MARKOV CHAINS WITH TRANSITION DELTA-MATRIX : ER , GODICITY CONDITIONS , INVARIANT PROBABILITY MEASURES AND APPLICATIONS

The authors find some structural propert}es of both types of Markov chains and develop a s}mple test for their }rreducibil}ty and apeHod}city. Necessary and sufficient conditions for the ergodicity of both chains are found in the article n two equivalent vers}ons. According to one of them, these cond}tions are expressed in terms of certain restrictions }mposed on the generating functions A{(z) of the elements of the {th row of the transition matt}x, {0, 1,2,...; n the other vers}on they are connected with the characterization of the roots of a certain associated function in the unit dsc of the complex plane. The nvaHant probab}l}ty measures of Markov chains of both knds are found n terms of generating functions. It }s shown that the general method n some important special cases can be smp]fied and ye]ds convenient and, sometimes, expl}c}t results.


INTRODUCTION
In this article a general analytical method is proposed for the analysis of discrete Mar- kov processes with, so-called, Am, n or A' transition matrices 2 ("delta-matrices") which are frequently encountered in applications.(These Markov processes were first introduced in the earlier work of the authors [3]).A description of a delta-matrix is given in the following defini- tions.

Definition 1.2:
If a matrix A -, 3,(ai:3 is either a _Am, n-or a ZX'm,n -matrix then it is called a delta-matrix. Thus, the transition probability matrix of Markov chains considered in the article has or the form of its transpose A'.Many discrete stochastic processes encountered in applications have transition matrices which are special cases of A (or A'): imbedded Markov chains describing the evolution of the queue in queueing systems MX/GY/1, GX/MY/1, GX/MY/n with bulk arrivals, and batch service, with state dependent parameters, with a threshold, with warm-up, with switching, with hysteresis service, with queue buildup, with preliminary service, and with vacations of the server.As examples, there are also Markov chains describing the state of a storage in the theory of inventory control, or the state of a dam (see, for instance, [2,4,9]).
In this paper the authors consider some properties of stochastic Am, n and A' -matrices and their applications to the analysis of the corresponding discrete Markov processes.A simple sufficient condition for the ergodicity of a finite Markov chain with 2A, denotes the transpose of A m n" m,n transition Z,m, n (Z'm,n)-matrix is found.For processes with an infinite number of states the corresponding necessary and sufficient condition is determined in two equivalent versions (Sections 3,4).According to one of them, this condition is expressed in terms of certain restrictions imposed on the generating functions Ai(z = aifi3 = 0,1,2,...; in the other j=o version it is connected with the existence and characterization of the roots of the function z m---K(z), K(z)= ki zi in the closed unit disk of the complex plane. 3In Section 5, the i=0 authors consider the problem of finding the invariant probability measure of infinite Markov chains with transition delta-matrix.It is shown that in the case of a Am, n-matrix this problem can be reduced to the problem of finding a unique solution of a linear (n + 1) x (n + 1) system of equations whose coefficients may contain the roots of the function z m-K(z).In some special cases this method can be considerably simplified and yields convenient and, sometimes, explicit results (see .In Section 6, the authors analyze the problem of finding the invariant probability measure of a Markov chain with transition A' n-matrix. It is shown that m the generating function of the stationary distribution of this chain can be implicitly found in terms of the first (n + 1) invariant probabilities, which, in turn, form a unique solution of the provided system of (n + 1) linear equations. Usng this approach the authors succeed in finding a relatively simple, compact and explicit expression for the generating function of the stationary probabilities in a special case corresponding to a GX/MY/1 bulk queueing system with continuously operating server.The method used in this section is based on the employment of Liouville's theorem for analytic functions of a complex variable.In the form of a Pdemann boundary value problem, this method was previously introduced and developed by one of the authors in [6,7].

DELTA-MATRICES AND THEIR PROPERTIES
A general definition of a delta-matrix is given in Section 1.In this section, keeping in mind specific features of processes encountered in applications, we will introduce some additional notions and then mention some properties of delta-matrices.
First we will define a positive N-homogeneous Am, n(A'm,n)-matrix.Discrete Markov processes with transition matrices of these two types are typical for queueing systems, inventory, and dam models.Let A = (aij) be a stochastic Am, n-matrix (resp., A'm, n'matrix)" If there exists a number N (N >_ n) such that aij = kj_i+rn for > N and j >_ i-m (resp., 3The results obtained in the previous work [3] of the authors were revised and included in Sections 3 and 4. aij-ki-j + m for j > N and >_ j-m) then the matrix A is called a N.homogeneous Am, n- matrix (resp.an N.homogeneous zX' n-matriz) If N = n, the matrix A is called homogeneous.
Another definition of an N-homogeneous Am, n-matrix (that is more convenient in the case of infinite matrices) can be given in the following way.Let Ai(z = _aijz3 be the generat- j=o ing function of elements aij of the ith row of a stochastic matrix A = (aij i, j = 0,1, 2,...).The set of functions Ai(z), i-" 0, 1,2,..., completely determines the matrix A.
r"O Definition 2.2 is equivalent to Definition 2.1.

Definition 2.3:
A stochastic Am, n-matrix (resp., A'm, n-matrix) is called positive if aij > 0 for < n and any j, and for > n, j >_ i-rn (resp., j < n and any i, and for j > n, i>_j-m).
We will also need a more general version of a positive delta-matrix which is given by the following definition.
Now let us point out some properties of Am, n-matrices (analogous properties hold for A' matrices) Property 1: Let A and B be two infinite essential (resp., positive) Am, n-and A .matrices, respectively.Then AB is an infinite essential (resp., positive This fact can be shown by direct verification.
The following statement is an immediate consequence of Property 1.

Property 2:
If A is an essential (resp., positive) infinite Am, n-matrix then for any integer k, k > 0 the matrix A k is an essential (resp., positive) infinite Akm, n+(k_l)m- matrix.
Applying this result to a finite positive Am, n-matrix we obtain: Property 3: Every finite positive Am, n-matrix is primitive (i.e. a matrix M for which there is a k > 1 such that ]lIk contains strictly positive entries).
Based on Property 3, a simple test for the ergodicity of a Markov chain with a finite transition delta-matrix will be established.
In the next section we will obtain necessary and sufficient conditions for the ergodicity of Markov chains {r} and {r} with finite and infinite transition delta-matrices.
3. NECESSARY AND SUFFICIENT CONDITIONS FOR.TIIE ER.GODICITY OF MArKOV CHAINS WITH TRANSITION DELTA-MATR/X In this section we first mention a simple sufficient condition for the ergodicity of a finite Markov chain with transition probability delta-matrix.
Theorem 3.1: Every Markov chain whose ransiion matrix can be represented as a finite positive dela-marix is ergodic. Proof: The statement of the theorem follows directly from Property 3 of the previous section.El Now we consider discrete Markov processes with an infinite number of states.First we prove that under some natural conditions, any Markov chain with transition delta-matrix is irreducible and aperiodic.A Markov chain {k} is called queue-type if (a) {k} takes on only nonnegative integers, (b) 0 < P{k+l > k = i} < 1, = 0,1,2,..., (c) 0 < P{k + < i} < 1, = 1,2,3,..., k = 0,1,2,..., (3.1) Theorem 3.2: A "queue-type" Markov chain {k} is irreducible and aperiodic. Proof: Due to (3.1), {k} can reach state {0} starting from any state.Therefore, {0} must belong to every class of essential states.Since these classes are classes of equivalence, then having a common element means that they coincide; therefore, {k} has only one class of essential states.
Remark 3.1: The fact that a Markov chain has only one class of essential states does not mean that all the states belong to this class (in which case the steady state probabili- ties are positive).Consider, for example, a standard MX/GY/1 bulk queuing system in which customers arrive in batches of two and the server's capacity is also two.The imbedded Markov chain {,}, where (n is the number of customers in the system at moments of successive service completions, is aperiodic and irreducible but only even states are essential.
Remark 3.2: Condition (b) of Definition 3.1 guarantees that infinitely many states can be reached starting from any state (say, {0}), so that the class of essential states is infinite.
Next we will establish the main result of this section.For the sake of simplicity, we will assume here and later that all considered Am, n-matrices are homogeneous (unless stated otherwise).However, all results obtained can be easily extended to the general case of N-homo- geneous Am, n-matrices, N > n.

3.2)
Setting zj--j, j-0, 1,2,... and using the definition of a homoge- neous zX m, n -matrix we obtain: aijxj-x = j=0 By the corollary of Moustafa [8] to Foster's lemma we conclude that (3.2) is sufficient for the ergodicity of the chain {(r}" Necessity: Suppose, on the contrary, that K'(1)>_ m and the chain {(r} is ergodic.Then there are stationary probabilities Pi, not all zero, such that PJ = E Piaij j = 0,1,2,... (3.3) i=0 or P-PA, where P-(Po, P, P,--')" (3.4)   Multiplying both sides of (3.4) from the right by vector 0, 0,.-.,0, 1,z, z2,...)T, we obtain: Taking the limit of both sides of (3.5) as z--,1 we obtain: While the left-hand side of (3.6) is supposed to be nonpositive, the right-hand side is a sum of nonnegative terms, and so (3.6) means that both sides must be zero.Taking into account that {r} is queue-type, we conclude that there must be an essential state 0 < n + 1 and J0 >n + 1 such that aioJo > 0 (see Property (b) of Definition 3.1).On the other hand, the right-hand side of (3.6) being zero means that J0 can only be n + 1, for otherwise the first sum in the right-hand side will be positive.It follows that n + 1 is an essential state and Pn + 1 > 0, and so the right- hand side being zero requires that n m--1 Since E pi(m-K'(1)) =0 and Pn+l > 0, K'(1) must be equal to m.This is i=n+l possible only if k m = 1, which contradicts to inequality (c) of Definition 3.1.This contradiction proves part B of the theorem.
theorem in case of a finite limit we obtain If K'(1)> 1 then by using a Tauberian lira D (i)K(z)-zm lira K(z)z m = 1 +m <0.-0 1-g'( In this case there exist M < 0 and r such that Z aijxj-x < > 7".
The conditions of ergodicity established in Section 3 are closely connected with the number and location of the roots of the function z m-K(z).More precisely, the following theorem is valid.

No
If K'(1)< m, then the function zm-K(z) has exactly rn roots (counting multiplicities) in the closed unit disk +.The roots lying on the boundary F are simple and, for some integer r, are all rlh roots of 1. /f g'(1)> m, then lhe function z m-K(z) has exactly rn roots (counting multiplicities) in the open unit disk F+; on the boundary F there can be r additional simple roots which are all the rth roots of 1, where r is an integer, and 1 <_ r <_ m.
We will need the following auxiliary result.
The function z m-K(z) has roots on F if and only if there exists a common divisor r of m and all such that k 5 O.If this condition is satisfied, then all roots on F with this property coincide with the roots of the equation z r-1-O, where r is the mazimal number having this property. Proof: Suppose that z 0 is a root of the equation 1.Then g(zo) znl = 1.On the other hand, K(zo) i=0 _. kilzol i=0 =1 z m-K(z)=O such that and equality is attained if and only if z 0 = zol = l for any i, such that k :/: 0 and z n = 1.
The realization of both of these conditions simultaneously is possible only if the conditions of t' 1.
the lemma are satisfied.In this case, obviously, z 0 Suppose now that the conditions of the lemma are satisfied.the equation z r-1 0, we obtain K(zo)-1 = z which implies that the root z 0 is a root of the equation K(z) z m = O.
Then for any root z o of Now we return to the proof of the theorem.

A.
First we suppose that the number r, appearing in the Lemma 4.1, is 1.Then the function z m-K(z) has only one root on the unit circle.This root is equal to 1 and it is simple, since K'(1) m.We will prove that in this case z m-g(z) has exactly m-1 roots in Consider an auxiliary function: Clearly, f(z) 5/= 0 for all z, z = 1 since the numerator of the expression for f(z) may be zero only if z-1, but f(1)-m-g'(1)>0.Let Indrf(z denote the difference between the number of the roots and the number of the poles of the function f(z) in I" +.By the argument principle Indrf(z = --rArArgf(z) Indr.[1 z-rag(z)]---Indr(1 z-1) where ArArgf(z is the increment of the argument of f(z) when the argument of z = e ' increases from 0 to 27r.
Thus, the total number of the roots (including z = 1) of the function z m-K(z) in the unit disk + is equal to m.Now consider the general case r > 1. Introduce the function m oo F(z)-z -e'-E kizL From the definition of r it follows that all exponents of z in (4.2) are integers.Applying the previous reasoning to (4.2) we see that F(z) has exactly --1 roots in F +, and one root z = 1 on the boundary.It is clear that the set of all rth roots of F(z) gives us all roots of z m-K(z).
On the other hand, any root of z m-K(z) raised to the rth power is, obviously, a root of F(z).Therefore, the set of all roots of z m-K(z) is described completely.The number of them is m; m-r are in the region F +, and r roots are on the boundary.
Suppose now that z 0 is a root of z m-K(z) such that z0l 1.Then, IK'(z0) = Eiki zi-1 = zo i=l It follows that K'(zo) raze-Part A of the theorem is proved.
which implies that all roots on the boundary are simple.
Therefore, the inequality Ig(z) < zml is correct for any z, such that zl = P-Using Rouche's theorem we obtain that z m-K(z) has exactly m roots in the region zl < 1.The number of roots on the boundary, as before, depends on the value of r.If r > 1 then by Lemma 4.1 there are r roots on the boundary which represent a set of all rth roots of 1.The simplicity of these roots can be proved as before.

FINDING THE ERGODIC DISTRIBUTION OF A MARKOV CHAIN
WITH TRANSITION Am,a-MATRIX Let {r}, r-0,1,2,... be a queue-type Markov chain with transition homogeneous Am, n -matrix A = (aij), i, j = 0,1, 2, Suppose that Ai(z), = 0, 1,2,...,N and K(z) satisfy necessary and sufficient conditions for the ergodicity of {r} established in Section 3. Then the invariant probability measure P = {P0,Pl, P2,.'-} of the matrix A exists and represents the only solution of the matrix equation P-PA.In the following theorems P is found in terms of the generating function P(z) piz'.
'-0 Theorem 5.1: Under conditions (3.1) and (3.2), the generating function P(z) = E Pi z' of the ergodic distribution of a queue-type Markov chain {r}, r-0,1,2,... with transition homogeneous Am, n-matrix Aij= {aij}, i,j=O,1,2,.., is determined by the following relations: E Pi[Ai(z)zm-K(z)zi] P(z) i= o zmg(z) The unknown probabilities Po, Pl,'",Pn on the right-hand side of (5.1) form a unique solution of the system of n + 1 linear equations: 1)] = m-g' (1) (5.3) i--0 where z r are the roots of z n+ 1-zn-m + l g(z) in the region +\{1} = {z, z[ <_ 1,z 7 1} R with their multiplicities r r such that r r -n.rmI Proof: Taking advantage of particular features of the homogeneous Am, n matrix A and the matrix equation P-PA after elementary transformations we obtain: In order to find n + 1 relations necessary for determining the unknown probabilities on the right-hand side of (5.4), we represent this relation in the following form" E Pi[Ai(z) zi] E pizi-n-l= i=0 I(,(z) (5.5) z n + 1_ z n "m + i i=n+l Since the function on the left-hand side of (5.5) is, analytic in the region I "+ and continuous on the boundary F of this region, so must be the function on the right-hand side of (5.5).On the other hand, due to Theorem 4.1 the function z n+ 1_ z n-ra + 1K(z) has exactly n + 1 roots in + (including the simple root 1).Using the analyticity of the function on the right-hand side of (5.5) and condition P(1)= 1 we obtain (5.2)and (5.3).
It can be proved that the system of equations (5.2)-( 5.3) has a unique solution.
It also follows from (5.3) that Pi = 1.Therefore, we have two different absolutely 0 summable solutions of the matrix equation X = XA with (X, 1)= 1.This contradicts to the Chung Kai-Lai theorem [5].
El Pmark 1: The results of Theorem 5.1 can be easily extended to the case of Nhomogeneous Am, n-matrices, N > n.It is obvious that n-m+l roots of the function z TM -z n-m+ 1K(z) are zeros.Utilizing these roots and the anMyticity of the right-hand side of (5.5) we obtain, as one would expect, the first n-m + 1 component-wise equations of the mat- rix equation P = PA.This part of the system (5.3)-(5.4) in some cases can be simplified.
One of these special cases occurs when the transition N-homogeneous Am, n-matrix of the chain {r} is essential.. Due to the structural properties of this matrix, every pj, j = n + 1, n + 2, ..., N can be uniquely expressed as a linear combination of Pi, = 0,1,2,...,n which enables us to reduce the number of equations in (5.2)-(5.3).This fact, in combination with the so-called "method of continuation" [1], makes it possible in some cases to obtain simple and even explicit results for the generating function P(z) and other characteristics of the ergodic distri- bution of {r} (see Example 4).
Another special case is a Markov chain with, a so-called transition Am-matrix, which was first introduced and studied by one of the authors in [1].A Am-matrix is a special case of Am, n-matrix when n-m and is also frequently encountered in applications (see Examples 1, 2, 3).Using Theorem 5.1 for n = m, we can find an expression for the generating function of the ergodic distribution of a Markov chain with transition Am-matrix: m E Pi(zmAi(z) zig(z)) where the unknown probabilities on the right-hand side of (5.7) can be found from (5.2)- (5.3)where n = m.However, in this case, since the number of the unknown probabilities is only m + 1, it is enough to express Pm in terms of Po, Pl,'",Pm-1 to equate the number of the roots of z m-K(z) and the number of the unknown probabilities on the right-hand side of (5.7).It follows from the matrix equation P = PA that ml l-a00 ci = ai--- (5.8) Pm oqPi, where c 0 am0 -, amo, .., t'-0 Therefore, eliminating Pm from (5.5) we conclude that all probabilities P0, Pl, P2,-", Pro-1 can be found from the condition of the absence of poles of the function in the closed uni disk 1 + of the complex plane.The uniqueness of the solution of the corresponding system of equations is proved in Theorem 15.1.

EXAMPLES AND SPECIAL CASES
First we introduce an auxiliary Markov chain {r}, r = 1,2,... and a polynomial R(z) which will play an important role in the analysis of some practical problems.
We denote this chain by {r}" The transition matrix of {r} is a homogeneous Am-matrix.m-1 m Assume that 0 < k < k < 1 and K'(1) < m.Due to Theorems a.3 and 3.4, the ,=0 i=0 chain {r} is irreducible, aperiodic and ergodic.According to (5.7), the generating function of the stationary distribution {Tr0, 71"1, 71"2,...} of the chain {r} is (5.10) ,=0 i=m Since the left-hand side of (5.10) is a Taylor series with absolutely summable coefficients, all roots of R(z) are the roots of z m-K(z) in + with the same multiplicities.Therefore, R(z) is completely determined up to a constant factor fixed by R'(1)-m---K'(1) that follows from (5.10).As was mentioned above, the polynomial R(z) can be used in the analysis of some practical problems.One of them is the following lemma.Let [i, = 0,1,2,...,n be any solution of the system of equations (5.2)-(5.3).Then the function n (z) Z [i[zmAi(z) ziK(z)] [zm-K(z)]-1 i=0 can be expanded in a Maclaurin series in z with absolutely summable coefficients. Proof: We can represent ab(z) in the form where Due to (5.2), f(z) is analytic on r + and continuous on r.Let us rewrite f(z) as follows: f(z) = (z 1)[R(z)]-1 i[Ai(z) zi](z 1) 1. i--0 The function (-1)[R()]-1 is rational, with all its poles belonging to I '+ and so it can be expanded into a Maclaurin series in with absolutely summable coefficients.In other words, Maclaurin series is absolutely convergent on I'.
On the other hand, all the functions [Ai(z)---1](z-1) -1, i= 0,1,...,n under the original assumptions that A(1) < cx, are expandable into Maclaurin series in z with absolutely summable coefficients, or equivalently the series are absolutely convergent on F.
Consequently, the same is true for Therefore, f(z) on F equals to the product of two series in z and in with absolutely summable coefficients which is a Laurent series with absolutely summable coefficients.
However, since f(z) is analytic in I' + and continuous on I' the principal part of this Laurent series must be zero.

El Example 1:
Consider a M/GY/1 queueing system with bulk service, ordinary input, and additional exponential service phase provided by the server when the number of customers in the system at the beginning of a service act is less than the server capacity m (it is supposed that the server is never idle).The imbedded Markov chain {r} describing the number of customers in the system at n + 0 where tn, n = 1,2,... are successive moments of service completions, has the transition matrix of A m type with g(z)7(7 + z)-, < m Ai(z) zi-mK(z), i>_m where K(z)= B*(A-Az), B is the distribution function of service time, B* is the Laplace- Stieltjes transform of B, A is the intensity of Poisson arrival process, and 7 is the rate of the additional phase.Condition A 0 guarantees that k > 0 for = 0, 1,2,... and, therefore, the chain {r} is queue-type.Due to Theorem 3.3 the chain is ergodic if and only if K'(1)< m.
Using relation (5.7), it can be shown that m--1 It follows that pi[z mzi(7 + A-Az)7-1], an mth degree polynomial in z, must have the i=0 same roots with the same multiplicities as zm-K(z)in P+\{1} and must assume value m-K'(1) at z-1.Therefore, this polynomial must be equal to R(z) and we get: Retaining all assumptions of Example 1, suppose in addition that customers can also arrive in pairs, so that the generating function of the arriving groups of customers is a(z) pz + qz2, p + q = 1.Then g()'r(7 +pz-qz ) -, < m Ai(z) z inK(z),
z m K ( Z ' ! ( z ) t , Repeating the arguments of Example 1, we conclude that the sum in (5.11) must be divisible by R(z), and, therefore, being an (m+ 1)th degree polynomial, can be factored into the product of R(z) and a first degree polynomial that equals 1 at z = 1" m-1 E Pi [zm zi7 1(7 + A Apz Aqz2)] (Z C)(1 C)-1R(Z), (5.12) i=0 where c 7(: 1) is a constant.To find c, substitute for z respective roots fll and f12 of the identically distributed random variables and that Yk "-min{Zk + Xk, m}, where Z k is the supply of water at instant k + 0 and m is a constant.A single-level control policy requires that for all k { } { G(z), if j = 0, 1,2,...,n P X<_z[Zk=j K(z), ifj=n+l,n+2,... and so that r if Z k = 0,1,2,...,n m if Z k-n+ 1, n+2,...,m > r, min{Zk+Xk, r}, Yk min{Zk + Xk, m}, if Z k = 0,1, 2,. . n if Z k = n + 1,n + 2,...Under these conditions {Zk) k-0,1,2,... constitutes a homogeneous Markov chain with the following transition Am, n-matrix: Theorem 5.1 we obtain the following expression for the generating function P(z) of the ergodic distribution of the chain (Zk) k 0,1,2,..The expression in the right-hand side of (5.16) can be simplified by noticing that if G(z) = K(z) and m-r, the relation (5.16) turns into Multiplying both sides of the matrix equation P = PA from the right by vector (1, z, z2,...)T, z = 1 and using (6.1) we get: o i+m P(z) Po = E Pi E ki + m j zj, z = 1.Since zmQ()qiz m-i, the left-hand side of (6.4) is analytic in F + and i=O continuous on I'.
On the other hand, the right-hand side of (6.4) by the definition of Q(z) and if(z) and due to the obvious relation P0 = (I)(1) is analytic in I'-and continuous on F. By Liouville's theorem, it means that (6.4) holding true on I' is only possible if both sides are identically equal to the same constant, say, C. Therefore, P(z)zmQ(l) C. Since P(1) 1, C = Q (1) which yields (6.2).Now we formulate the main result of this section.Theorem 6.2: The generating function of the stationary distribution of a Markov chain {'r} with transition homogeneous A'm, n "matriz A = (aij) is determined by P() = E pizi-n + lilP n +1 -i qj-i-j[mO() -1 (,) i=0 j=O The unknown probabilities Po, Pl,"',Pn on the right-hand side of (6.5) form a unique solution of the system of (n + 1) linear equations Pj ":" E Piaij--E Pn + l ibij, J 1,2,...,n (6.6) Multiplying both parts of the matrix equations P = PA from the right by vector (0,0,...,0,1,z, z2,...)T, zl = 1, and using the definition of a homogeneous Am, n- The right-hand side of (6.9) is, obviously, analytic in F + and continuous on F. At the same time the right-hand side of (6.9) is analytic in F-and continuous on F. To show this fact, i.t is enough to notice that the first part of the right-hand side of (6.9) is a polynomial in while the function (z) in the second part is, by inspection, a Maclaurin series in with absolutely summable coefficients.Since the left-hand side of (6.9) exists at z-1, so does the right-hand side of it.Also, by definition of Q(z), Q()/(z -m-K(1-e)) is obviously analytic in F-.Now, applying again the Liouville's theorem to (6.9), we conclude that both sides of this relation must be identically equal to the same constant C. The limit of the expression in the right- hand side of (6.9) as z---,oo is 0, because if(z) contains only negative powers of z.Therefore, C 0 and the left-hand side of (6.9) gives which yields (6.5).Applying operators D l), l-O, 1,2,... to (6.10), we obtain 3=0 r=O Therefore, k r O, r O, 1,..., m-1.