions to difference equations using generatingfunctions. Complete solutions are provided for difference equations through thefourth order. An approach is offered that will in general solve the kth orderdifference equation, for k > 1.

In addition, this textbook makes two additional contributions. The first isthe consistent emphasis on public health applications of the difference equationapproach. Issues in hydrology, health services, cardiology, and vital statusascertainment in clinical trials present themselves as problems in differenceequations. We recognize this natural formulation and provide their solutionusirig difference equation techniques. Once the solution is obtained, it isdiscussed and interpreted in a public health context.

The second contribution of this book is the focus on the generatingfunction approach to difference equation solutions. This book provides athorough examination of the role of generating functions to understand andsolve difference equations. The generating function approach to differenceequation solutions is presented throughout the text with applications in the mostsimple to the most complex equations.

Although mention is made of computer-based solutions to differenceequations, minimal emphasis is placed on computing. Software use is notemphasized and, in general, computing is used not as a tool to solve theequations, but only to explore the solution identified through the generatingfunction approach. This computing de-emphasis supports our purpose to providethe opportunity for students to improve their own mathematical skills throughuse of generating function arguments. In addition, computing software, likemost software, is ephemeral in nature. A book focused on the application of oneclass of software ties its utility to the short life expectancy of the software. Tothe contrary, the time-tested generating function approach to differenceequations is an elegant, stable, revealing approach to their solution. An analyticsolution to a system of difference equations as identified through generatingfunctions is a full revelation of the character of the equation.

Students in undergraduate or graduate programs in statistics, biostatistics,operations research, stochastic processes, and epidemiology compose the targetaudience for this book, as well as researchers whose work involves solutions to

Introduction xi

difference equations. It is envisioned that this book will be the foundation textfor a course on difference equations in biostatistics, mathematics, engineering,and statistics. Difference equations are introduced at a very introductory level. Acourse in calculus is strongly recommended, and students so equipped will beable to practically take the tools discussed and developed in this text to solvecomplicated difference equations.

Chapter 1 provides a general introduction to the difference equationconcept, establishes notation, and develops the iterative solution useful for first-order equations. Chapters 2 and 3 discuss the generating function concept anddevelop the tools necessary to successfully implement this approach. Chapter 2provides an introduction to generating functions, and extensively developstechniques in manipulating generating functions, sharpening the mathematicalskills and powers of observation of the students. Chapter 2 also provides thecomplete derivation for generating functions from commonly used discreteprobability distribution functions. Chapter 3 provides a complete comprehensivediscussion of the use of generating function inversion procedures when thedenominator of the generating function is a polynomial. Chapter 4 combines theresults of the first three chapters, demonstrating through discussion andnumerous examples, the utility of the generating function approach in solvinghomogeneous and nonhomogenous difference equations. Chapter 5 addressesthe complicated topic of difference equations with variable coefficients.

Chapter 6 introduces a collection of difference equation systems developedby the authors which are useful in public health research. Many of theseapplications are based in run theory, which the authors have found is a naturalsetting for several important issues in public health. The public health problemis often naturally expressed using run theory and a family of differenceequations is provided to describe the recursive elements of the public healthproblem. These difference equations are motivated and solved in their entiretyusing the generating function approach. Chapters 7-10 applies these systems toproblems in public health. Chapter 7 applies difference equations to droughtpredictions. Chapter 8 explores the applicability of difference equations topredict the occurrence of life-saving cardiac arrhythmias. In Chapter 9,difference equations are provided to describe patterns of missed visits in arandomized controlled clinical trial. Chapter 10 applies difference equations tohealth services organization plans for cash flow management at smallambulatory clinics. In each application, the solutions of the equations areformulated in a way to provide additional insight into the public health aspect of

Introduction

the problem. Finally, the application of difference equations to epidemiologicmodels is provided through the treatment of immigration, birth, death, andemigration models in Chapters 11 and 12. Solutions for combinations of thesemodels are also provided.

Difference Equations with Public Health Applications provides solidgrounding for students in stochastic processes with direct meaningfulapplications to current public health issues.

Lemuel A. MoyeAsha Seth Kapadia

Contents

Series Introduction vPreface viiIntroduction ix

1. Difference Equations: Structure and Function 1

2. Generating Functions I: Inversion Principles 21

3. Generating Functions II: Coefficient Collection 65

4. Difference Equations: Invoking the Generating Functions 91

5. Difference Equations: Variable Coefficients 145

6. Difference Equations: Run Theory 187

7. Difference Equations: Drought Prediction 215

Contents xiv

8. Difference Equations: Rhythm Disturbances 243

9. Difference Equations: Follow-up Losses in Clinical Trials 261

10. Difference Equations: Business Predictions 277

11. Difference Equations in Epidemiology: Basic Models 291

12. Difference Equations in Epidemiology: Advanced Models 325

Index 389

1Difference Equations: Structure andFunction

1.1 Sequences and Difference Equations

1.1.1 Sequence definitions and embedded relationshipsTo understand the concept and to appreciate the utility of difference equations,we must begin with the idea of a sequence of numbers. A sequence is merely acollection of numbers that are indexed by integers. An example of a finitesequence is {5, 7, -0.3, 0.37, -9 }, which is a sequence representing acollection of five numbers. By describing this sequence as being indexed by theintegers, we are only saying that it is easy to identify the members of thesequence. Thus, we can identify the 2nd member in the sequence or, as anotherexample, the 5th member. This is made even more explicit by identifying themembers as yi, y^, yi, y^ and y5 respectively. Of course, sequences can be

2 Chapter 1

infinite as well. Consider for example the sequence /, i/ i/ i/ i/ i/ \\l'/2'A' 78'/16 '/32 '-/

This sequence contains an infinite number of objects or elements. However,even though there are an infinite number of these objects, they are indexed byintegers, and can therefore have a counting order applied to them. Thus,although we cannot say exactly how many elements there are in this sequence,we can just as easily find the 8th element or the 1093rd element.* Since we willbe working with sequences, it will be useful to refer to their elements in ageneral way. The manner we will use to denote a sequence in general will be byusing the variable y with an integer subscript. In this case the notation yi, y2,y3, ... ,yk, ... represents a general sequence of numbers. We are often interestedin discovering the values of the individual members of the sequence, andtherefore must use whatever tools we have to allow us to discover their values.If the individual elements in the sequence are independent of each other, i.e.,knowledge of one number (or a collection of elements) tells us nothing about thevalue of the element in question, it is very difficult to predict the value of thesequence element. However, many sequences are such that there existsembedded relationships between the elements; the elements are not independent,but linked together by an underlying structure. Difference equations areequations that describe the underlying structure or relationship between thesequence element; solving this family of equations means using the informationabout the sequence element interrelationship that is contained in the family toreveal the identity of members of the sequence.

1.1.2 General definitions and terminologyIn its most general form a difference equation can be written as

Po(k)yk+n +P,(k)yk+n-1 +P2(k)yk + n_2 +-+Pn(k)yk =R(k) (l.i)which is the most general form of difference equations. It consists of termsinvolving members of the {yk} sequence, and, in addition, coefficients such asPj(k), of the elements of the {yk} sequence in the equation. These coefficientsmay or may not be a function of k. When the coefficients are not functions of k,the difference equation has constant coefficients. Difference equations withcoefficients that are functions of k are described as difference equations withvariable coefficients. In general, difference equations with constant coefficientsare easier to solve then those difference equations that have nonconstantcoefficients.

Sequences which are infinite but not countable would be, for example, all of thenumbers between 0 and 1.

Difference Equations: Structure and Function 3

Begin with a brief examination of the right side of equation (1.1) tointroduce additional terminology. Note that every term on the left side of theequation involves a member of the {yk} sequence. If the term R(k) on the rightside of the equation is equal to zero, then the difference equation ishomogeneous. If the right side of the equation is not zero, then equation (1.1)becomes a nonhomogeneow difference equation. For example, the family ofdifference equations

yk .z=6yk + , -3yk (1.2)

for k an integer from zero to infinity is homogeneous since each term in theequation is a fimction of a member of the sequence {yk}. The equation

y k + 2 =6y k + 1 -3y k +12 (1.3)

for k = 1,2,..., oo is a nonhomogeneous one because of the inclusion of the term12.

Finally, the order of a family of difference equations is the difference insequence location between the term with the greatest index and the term with thesmallest index of the {y1? y2, y3, ...,yk, } sequence represented in the equation.In equation (1.1) the member of the sequence {yk} with the largest subscript isyk+n and the member of the sequence with the smallest subscript in equation(1.1) is yk. Thus the order of this difference equation is k + n - k = n, andequation (1.1) is characterized as an nth order difference equation. If R(k) = 0,we describe equation (1.1) as an nth order homogenous difference equation, andifR(k) & 0 equation (1.1) is an nth order nonhomogeneous difference equation.

As an example, the family of difference equations described by

yk + 2=6yk + 1-3yk (1.4)

for k = 0, 1,2, ..., oo where yo and yi are known constants is a family of second-order homogeneous difference equation with constant coefficients, while

l) (1.5)

would be designated as a fourth-order, nonhomogeneous family of differenceequations with variable coefficients.

4 Chapter 1

1.2 Families of Difference EquationsAn important distinction between difference equations and other equationsencountered earlier in mathematics is that difference equations represent apersistent relationship among the sequence members. Since these relationshipscontinue as we move deeper into the sequence, it is clear that the "differenceequation" is not just one equation that must be solved for a small number ofunknown variables. Difference equations are in fact better described as familiesof equations. An example of a fairly simple family of difference equations isgiven in the following equation

Yk+i = 2yk -2, k an integer > 0, y0 known (1.6)

Recognize this as a first order, nonhomogeneous difference equation withconstant coefficients. We must now expand our view to see that thisformulation has three components, each of which is required to uniquely identifythis family of difference equations. The first is the mathematical relationshipthat links one member of the sequence to other sequence members-this isprovided by the essence of the mathematical equation. However, the secondcomponent is the range over the sequence for which the difference equationsapply. The relationship yk+1 = 2yk 2 serves to summarize the following set ofequations:

y, =2y 0 -2y 2 = 2 y , - 2y' =

*y'-* d.7)y 4 = 2 y 3 - 2

y 5 = 2 y 4 - 2y 6 =2y 5 -2

The key to the family of equations is that the same equation is used to link onemember of the sequence (here yk+1) to at least one other member of thesequence. It is this embedded relationship among the sequence members thatallows one to find the value for each member of the sequence.

Rather than be forced to write this sequence of 100 equations out one byone, we can write them succinctly by using the index k, and consideration of thisindex brings us to the second component of equation (1.6)'s formulation:namely, the range of sequence members for which the relationship is true. Forequation (1.6) the sequence is only true for y0, yl5 y2,...,yioo- Some otherrelationship may hold for sequence members with indices greater than 100. Ifthe sequence of integers is finite, the number of equations that need to be solved


is finite. An infinite sequence will produce an infinite number of equations thatrequire solution.

The third and last component of the equation for a family of differenceequations is the initial or boundary condition. For example, for k=0, the firstmember in the family of difference equations (1.6) reduces to yi = 2yo - 2.Since there is no equation relating yo to another sequence member, it is assumedthat y0 is a known constant. This is called a boundary condition of thedifference equation family. The boundary conditions are necessary to anchor thesequence for a particular solution. For example, if y0 = 3, the sequencegoverned by the set of difference equations yk+1 = 2yk - 2 is {3, 4, 6, 10, 18,... }If, on the other hand, the sequence began with a different boundary condition y0= -1, the sequence becomes {-1, -4, -10, -22, -46, ...}. Thus, the samedifference equation, with different boundary conditions, can produce verydifferent sequences. Each of these three components is required to uniquelyspecify the difference equation.

1.2.1 ExamplesThere are infinite number of families of difference equations. Some of theexamples are given below.

Example 1 :+cyk (1.8)

for k = any integer from 0 to 20: a, b, and c are known constants and yo, yi, andy2 are known.

Example 2:

for k = 0, 1, 2,..., 10: y0 a known constant

Example 3:3kyk+2=yk-0.952kyk+1 (1.10)

for k = 0,...,oo : y0 and y, are known.

Each of these three examples represents a difference equation family with itsown unique solution depending on the relationship, the range of the index, andthe boundary conditions. Thus, these families of equations represent uniquesequences.

6 Chapter 1

1.3 Solutions to Difference EquationsBeing able to first formulate a public health problem in terms of a family ofdifference equations, and then find the solution to the family, is a useful skill forwhich we will develop powerful tools from first principles. There are severalapproaches to the solutions of difference equations. Chapters 2 and 3 willdevelop some analytical tools useful in their solution. However, two otherrather intuitive approaches should be considered as well, and can themselves beuseful and constructive: The first is the iterative approach to the solution ofdifference equations, and the second is the use of mathematical induction. Bothare briefly discussed below.

1.3.1 Iterative solutionsThe temptation is almost irresistible to solve difference equations iteratively. Tobegin, consider the following simple set of relationships in which the task is toexplicitly find all values of the sequence {yk} where a is a constant

y M =ay k (i.n)for k = 0, 1 ,2, . . . , oo : y0 a known constant. This first-order, homogeneous family

of difference equations represents a relatively easy set of relationships, and onecan envision starting with the first elements of the sequence for which the valuesare known, then using the difference equations to move further into thesequence, utilizing the recently discovered value of the k* element to helpidentify the k+ 1st value.

y 2 =ay, =a(ay0) = a2y0V 3 =av 2 =a(a 2 v 0 ) = a3y0 (1.12)V 4 =ay 3 =a(a 3 y 0 ) = a4y0

If we continue to develop the sequence elements, it can be seen that the generalsolution for this family of difference equations is yk = a1^- The logic of theunderlying solution is embedded in the family of difference equationsthemselves, and by moving through the recursive equations one at a time, weallow the difference equation family essentially to solve itself.

As another example of the ease of this approach, consider the family ofdifference equations


y k + 1 = a y k + b (1.13)

for all k > 0; y0 known. Proceeding as in the earlier example, the examinationof the values of the elements of the sequence of {y^} can proceed in ananalogous fashion. From equation (1.13), the next value in the {y^} sequenceyk+i is obtained by multiplying the previous value in the sequence y^ by theknown constant a, and then adding to this product the known constant b.Equation (1.13) provides the necessary information to evaluate each of the y l5

y2 =ayj +b = a(ay0+b) + b = a2y0 +ab + by3 = ay2 +b = a(a2y0 +ab + b)+b = a3y0 2

y4 =ay3 +b = a(a3y0+

and so on. Again the solution is obtained through an iterative approach.Unfortunately, this intuitive approach to the solution, which is very useful

for first-order difference equations that were described above, becomessomewhat complicated if the order is increased. Consider, for example, thesecond-order, nonhomogeneous family of difference equations

y k + 2 =ay k + 1 +by k +d (1.15)

for k = 0,l,2,...,oo; y0, y, are known. The iterative approach begins easilyenough but rapidly becomes complicated.

y 2 =ay ,+by 0 +d

= a2y, + aby0 + ad + byt + dy4 =ay 3 +by 2 +d = a(a2y,+aby0+ad + by1 +d) (1-16)

+ b(ay, +by0 +d) + d= a3y, + a2by0 +a2d+aby, +ad + aby, +b2y0 +bd + d= (a3+2ab)y,+(a2b + b 2 )y 0 +(a 2 +a+b+l)d

From the developing pattern it can be seen, that, in general, yk will alwaysbe written as a function of the constant coefficients a, b, and d, as well as interms of the boundary conditions (i.e., the values of yo and yi). Recognizing

8 Chapter 1

these terms is by itself not sufficient for attaining the actual solution. It isdifficult to see a pattern emerging since yk becomes an increasingly complicatedfunction of a and b as k increases. The situation becomes even morecomplicated for third-order families of difference equations. Although onlyalgebra is required for the iterative solutions of these equations, this algebraitself becomes more complicated as solutions are found for values deep in the{yk} sequence. Thus the iterative approach will always work, but it becomesdifficult in complicated difference equations to envision the general solution forany sequence member yk

1.3.2 The induction methodOne way to solve these simple equations is to guess the solution and then provethat the guessed solution is correct through the use of induction. Briefly, theinduction argument outlines a simple sequence of three steps in the proof of anassertion about the non-negative integers. The steps are

(1) Demonstrate the assertion is true for k = I(2) Assume the solution is true for k(3) Use (1) and (2) to prove that the assertion is true for k+1This time-tested tool has been very useful in proving assertions in algebra forintegers. We will use as a simple example an examination of a possible solutionto the nonhomogeneous, first-order equation

y k + I = a y k + b (1.17)

for all k > 0; y0 a known constant. Assume the solution

k-ly k = a k y 0 + b a j (1.18)

To begin the proof by induction, first check if this solution is true for k=l

y i = a y 0 + b = ay0+biy (1.19)

so the assertion is true for k = 1.The next step in the induction argument is to assume that equation (1.17) is truefor k (i.e., assume that the assertion is true for some non-negative integer k), and


then, using this assumption, prove that the proposed solution is true for k = k+1.The task is to prove that

yk + 1=ak+'yo+bI> J (1.20)j=o

This is accomplished easily with simple algebra

(1-21)= ak+1y0 + ba a j + b = ak+1y0 + b a1

j=0 j=0

so the assertion is true for k = k+1 and we have the solution for the differenceequation family. Of course, an obstacle to this approach is the requirement offirst having an idea of the solution of the family of equations. Induction does notwork without a good guess at the solution. Looking back at our work on thesecond-order, nonhomogeneous family of difference equations with constantcoefficients given by

y k + 2 = a y k + 1 + b y k + d (1.22)

for k = 0,1,2,...,00; y0, yl are known it is difficult to guess at the solution. Ingeneral guessing will not serve us well, and we will use the generating functionargument to be developed in the next two chapters to guide us to the solutions.

1.4 Some ApplicationsDifference equation families can be elegant mathematical expressions ofrelationships that extend themselves through a sequence. There are manyexamples of their applications. Some of the more classic ones occur in Goldberg[1], Feller [2], Saaty [3], Chiang [4], and Bailey [5]. They do in fact appear inpublic health. Here are three illustrations.

1.4.1 Health services researchConsider a situation in which you are hired as a manager of a medical complex.This organization includes physicians, nurses, physician assistants, andadministrators. One of the new manager's first responsibilities is to help resolvean issue that has become an important problem for the employees of the

10 Chapter 1

company who have been employed the longest, and have the company's bestand greatest experience. The standing company policy has been to award eachcompany employee an increase in their hourly wages. The increase comes froma table of hourly rates. The table has ten rows, each row representing the numberof years of service. For each row the new hourly wage is supplied. In thiscompany, several of the employees have been in the organization for more thanten years (Table 1.1).

Table 1.1 Observed Clinic Hourly Wage by Years of Service

Year of Service Wage

12345678910

6.006.487.007.568.168.829.5210.2811.1111.99

However, since the table only contains ten entries, those employees who havebeen in the organization for more than ten years cease to get yearly increments.The assignment is to compute the correct adjustment in wages for theselongstanding employees.*

A review of Table 1.1 quickly reveals that the increase in hourly wagerepresents about an 8% increase over the previous year's hourly wage. Yourtask is to represent the worker's hourly wage for this year in terms of last year'shourly wage. Choose yk as the hourly wage for the k* year. Then the hourlywage an individual makes is indexed by the number of years he/she has worked,and a worker who has worked for k years has the sequence of yearly hourlywages YI, y2, y3, y4, y5, y6, ... depicting their history of hourly wages. What is

The employees in this friendly group did not ask for back wages, or for interestassociated with the lost wages, only for the immediate adjustment of their wages to thecorrect rate.


required is a relationship between the hourly wage for the k+l th year, and thehourly wage for a salary for the previous year. You write

yk + 1=l-08yk (1.23)

for k = 1 to oo and yi a fixed, known, constant. Note that this equation is afirst-order homogeneous equation with constant coefficients, and is the same asthe first difference equation that was solved using the iterative approach. Thesolution for that equation was yk = a"Vo for k = 1,2,3,4,5, ...

which in the circumstances of this clinic becomes

Table 1.2 Predicted Clinic Hourly Wage by Years of Service

Year of Service Wage

1234567891011121314151617181920

6.006.487.007.568.168.829.5210.2811.1111.9912.9513.9915.1116.3217.6219.0320.5622.2023.9825.89

y t=(l.08)k"y, (1.24)

12 Chapter 1

k = 1 to co and you can complete the table for the senior employees.Obtaining y} from the first row of Table 1.2, compute the hourly wage for

an employee who is in their twelfth year of service is y12 = (1.08)''(6.00) =$13.99 per hour. Of course, employees who have been there for many years,would have exhausted this company's payroll. For example, an employee whohas been working for forty years would have an hourly wage of y40 =(1.08)39(6.00) = $120.69. Fortunately for the organization, there was noemployee with more than fifteen years of service.

7.4.2 Screening clinic paymentsAnother useful example of the derivation of a popular (indeed, infamous) familyof difference equations involves debt repayment. Consider the construction of awomen's clinic in an underserved area, which will play a major role in screeningwomen for breast and cervical cancer. The clinic owners arranged for a businessloan of $500,000 for equipment and operations costs to be paid back in 30 years.If the clinic has arranged for an annual interest rate of r, then what must themonthly payments be over the next thirty years?

As before, we do not start with the difference equation to solve. Instead, webegin with the issue and attempt to formulate it as a difference equation. Weexpect that there is a recursive relationship built into the sequence of monthlyinterest payments and monthly principal payments that needs to be discovered.Let's start our approach to this problem by introducing some helpful notation.Let mk be the total monthly payment that is due. It is assembled from twocomponents: The first is the interest payment for that month, ik; the secondcomponent is the principle that is due for the kth month,* pk. Then

m k = i k + p k (1.25)

fork =1,2, 3,...,360.Begin by finding it, the interest payment for the first month. Recall that theamount of interest due in any month is based on the unpaid principle. Since inthe first month, no principle has yet been paid,

i.=500,00o(!1 (1.26)1 U200J

The term is the annual interest rate divided by 100 (to get the annual rate1200

in percentage points) and then divided by 12 (to obtain the monthly rate). If r =

For simplicity, we will ignore premium mortgage insurance, property insurance, andtaxes which would increase the monthly payment even more


1212, then it is . Compute the interest payment due in the second(100) -12payment month as

i2=(500,000-p.) (1.27)2 v Vl> U200J

Another critical feature of this system of interrelationships is that the monthlypayments are constant. Even though the interest payments change each monthand the amount of the principle changes with each payment, the monthlypayment itself (which is the sum of these payments) remains constant. That is,Hi] = m2 = m3 = . . . = m360, as there are 360 monthly payments during the term ofthe thirty year loan. Thus

> (1.28)i = i 2 + p 2

and therefore

500,000] 5 I + p. = (500, 000 -p,) + p, (1.29)U200J ' V Hl'l200 2

This suggests that we can solve for the 2nd month's principle, p2 in terms of pito find

1200 + rp, = p, (1-30)2 1200 '

We can proceed to the next step in the sequence of payments, now evaluatingthe relationship between p3 and p2

m, =nv,.

2 3. 0.31)i 2 + p 2 = i 3 + p 3

which becomes

(500000 -p,)-^ + p, =(500000-?! -p,)-^ + p3 C1-32)V '

H2 V '

2/1200 H3

Solving p3 in terms of p2

14 Chapter 1

(0.32)1200

and, in general1200+ r (0-33)

for k = 1,2, ..., 360. So, the sequence of 360 principle payments has embeddedin them a relationship reflected by a difference equation. Recognize thisequation as a first-order, homogeneous equation with constant coefficients.

Furthermore, letting = a , it can be seen that equation (0.33) is anotherform of equation y^+i = ay^, for which it is known the solution is yk = a k y , .Applying this principle here, we find

(0-34)1200

However, pi is not yet known. Since we will need a tool from Chapter 2 to findthis, we must postpone our solution until later. As it turns out, knowing p, is theone piece of the puzzle that is needed. Since i j is already known the interestpayment for the first month, compute m, from nil = ij + pt to compute themonthly payment*. We will return to a complete solution for this problem inchapter three. The point here is that it can take some work and ingenuity toidentify the appropriate family of difference equations.

1.4.3 Chronobiology and mixed difference equationsIt is sometimes of interest to combine information for a system that includes twofamilies of difference equations. The solutions provided for this procedure willdemonstrate the adaptability of the difference equation perspective. Consider asystem that describes the likelihood that a woman is most likely to ovulate on agiven day. There is important variability in the day a woman in the populationwill ovulate. Assume that the ovarian cycle is regular and of 28 days duration(this is different than a calendar month). The women's likelihood of ovulationincreases as the day comes closer to the middle of the cycle. Let yk be thelikelihood that a woman will ovulate on the kth day, k = 1, 2, 3, ..., 28. Supposeas a first approximation, the relationship is

A general solution for this equation will be presented in Chapter 3, after the concept ofgenerating functions is introduced in Chapter 2.


y k =1 .46y k _ , (0.36)

for k = 2, 3,..., 28 (let y,=1.46). Note, that equation (0.36) states that thelikelihood of ovulation increases, and, is in fact a multiple of the previous day'sovulation likelihood. However, this family of equations has two problemsrelated to the unbounded nature of yk in this relationship. The first is that theselikelihoods are not probabilities and do not sum to one across the 28 day cycleperiod. The second problem is that it is known that the likelihood of ovulationdoes not increase monotonically-rather, it peaks at the middle of the cycle anddecreases thereafter until the cycle ends. Thus, the system requires a model thatallows the ovulation likelihood to increase to a particular day in the month, andthen begin to decrease. One way to build this into the difference equation familywould be to let the form of the difference equation relationship be a function ofthe day. The difference equation will be

yk =1.46yk , f o r 2 < k < 1 4J k ^k - i

(Q37)= 0.47yk_, f o r ! 5 < k < 2 8

where yi is a known constant. This system is actually really a hybrid of twodifference equation families. For k< 14, the likelihood monotonically increases.However, for k > 14, the relationship between yk and yk.i changes, with adecreasing value of the likelihood of ovulation as the days progress to those laterin the 28-day cycle.

Using the iterative approach to identify a solution to this family ofdifference equations.

y 2 =( l .46)y , : y 3 =( l . 46 ) 2 y i :L y]4 =(l.46)'3 Yl (0.38)

When k = 15, use the second summand of equation (0.37) to find y,5 = 0.47y,4 =0.47(1.4613yi) and proceed on to the end of the 28-day cycle

y,6=(0.47)2( l .46) l 3 y ]y,7=(0.47)3(l.46)'3y i (0.39)y,8=(0.47)4(l .46) '3 y i

This evaluation reveals that the solution for yk for k > 15 has two components.The first reflects the influence of the first 14 days of the month, as measured by(1.46)13yi- The second component reflects the influence of the sequence

16 Chapter 1

members {yk} for k > 15. This component is calculated to be (0.47)k~13. We cansee that the general solution for this problem demonstrates that

yk=(l.46k- ')y I f o r l < k < 1 4= (l.4613)(0.47k-13)y, fo r !5


'Ovu

latio

n

=> s

?SI

U

l C,

0 0.15

1 "^

2 0.05O rrl k- l-as 1-as

allowing the conclusion that G(s) = > |ak}. Again, as in the earlier case,1-as l '

the inversion of G(s) required the identification of a one-to-one correspondencebetween the coefficient of sk and the sum of the series.

2.4 Recognizing Generating Functions - General Principles

Much of the effort in the inversion of a generating function involves recognitionof key components of G(s). In many of the examples we will be working within this text, G(s) will at first glance appear complicated. However, furtherinspection will reveal that G(s) is composed of component pieces, each of whichprovides a key to the inversion. Thus, although G(s) may be complicated, it willbe disassembled into component features, each of which may be easily inverted.In this section some tools and maneuvers will be developed that will be usefulin converting a complicated generating function to one which is more easilyinverted.

2.4.1 Scaling a generating function - the scaling toolAs we work through this chapter, we will build a list of generating functions andtheir inversions, as well as develop the skills in manipulating generatingfunctions. This first tool is very easy to apply. Consider the infinite series (|as| 1-as

Now multiply both sides by a known constant c to find

c + cas + caV + ca3s3 +... = caksk = c j>Vsk = = G(s) (2.29)t=o L-=O l as

From this see that > |cak}, or the generating function is the sum1-as l ; 1-as

of the infinite series whose kth term is caksk. Thus define

The scaling tool for generating function inversionLet G!(S) be a generating function such that G,(s) > {yk}. Let G2(s) = cG^s).ThenG2(s)> {cyk}.

The scaling tool will be a helpful device throughout this text. Some easymanipulations follow from the application of this simple scaling tool. Forexample, the scaling principle can be implemented at once to assess whether

G(s) = is a generating function,a + bs

1 1 / 7G(s) = -^- =

r . 7 1 = -^ T- = /? A (2-30)a + bs af l + ^ /s] l + %s l-(-b/)s

and now using the scaling tool to see that

a I a

2.4.2 Additive principle of generating function inversionTo continue increasing familiarity with generating functions, now consider twogenerating functions G,(s) and G2(s). Assume G,(s) > {yik} and G2(s) > {y2k}-

Generating Functions I: Inversion Principles 31

The question before us is whether G|(s) + G2(s) is a generating function.

k +y 2 JS (2.32)k=0 k=0 k=0

and we find that G,(s) + G2(s) > {y,k + y2k}. We can now state the

The additive principle of generating functionsAssume G,(s) > {y,k} and G2(s) > {y2k}. Then G,(s) + G2(s) > {ylk + y2k}

This principle will be most useful when dealing with a generating function thatis a sum. If each of the summands can be inverted then we have the inversion ofthe original G(s) in our hands, since all that is needed is to obtain the kth termfrom each of the summands. In a completely analogous fashion, it can be seenthatG,(s)-G2(s)> {y,k-y2k}.

2.4.3 Convolution principle of generating functionsUnfortunately, the product of generating functions is not the product of the k'hterms of each of the infinite series. However, we do not have to look too muchfarther in order to identify the terms of the infinite series that represent the truevalue of the kth term of the series when the series is the product of at least twoother series. The solution to this problem will simplify the inversion ofcomplicated forms of G(s), when G(s) can be written as the product of two othergenerating functions. Begin with two generating functions, G,(s) > {ak} andG2(s) > {bk} as follows

G,(s)G,(s) = (a0 +a s + a,s2 + a,s3 +L ) (b0 +b s + b,s2 +b,s3 +L )1 M - /

Here, we only need evaluate this product term by term and gather together thosecoefficients which are associated with like powers of s. A table which relates thepowers of s to the coefficient s to that power quickly reveals the pattern.

32 Chapter 2

kIn general write the coefficient of sk, y^ as yk = ^3jbk_j and so observe

j=0

Table 2.1 Powers and Coefficients of s for G l (s)G 2 (s)

Power Coefficient

0 aobo1 a0b! +a,b02 a0b2+a,b,+a2b03 a0b3+alb2+a2b1+a3b04 a0b4+a,b3+a2b2+a3b,+a4b0

j=0

Note that for each term in the summand, the sum of the subscripts is equal to k.This constant sum of subscripts, as j varies from 0 to k, is what makes thisparticular sum of terms a convolution. Thus, we may write

The Convolution Principle of Generating Function InversionBegin with two generating functions, GI(S) > {ak} and G2(s) >

The convolution principle will be extremely useful throughout our experience ofgenerating function inversion. For example, consider the generating function


G(s) = (2.36)

We know > (5k) and > ( 2 k ) . This is all that we need to apply thel-5s l j l-2s l J

convolution principle to see that

G(s) = (1-5SX1-2S) u,0 j (23?)

We can see the solution immediately upon visualization of G(s) as aconvolution. Proceeding, we can compute y0 - 1, yi = (1)(2) + (5)(1) = 7, y2 -(1)(4) + (5)(2) + (25)(1) =39, y3 = (0(8) + (5)(4) + (25)(2) + (125)0) = 203.The entire sequence can be generated in this fashion. Also, one need not buildup the sequence of solutions for y0> yi, y2, y3, ... , yk-i to find the solution to yk.Instead, using the generating function inversion principles, we can "jump"directly to the solution for yk. This is an attractive feature of generatingfunctions. We can, for example, compute the value of y&, without computing yothrough y5.

The convolution principle propagates nicely. For example, if we have threegenerating functions G!(S), G2(s), and G3(s) where G t(s) > {ak}, G2(s) > {bk},G3(s) > {ck} then we can demonstrate that

G,(s)G2(s)G,(s)

To begin this demonstration, first observe that

G,(s)G2(s)G3(s)= (a t ) +a | S + a 2 s 2 +a 3 s 3 +L ) (b0 +b i s + b2s2 +b3s3 +L ) (2.39)

(co H-C^ + CjS2 +C3S3 +L J

We can write the right side of equation (2.39) one product at a time.

34 Chapter 2

(a0 +a is + a2s2 +a3s3 +) (b0 +b s + b2s2 +b3s3

= a0b0+(a0b, +a,b0)s+(a0b2 +a,b, +a2b0)s2+(a0b3 +a^2 +a2b, + a3b0)s3

(2.40)

resulting in

j=0S +

j=o; S J + . . . + (2.41)

The coefficients of sk in equation (2.41) represents the inversion of Gj(s)G2(s).The triple generating function product can be written as

G1(s)G2(s)G,(s)

j=0

1 I f 2ZaJbH 9+ ZJb2-jj=o J LJ=0

(c0 +J=

s 3 + - (2.42)

and we just take another convolution of these remaining two sequences in thisproduct to show

G,(s)G2(s)G3(s) (2.43)

As another example of the productivity of this principle, when confronted witha generating function of the form

G(s) = - 1

First write

G(s) = - 1 V 1

(2.44)

(2.45)= G,(s)G2(s)G3(s)


and then use the convolution principle to see (note that a0 = b0 = c0 = 1)

G(s) = 7 w . w T HyTaibiCk i ,\ (2-46)' J k

-

J

2.4.4 Sliding tool of generating functions

Thus far we have considered generating functions in which the denominator is afunction of s but not the numerator. In this short section, we will see how tohandle generating functions where the numerator also contains terms involving

s. Begin with G(s) = and find1-bs

1 = 1 + bs + bV + bV + bV + ... (2.47)

1-bs

and we know that G(s) = > I b k } . We can multiply each side of1-bs l '

equation (2.47) by s to obtain

S = s + bs2 + bV + b3s4 + bV +... (2.48)

1-bs

here the coefficient of sk is not bk, but bk~'. Having the term involving s in thenumerator has essentially slid the necessary coefficient in the series one term tothe left. Thus, to invert

(2.49)l-4s

first recognize a subcomponent of G(s), > J4k } . The numerator of G(s)1 - 4s l '

however reveals that the kth term of the sequence {4k} is three terms to the right

36 Chapter 2

of the correct term. Thus, sliding three terms to the left reveals thatG(s) > {4k~3} As another example, consider the inversion of

(2.50)6-s 6-s

Inversion of the generating function in equation (2.50) is eased by the commondenominator in each of the terms on the right side of the equality. The inversionitself will involve the application of the scaling, summation, and slidingprinciples. Begin with the recognition that

-^T-HI T i ^ (2-5D6-sThe inversion of the factor (6-s)"1 common to each term of equation (2.50) iscomplete. Now use the sliding tool and summation principle to write

G(s) = - > - + - > - - (2.52)6-s 6-s ' f

The sliding tool works in the reverse direction as well. For example consider

= = _ >(?k+3ls'-2s' "('-*) '-* '2 ' p.53,

In this case the s"3 in the denominator forced a slide to the right, increasing therequired term in the {yk} sequence by 3.

2.5 Examples of Generating Function Inversion

In this section we demonstrate by example the inversion of several generatingfunctions using the tools and principles developed thus far. These examples willbe provided in increasing order of complexity. It is important that the readerunderstand each step of the process for every example.


2.5.1 Example 1

G(S) = (s-2)(s-l)

The convolution principle allows us to decompose this generating function as

G(s) = / J/ ,,=G.(s)G2(s) (2.55)

If each of GI(S) and G2(s) can be inverted, the convolution principle will permitus to construct the sequence {yk} associated with the inversion of G(s).Beginning with G^s)

G,(s) =1 s-2 2-s V2(2-s) 1-1/s

inverting the second generating function of equation (2.55) similarly

s-1 1-s(2.57)

We can now apply the convolution principle, the coefficient of sk in the product

of the sequences I j- j- > and {-1} is

G.(s)G,(s)> > T (-l) = >1^- = ->'\- =1- - (2.58)lW 2W [^UJ+'J 1^2J+1 2^UJ UJ j

Sof / -1 Ak+11

(2.59)

38 Chapter 2

2.5.2 Example 2To invert

G(s) = s2-s4 (2.60)

first proceed by recognizing that the numerator will utilize the sliding tool.

1G(s) = s

2-s4 (2.61)

Begin by inverting the last term using the convolution principle, observing

L(l-5s)(l-9s)

Now use the sliding tool and the summation principle to compute

(2.62)

s2-s4 k-2 k-4 vk-4-j

j=0 (2.63)

2.5.3 Example 3

Consider

G(s) =(s_3)(s-4)(s-5)

= ss-3As-4As-5

(2.64)

It can be seen from the denominator of G(s) that the convolution principle willbe required not once, but twice. Begin by considering each of the terms in thedenominator of equation (2.64). A simple application of the scaling principlereveals


3 - ( s ) 1 m K 3 A 3

Thus we see that

i f rnk+1l i f r iY + 1 l i I rnk+ ' l_i_> _ i I _ J_>J_ i land 1> - - I (2.66)s-3 ( 3 s-4 UJ s-5 I I s J f

What remains is to reassemble G(s) from the product of these terms, whilekeeping track of the coefficients of sk. Apply the convolution principle for thefirst time to see

(s-4)(s-5)and again to find

I

(s-3)(s-4)(s-5)

and invoking the sliding tool

f k-3k-3-i ^ 1 Y+1 f 1 V+' ( 1 N''-i-j-2]

. 4 5

40 Chapter 2

beginning with smaller values of the positive integer k and then building up tolarger values of k.

2.6.1 Inversion ofG(s) torn = 2The solution for n = 1 has already been provided. To see the solution for n = 2,only a small amount of algebra is required. Begin by writing

(2.70)1-s2 1-s-s

If the variable was a constant s, such as the constant a, the inversion procedurewould be straightforward as depicted in equation (2.70)

5 = 1 + as + aV +aV +aV +... > (ak| (2.71)1-as l ;

Now just substituting s for a in both the left side and the right side of equation(2.71) provides an interesting solution for G(s)

V +sY + sV+...1-s2 - - - - - - - ' - - ' - - (2.72)= l+s 2+s 4+s 6+s s+. . .

and see what the coefficients are. If k is odd, the coefficient of sk is zero. Onthe other hand, if k is even, then the coefficient of sk is one. We may write

(2-73)

The expression Ikmod2 =o is a combination of two useful functions. Let I^A be theindicator function, equal to one when x = A and equal to zero otherwise. Forexample, Ix=i,2,3 is a function of x, equal to 1, when x is equal to 1, 2 or 3, andequal to zero for all other values of x. The function k mod 2 is short for kmodulus 2, which is the integer remainder of k/2. For even values of k, k mod 2is equal to 0. For odd values of k, k mod 2 is equal to one.


We have seen that the inversion of G(s) = - requires us to include a1-s

one for each even power s, and a zero for an odd power of s. The functionIkmod2=o is precisely the function we need to carry out this task.

2.6.2 Inversion forn = 3

The inversion process for G(s) = -, proceeds analogously. Once we write1-s

G(s) asG(s) =

1-s3 l -s2-s= l + s3+s6+s9+... (2.74)

Using the indicator function and the modulus function to write

} (2.75)

2.6.3 The general solution for positive integer values of n

We can proceed with the general solution for - . as

G(s) = = - = 1 + s-'s + s2(n-"s2 + s3(n-'y1-s" l-s-'-s

= l + s n +s 2 n +s 3 n +. . . (2.76)

and we can use the indicator function and the modulus function to write

(2.77)1 S

42 Chapter 2

2.6.4 Additional examples

In this section, we will apply the experience developed thus far to relatedgenerating functions. Consider the generating function

G(s) = -1-= * - * (2.78)

We see that we can use the convolution principle for inversion of G(s).

__>/I ] : > {i \ (2.79)1-S3 knlod3- J _ g 4 kmod4-0

and invoking the convolution principle to write

/"< / \ * X"1 T TCj(s) 1> 4 / I 1I I s 111 s I t j=o

As another example, consider the generating function

0(8) = ^ (2.81)b-cs

Begin by writinga- a/

3 -

b -

b (2.82)

b-cs5 , c ,' -K S 1~\1S4^b 1 b

and proceed by writing

- = 1 + s"s + | -s4 I s2+| -s" I sj +...f . \M

- - (2.83)

= 1+s + - s'"+,b I b J I b2 fc310


and

,-^

Continue by invoking the scale principle to write

G(s) = b-cs5 b ba II c

(2.84)

(2.85)

For an additional example consider

G(s) = 3s-4(3s-2)(2-s3)

(2.86)

The work in this inversion is in the denominator. Once the denominator isinverted, use the sliding and scaling tools. Proceed by writing

1 -1 /2 -1

noting

31 ~ S2

Now invoking the convolution principle,

(2.87)

(2.88)

(3s-2)(2-s3) ' ||TJ-1 I(k-j)mod3=0 (2.89)

44 Chapter 2

And now invoking the scaling and addition principles complete the inversion

G(s) = 3s-4(3s-2)(2-s3)

hTI-I ^1 ->j=0-1

(2.90)(k-j)mod3=0

2.7 Generating Functions and Derivatives

Thus far as we have explored combinations of generating functions, eachmaneuver has broadened our ability to invert G(s), identifying the coefficientsyk of the sequence of interest. Continue this examination through the use ofderivatives. If G(s) is differentiable, then the derivative of this function withrespect to s is written as

dG(s)ds

= G'(s) (2.91)

Where G'(s) will imply that the derivative of G(s) is to be taken with respect tos. If G(s) = cs where c is a known constant, then G'(s)=c. If G(s) = sk, thenG'(s)=ksk~'. The only function of calculus used here is that of the chain rule,which the reader will recall, can be used to compute G'(s) when G(s) = g(f(s)).In this case G'(s) = G'(f(s))f (s). With these simple rules, we can further expandthe generating functions of interest for us to evaluate. Begin with the simplestgenerating function.

G(s) = = l + s+s2+s31-s

Taking a derivative on each side of the equality reveals

1

(2.92)

G'(s) = -^ -(1-s)(2.93)


and we see at once that - > {k +1}. Note this could be verified by using(1-s)

the product principle of generating functions on G(s) when written in the form'. We can multiply equation (2.93) by s to see

1-sAl-s

(1-s)2 (2.94)

sor -- > {k}. Of course, additional uses of the translation principle at this

(1-s)s2point leads to - > {k -1}, - > {k - 2}, - > {k - 3} and so on.

(1-s) (1-s) (1-s)Similarly, - > {k + 4} through an application of the translation process

s3(l-s)in the opposite direction.

We can continue with this development, taking another derivative from equation(2.93) to find

(1-s)2)-(k + l ) s k +- - - (2.95)

2which demonstrates that ->{(k + 2)-(k + l)}. Following the

(1-s)development above, one can see through the translation principle that

_??!_> (k-(k-l)} (2.96)

46 Chapter 2

and also that - > J2~' (k + 6) (k + 5)}. This process can proceeds4(l-s)

indefinitely. We will examine one more example.

3'2 2

= 3 - 2 - 1 + 4-3-2s + 5-4-3s2(l-s)4

k + l ) s k + - - - (2.97)

3 2and - > {(k + 3) (k + 2) (k +1)}. In fact, it is easy to see a pattern here.

(l-s)In general

r!(1-sr1 k! (2.98)

(k + r)!Taking advantage of the relationship that - = , we can writek!r! v r /

Finally, we can change the base series on which we apply the derivative. Forexample, it has been demonstrated that if G(s) = - , then G(s) > {k} .

(l-s)Following the pattern of taking a derivative on each side,G'(s) = j > |(k + 1)2 1 . We only need invoke the translation principle to

(l-s) ( >see that

G'(s) = > k2 (2.100)


This process involves taking a derivative followed by the use of the scaling tool.This may be continued one additional time. Continuing in this vein, a

straightforward demonstration reveals that(1-s)4

> j(k + l ) 3 > and a

s(s2+4s + l) , .translation reveals that --- > Ik \

0-s)4 { ]

The use of these principles applies to the notion of simplification andrecognition. The more experience one can get, the better the recognition skills,and the less simplification will be required. To invert

G(s) = 14(l-3s)(s-2)S3(l-as)3(4-bs)2

(2.101)

We recognize that the denominator will require the most work. The numeratormerely suggests that the final inversion will involve repetitive applications ofthe sliding tool and the principle of addition.Since the denominator of equation (2.101) contains the product of twopolynomials, the multiplication principle can be applied. Rewrite G(s) as

G(s) = -42s-1 + 98s-2- 28s~

(l-as)3(4-bs)2

= I -42s-1 + 98s-2 -28s-3(1-as)3 (4-bs)2

(2.102)

Evaluating the quotients first,

1(1-as)3

(2.103)

48 Chapter 2

by taking two successive derivatives of the sequence for which G(s) = -(1-as)

Proceeding in an analogous fashion for (4-bsy , write

716

'-5-(4-bs)2Applying the convolution principle reveals

|16

1(l-as)3

1(4-bs)2 j=o

-I-1614

and

G(s)>

k+l

j=0

k+2

16U

k+l-j

1 f b k+2-j16UJ

i fbY+3- J1614

(k-j+3)

(k-j+4)

(2.104)

(2.105)

(2.106)

A fine example of the derivative approach to the inversion of generatingfunctions is to find the generating function G(s) such that

G(s)>{kV} (2.107)


Begin with > (a k } and take a derivative of both sides with respect to s.1-as l '

This reveals - = V kaksk~' and therefore = V kaksk.(1-as)2 & (1-as)2 &

Taking a second derivative with respect to s and simplifying gives

a + a s =yk2aksk-i (2.108)

(1-as)3 h

2 2

Use of the sliding tool reveals - > {k2ak| .(1-as)

Another example of the principles developed in this section is the evaluation ofthe generating function

(2.109)|_cs+dfor integer n (> 0). The inversion of this generating function will require thetools of derivatives, the binomial theorem, as well as the sliding and scalingtools. Begin by writing G(s) as the product of two generating functions

=G,(s)G2(s) (2.110)

Recognize Gj(s) through a direct invocation of the binomial theorem. Write

akb-ksk (2.111)V'V k=oV k

The next task is to invert G2(s).

50 Chapter 2

i T _cs + d j

/d

.H'/dX

n

-M/Tl/dji

l-(~/d}\

The last term in equation (2.112) can be consideredapproach by taking consecutive derivatives.

11-as

a

G(s) =

G'(s) =

1

O^f"

Taking a second derivative reveals

2a

1

(1-as)2

(1-as)3

One additional derivative demonstrates

(k+2)(k

3a (k+3)(kp>-

(1-as)4

(I--)4 3!

1-as

(k+3)(k+2)(k

(2.112)

which we will

(2.113)

(2.114)

(2.115)

In general


1-as(k + n-l)(k + n-2)-(k

(n-1)!(k + n-1

1-asJ " \[ n-1

(2.116)

Note that the source of the combinatoric term is the result of the sequence ofsuccessive derivatives applied to the sequence {sk} in equations (2.113), (2.114),and (2.115), each of which retrieves an exponent of s, making this exponent acoefficient of s. Now write

Then,

cs + d j/d (2.117)

/d

d n-l

(n-1)!(2.118)

Note that

(2.119)

As demonstrated in problem 47 of this chapter when the generating functions00 J

G!(S) and G2(s) are defined as G,(s) = J]ajsJ and G2(s) = ^ bjSJ thenj=0 j=0

G,(s)G2(s)>min(J.k)

(2.120)

52 Chapter 2

Applying this result to the original problem

as + bG(s) =cs + d

n-la jb"

(2.121)

If G(s) = as + bc-ds

then write

G(s) =, ~in

as + b , , = (as + b)c-dsj V '

, -.

= (as + b)V '

IT 1

and, since

k c

(2.122)

(2.123)

invert using a convolution and the scaling tool

a + bsT

c-dsj n-l ajb (2.124)


r uinFinally, consider the inversion of ~^- for i > 0, n2 > 0, and n, ^ n2. By[c + dsfapplying the binomial theorem to the numerator, the derivative process to thedenominator, and a convolution, it can be shown that

as + bf[c + dsf

, n 2 - l (2.125)

2.8 Equivalence of Generating Function Arguments

At this point, it can be seen that for a given generating function G(s), we canfind several different approaches to the inversion of G(s). The possibility ofmultiple approaches provides a useful demonstration that each of twoapproaches will lead to the same infinite series, and a check of our work.

Consider the generating function G(s) =(a-bs)2 There are two useful

approaches to take for this inversion. The first might be inversion by theconvolution principle, recognizing that the denominator of G(s) is the product oftwo polynomials. However, a second approach would involve taking aderivative. Since G(s) produces only one value of yk for each k, the value of ykobtained using the derivative approach is equivalent to that of the "derivativeapproach." First consider the solution using the convolution approach

G(s) =(a-bs)

i V i

(2.126)

Evaluating the kth term of this sequence reveals

54 Chapter 2

a Ma

>

V(2.127)

and find that

G(s) = (a-bs)21 ( b

a V air 0-0 (2.128)

Using the derivative approach

a-bs J~ i_b/lYba a

(2.129)

and taking derivatives on both sides of equation (2.129) with respect to s,

(a-bs)b >UW( k + i )> (2.130)

which is equivalent to our earlier result, obtained through the application of theproduct principle.

2.9 Probability Generating Functions

The skillful use of generating functions is invaluable in solving families ofdifference equations. However, generating functions have developed in certainspecific circumstances. Consider the circumstance where the sequence {yk} areprobabilities. Then, using our definition of G(s) we see that

CO 00

G(s) - ]Tpksk where ^pk = 1 . In probability it is often useful to describe thek=0 k=0

relative frequency of an event in terms of an outcome of an experiment. Among


the most useful of these experiments are those for which outcomes are discrete(the integers or some subset of the integers). These discrete models have manyapplications, and it is often helpful to recognize the generating functionassociated with the models. The generating function associated with such amodel is described as the probability generating function. We will continue torefer to these probability generating functions using the nomenclature G(s). Thissection will describe the use of probability generating functions and derive theprobability generating function for the commonly used discrete distributions.Note that in this context of probability, the generating function can beconsidered an expectation, i.e., G(s) = E[sx]

2.9.1 Bernoulli distribution

Consider one of the simplest of experimental probability models, the Bernoullitrial. Here, denote X as the outcome of an experiment which can have only twooutcomes. Let the first outcome be X = 1 which occurs with probability p, andthe second outcome X = 0, which occurs with probability q = 1 - p. Although asequence of Bernoulli trials are a sequence of these experiments, each outcomeis independent of the outcome of any other experiment in the sequence, and theconcern is with the occurrence of only one trial. Observe at once that theprobability generating function is

Note that in this computation we can collapse the range over which the sum isexpressed from the initial range of 0 to oo to the range of 0 to 1 since Pk=0 fork=2,3,4, ...

2.9.2 Binomial distribution

The binomial model represents an experiment that is the sum of independentBernoulli trials. Here, for k = 0, 1,2, . . . , n

(2.132)

56 Chapter 2

The generating function for this model is

(2 ]33)/ x k / , xn-k(PS) O-P)

now using the binomial theorem for the last step.

^/V (2.134)

2.9.3 The geometric distribution

The geometric distribution continues to use the Bernoulli trial paradigm, butaddresses a different event. The question that arises in the geometricdistribution is "what is the probability that the first success occurs after kfailures, k = 0, 1, 2, ...,?" If X is the trial on which the first success occurs,then

P[X = k] = q k p (2.135)

We need to convince ourselves that this really does sum to one over the entirerange of events (i.e., all non-negative integers). Work in this chapter hasprepared us for this simple calculation.

= 1 (2.136)

and computing the generating function as follows

G(s) = fVqkp = P1 - qs J 1 - qs

(2.137)


2.9.4 Negative binomial distribution

As a generalization of the geometric distribution it is of interest to compute theprobability that the rth success occurs on the kth trial. It is useful to think of thisevent as the event of there being r - 1 successes in k - 1 trials, followed by oneadditional trial for which there is a success. Then, if X is the probability that thekth success occurs on the rth trial, then

P[X = k]= I _ |pr"qK~r |p =

The probability generating function can be derived as follows

Continuing

1(1-qsy

The last equality in equation (2.140) comes from equation (2.99). Thus

G(S) =PS

1-qs(2.141)

This development is described in Woodroofe [2].

2.9.5 Poisson distribution

The Poisson distribution is used to describe the occurrence of independentevents that are rare. A classic example is the arrival of particles to a Geiger

58 Chapter 2

counter. Suppose the average arrival rate in the time interval (t, t + At) is XAt. IfX is the number of particles which arrive in time interval (t, t + At) then

(2.142)

for all non-negative integers k. One key for this being a proper probabilitydistribution is the relationship

V = ex (2.143)t-1 Hk=o K:

With this result it is easy to show that the sum of Poisson probabilities is one.

= k] = e-

XAt = e -^T =

e-* -eXA< = 1 (2.144)

and the generating function follows analogously

G(s) = fV P[X - k] = JT skk!

t' If |k 0

_

eXAt(s-l)

These are the probability generating functions for the most commonly useddiscrete probability distributions. They will be important in the use ofdifference equations to identify the probability distribution of the number ofpatients with a disease in a population. Although the development of theseequations in Chapters 11 and 12 of this book will appear complex, therecognition of the resultant generating functions as the probability generatingfunctions of the probability distributions developed in this section will add toour understanding of the spread of a the disease process.


Problems

Using the scaling tool and the sliding tool, invert the following nine generatingfunctions. The solution should provide all of the information necessary toidentify each member of the infinite sequence {yk}.

1. G(s) =

2. G(s) =

3. G(s) =

4. G(s) =

5. G(s) =

6. G(s) =

7. G(s) =

3s-71

2 + 6s1

5s+ 91

3s-71

8. G(s) =

s-93

3 + 2s12

2s-7

7

9. G(s) =3s + 5

1l + 3s

Invert the following generating functions:

60 Chapter 2

10. G(s) = (s + l)(s-l)

ILG(S)=


22. G(s)=(s-2)2(s

+

4s223.G(s) =

24.G(s) =(2s-2)3

6

25.G(s) = - 13

26.G(s) = 1

27.G(s)= 3

28.G(s) =

29.0(8) = ^

30.G(s) = I

31.G(s)= 7

32.G(s) =

33.G(s) =

34.G(s) =

35.G(s) =

(9+6s)32

(3 +7s)23

(12s+1)45

(3s+ 4)2

62 Chapter 2

36.G(s) =(9s

37. G(s) =(5

+ 2s)2

Invert the following generating functions using the convolution principle and thesliding tool:

38. G(s) = -i

39.G(s) = 7

2)(s +

s2

40. G(s) =

41. G(s) =

42. G(s) =

2s5

(2s + 3)(s-8)6s3

(s-3)(s + 3)3s

45. Verify by using both the derivative approach and the product principle thatif

(a + bs)3. then f,(k) = f2(k).

46. In section 2.8 it was pointed out that if x is a random variable which followsa negative binomial distribution then its probability generating functionG(s) is


G(s) =1 -qs

Applying successive derivatives to G(s) and the scaling tool, compute theprobability P[X=k] for r = 4, 5, and 6.

47. Consider this lemma to the convolution principle. Consider two generating00 J

functions G,(s) = ^ajSj and G2(s) = ^bjSJ . Then provej=o j=o

f m i n ( k . J )G,(s)G2(s)> a,,.

[ j=o

48. It was shown is section 2.95 that

k!

Show that

k!

b

References1. Goldberg S. Introduction to Difference Equations. New York. John Wiley

and Sons. 1958.

3Generating Functions II: CoefficientCollection

3.1 Introduction

The procedures developed in Chapter 2 have provided a collection of importanttools for inverting generating functions. Already, there are many generatingfunctions that we can invert using a combination of the scaling and sliding tools,in combination with the addition and convolution principles. These toolsinclude the ability to identify and manipulate the generating functions of manycommonly used discrete probability distributions. However, there is animportant class of generating functions that we have not attempted to invert, butthat occur very frequently in the solution of difference equations. In fact, thisclass is among the most commonly occurring difference equations that oneencounters. In addition, they have historically provided important complicationsin the solution of difference equations. This class of generating functions arethose that contain general polynomials in the denominator, and will be thesubject of this entire chapter. At the conclusion of the chapter, the reader shouldbe able to invert a generating function whose denominator is a polynomial in s.

65

66 Chapter 3

3.2 Polynomial GeneratorsDefining a polynomial generator as a generating function that has a polynomialin s in its denominator, written as

G(s)= - (3.1)a0+a,s + a2s'+a3s +... + aps"

The denominator is a polynomial of order p (p an integer > 0). The followingrepresent examples of these polynomial generators

G(S) = !T17; (3'2)

4s-1 (3.3)-3s2+4s-l

Each of equations (3.2), (3.3), and (3.4) contains a polynomial denominator andeach represents an infinite sequence {yk} that must be identified.

Chapter 2 provides some background in the inversion of polynomialgenerators. We know from Chapter 2 how to invert the generating function inequation (3.2). For equation (3.3), since -3s2 +4s-l = (3s-l)(l-s), willenable the use of the principle of generating function convolutions to invertG(s). However, no clue is available as to how to invert the generating functionin equation (3.4) unless the denominator can be factored. If the denominator canbe factored into a number of simpler polynomials of the form (a + bs)m, theconvolution principle of generating function inversion discussed in Chapter 2could be invoked. If this cannot be done, there will be a need to create someadditional tools to invert these more complicated polynomial denominators.

We will begin the study of polynomial generators, beginning with thesimplest of polynomials, building up to more complicated polynomials.

3.3 Second-Order Polynomials

3.3.1 The role of factorizationA clear example of the generating function decomposition is in the factoring of acomplicated polynomial into the product of at least two terms. Consider as a firstexample:

Generating Functions II: Coefficient Collection 67

G(s)= (3.5)6s -5s + l

From what has been seen and accomplished thus far, little is known about howto proceed with the inversion of G(s) as written. However, the ability to factorthe denominator opens up some additional options that would allow completionof the inversion procedure. Rewrite equation (3.5) as

G(s) = 2

l = ' = (}{} 0.6)6s2-5s + l (l

Once the factoring is complete, there are two major procedures that can befollowed to complete the inversion. The initial factorization is the importantfirst step.

3.3.2 The use of partial fractionsAn important traditional tool in the inversion of polynomials is the use of partialfractions [1]. This procedure allows us to write a ratio with a polynomial in thedenominator as sum of several ratios of polynomials, each resultant polynomialbeing of lower order than the original. From equation (3.7)

A 0.7)

6s -5s + l (l-3s)(l-2s) l-3s l-2s

Note that the inversion may be completed if the constants A and B could bedetermined. Begin by multiplying both sides by (1 - 3s)(l - 2s) to find

l = A(l-2s) + B(l-3s) (3.8)

Since this must be true for all s, choose convenient values for s to solve for thequantities A and B. For example, letting s =1/2

(39)1=

0rB = -2

Similarly, allowing s = 1/3 in equation (3.8) removes the term involving B, andgives A = 3. Thus rewriting equation (3.7) as

68 Chapter 3

G(s) = - = - - - (3.10)2s) l-3s l-2s

and proceeding with the inversion

1G(s) = 6s2-5s+l (l-3s)(l-2s) l-3s l-2s (3.11)|3k+1-2k+1}

3.3.3 The use of convolutionsThe fact that G(s) from equation (3.5) can be factored allows applying theconvolution argument directly.

G(s) = 26s2-5s + l

Factorization makes the process much easier.

3.3.4 Comparison of partial fractions vs. convolutionsOf course, either of these two processes is applicable, provided the answers areequivalent. As a check, let k =3. From equation (3.11) y3 = 34 - 24 = 65.From equation (3.12)

y3 =323+3122+3221+332= 8 + 12 + 18 + 27 = 65 (3.13)Equations (3.11) and equation (3.12) are equivalent.* Of the two, however, theform of the answer provided by partial fractions is the easiest to understand.

~ - = .--*? = V aV-J,a * ba-b a-b


3.3.5 Coefficient collectionYet another procedure is the method of coefficient collection. This procedure isa very general approach using the binomial and/or multinomial theorems.Return to the example provided in equation (3.12). Rather than factor thedenominator, rewrite it as

6s2-5s + l l-(5-6s)s l-wss(3-14)

where ws = 5 - 6s. If we invert , treating ws as a constant,l-wss

1> (wk } permitting the observation that the kth term of the sequence isl-wss l ;

wsksk = (5 - 6s)ksk. However, although this finding is correct, it is not helpful,

since powers of s are produced from the (5 - 6s) term in the denominator. Weuse the notation >s to denote that the inversion is incomplete since ws is afunction of s. The difficulty with this incomplete inversion becomes apparent atonce. As an example, examine the coefficient for the k = 3 term. Note that

(5 - 6s)3 s3 = (l 25 - 450s + 540s2 + 2 1 6s3 ) s3V '

V ' (3.15)

= 1 25s3 - 450s4 + 540s5 + 2 1 6s6

So an examination of the k = 3 term does not yield only coefficients of s3, butcoefficients of other powers of s as well. In addition, just as the k = 3 termyields powers other than those of s3, it will also be true that powers of s3 willarise from the examination of other terms in this expression other than k = 3.For example, expansion of the k = 2 term reveals 25s2 - 60s3 + 36s4. We aregoing to have to find a way to collect all of the coefficients from every place inthis sequence where s3 occurs. This will be a two-step process, and in general isvery straightforward.

To begin this process, first find a more succinct way to identify all of thepowers of s from the term (5 - 6s)ksk. For this, return to the binomial theorem,which states that, for any constants a and b and any positive integer k

(3.16)

70 Chapter 3

where - -. '- r- . Applying the binomial theorem to (5 - 6s)kUJ J!(k-J)!

(5-6s)k = 6 s y 5 k - = ( - 6 ) J 5 k - V (3.17)j= V J ) j=o V J

and

This transformation gives exactly the coefficients of sk that are produced by thekth term of the sequence (5 - 6s)ksk. For k = 0, the only exponent of s is for k = 0and j = 0, and therefore the only coefficient of s produced is one. When k = 1, jcan be either 0 or 1. For j = 0, sk+j = s1 and for j = 1, sk+j = s2. Continuing withthese observations allows the generation of a table (Table 3.1) from which apattern of the appearances of exponents of s may be noted.

Thus, the coefficients of s3 will be

-eys '+j )(-6)53=-60 + 375 = 315 (3.19)

The process followed here for s3 is the process that should be followed ingeneral to collect the terms sk . The pattern by which powers of sk are generatedis therefore identified. As defined in Chapter 2, consider the indicator function


Table 3.1 Relationship Between Values of k in the Seriesand the Power of s Produced

k

0112223333

j0010120123

(k+j) Power of s0122343456

In general, all of the coefficients of sk are represented by the expression

(3.21)

The indicator variable accumulates all of the appropriate (m, j) combinations.The above equation indicates that terms of the form (-6)J5m~j for which m+ j = k and j < m are the only ones that can be collected. For example, toidentify the coefficients of s4 from equation (3.21), the (m, j) combinations ofinterest are (4, 0), (3, 1), and (2, 2). Collecting coefficients

= (1)(1)(625) - (3)(6)(25) + (1)(36)(125) (3.22)= 4675

Thus, the inversion may be written as

72 Chapter 3

(3-

23)

and the indicator function indicates precisely which of the summands in thedouble summation of equation (3.23) need to be evaluated to bring together allof the terms involving sk.

In general, using this process of collecting coefficients, any second-order

polynomial generator G(s) of the form - may be inverted. First

rewrite G(s) as follows.

G(s) =1 1

1 a0 a0a0+a is + a2s l(ao+aiS+a2S2) 1 + aLs + aV

ao ao ao

1 1ao ao

(3.24)

1 "7 /s l- +-2-s (-

Rewriting G(s) as in equation (3.24) shows that the inversion of G(s) requires( V

that collection of terms for the sequence whose kth term is H-s (-l)ksk,lao ao )

i.e.

G(s) >sa.

L+^S (_1)k (3.25)ao ao

The job now is to collect all of the coefficients of sk in an orderly manner.Begin by proceeding as before, invoking the binomial theorem for L H -s

lao aoas follows.

k

(3.26)


Holding aside for a moment the constants and (-l)k from equation (3.25),ao

note that G(s) generates the sequence

" ' sV = _Lyfklav-+j (327)ak^uJa2a ' s

Focus on the last expression in equation (3.27). An examination of this termreveals that powers of s are generated not just from an examination of k but froman evaluation of k+j. Thus, there are different combinations of (k, j) pairs thatwill generate the same power of s. For example, (1,1) and (2, 0) each generate

1 (kV the power of 2. Thus, for every (k,j) combination, the term a^a, J mustao UJ

be included as a coefficient of that power of s. These observation lead to theconclusion that the task of gathering powers of s (termed here collectingcoefficients of the powers of s), is one of organization. The last expression inequation (3.27) displays the method for gathering the coefficients of sk. Definem and j such that 0 < m < k, and 0 < j < m, and note that the term . aJ2a,

m~

j will contribute to the coefficient of sk whenever m + j = k. The

sum of these coefficient terms can be expressed as

yy_L/mlaja - -+^s (-1)

74 Chapter 3

v v iiJn"-JTw am+i 2. : a2ai Wt

=0 s {(6-s)kl (3.31)

S fis 4- 1 1 (fl sVl ' '

_1s2-~6s + l l-(6-

This notation means that each term of the sequence00 00

G(s) = ^ yksk -^(6-s) s k . Now use the binomial theorem to write

DJ6k-JsJ (3.32)j = o v J yThus, G(s) may be written as

k=0 j=0 V J


What remains is to collect the coefficients of each unique power of s. Begin bycreating the integer valued variables m and j such that 0 < m < k, 0 < j < m,observing that the . (-l)J6m~j will be a coefficient for sk whenever m + j = k.Thus the coefficient of sk from the rightmost expression in equation (3.33) is

n \ (-l)J6m~JIm+J=k The inversion of G(s) is now complete.

=0 j=0 V. J

When ao = 0, the process equation (3.30) breaks down because ]/ is not/ao

defined. However, when ao = 0, the process of collecting coefficients for asecond-order polynomial need not be invoked, and the inversion of G(s)simplifies. In this case, we observe

G(s)=-r \ t \a2s +a,s s(a2s + a, j (a2s+aj

The denominator becomes

I// a,

l/(a2s+a i) i-a(3-36)

Use the sliding rule to see that, when EI ^ 0,

G(s) =a2s +a,s

-a. (3.37)

76 Chapter 3

3.4 Third-Order Polynomials

In this section we consider polynomial generators of the form

G(s) = -\ (3.38)a3s +a2s +a1s + a0

It can be seen that these equations will be handled in the same fashion as thesecond-order equations. There are, in general, three methods of inverting thisgenerating function. Two of them (method of partial fractions and the principleof convolutions) lead to easily articulated solutions as long as the function canbe factored. Since most cubic equations cannot be factored, however, the utilityof these approaches is, in general, limited. After providing examples of each ofthe partial fraction and convolution approach, the balance of the discussion inthis section will be on the method of coefficient collection.

3.4.1 Partial fractionsConsider the third-order polynomial generating function

-s3 +s2 +!4s-24 ~ (s-3)(2-s)(s + 4)

Since G(s) can be factored, apply the method of partial fractions and proceed asin the previous section, writing

1 A B C

(3_4Q)(s-3)(2-s)(s + 4) s-3 2-s s + 4

Multiply throughout by the three terms in the denominator,

1 = A(2-s)(s + 4) + B(s-3)(s + 4) + C(s-3)(2-s) (3.41)

The task is somewhat more complicated since each of the constants multipliesnot one but two terms. Letting s = 2 removes the a and c terms, and from theresulting equation find B = -1/6. Similarly, s = 3 leads to the conclusion that A= -1/7, and finally s =-4 leads to C = -1/42. Equation (3.41) can now be writtenas


42(s-3)(2-s)(s + 4) (s-3) (2-s) (s + 4)

Invoking the addition principle reveal

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