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The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
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Cited by 14 (1 self)
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The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
BirthDeath Processes and Associated Polynomials
, 2001
"... We consider birthdeath processes on the nonnegative integers and the corresponding sequences of orthogonal polynomials called birthdeath polynomials. The sequence of associated polynomials linked with a sequence of birthdeath polynomials and its orthogonalizing measure can be used in the analysis ..."
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Cited by 7 (3 self)
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We consider birthdeath processes on the nonnegative integers and the corresponding sequences of orthogonal polynomials called birthdeath polynomials. The sequence of associated polynomials linked with a sequence of birthdeath polynomials and its orthogonalizing measure can be used in the analysis of the underlying birthdeath process in several ways. We briefly review the known applications of associated polynomials, which concern transition and firstentrance time probabilities, and establish some new results in this vein. In particular, our findings indicate how the prevalence of recurrence or #recurrence in a birthdeath process can be recognized from certain properties of the orthogonalizing measure for the associated polynomials. Keywords and phrases: birthdeath process, spectral measure, firstentrance time, recurrence, #recurrence, orthogonal polynomials, associated polynomials 2000 Mathematics Subject Classification: Primary 60J80, Secondary 42C05 1
On Associated Polynomials and Decay Rates for BirthDeath Processes
, 2001
"... We consider sequences of orthogonal polynomials and pursue the question of how (partial) knowledge of the orthogonalizing measure for the associated polynomials can lead to information about the orthogonalizing measure for the original polynomials. In particular, we relate the supports of the two m ..."
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Cited by 6 (3 self)
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We consider sequences of orthogonal polynomials and pursue the question of how (partial) knowledge of the orthogonalizing measure for the associated polynomials can lead to information about the orthogonalizing measure for the original polynomials. In particular, we relate the supports of the two measures, and their moments. As an application we analyse the relation between two decay rates connected with a birthdeath process. Keywords and phrases: orthogonal polynomials, associated polynomials, numerator polynomials, birthdeath process, decay rate, rate of convergence, firstentrance time 2000 Mathematics Subject Classification: Primary 42C05, Secondary 60J80 1
Quasistationary Distributions For LevelDependent QuasiBirthAndDeath Processes
 Stochastic Models (Special Issue in Honour of Marcel Neuts
, 1998
"... In this paper we provide a complete quasistationary analysis for the class of leveldependent, discretetime quasibirthanddeath processes (QBDs) for which level zero has been collapsed into an absorbing state. We show that the form of a quasistationary distribution depends upon whether the eigenv ..."
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Cited by 3 (0 self)
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In this paper we provide a complete quasistationary analysis for the class of leveldependent, discretetime quasibirthanddeath processes (QBDs) for which level zero has been collapsed into an absorbing state. We show that the form of a quasistationary distribution depends upon whether the eigenvalue of a certain matrix is equal to one or less than one. Furthermore, we show how to calculate the convergence norm of such a QBD and observe that the QBD is recurrent in the rst case mentioned above and transient in the second case. The further classication of an recurrent QBD as positive or null depends on whether the convergence radius 2 of the modied QBD in which level one is collapsed into an absorbing state is strictly greater than, or equal to, . In the rst of these cases the QBD is positive, while in the second case the QBD may be positive or null. Key words: Quasibirthanddeath process, Quasistationary distributions, Limiting conditional distribution...
The Waiting Time Distribution for the M/M/m Queue
 In Proceedings of IEE Communications
, 2003
"... This paper presents a novel method for the calculation of the waiting time distribution function for the M/M/m queue. It is shown that the conditional waiting time obeys an Erlang distribution with rate mµ, where µ is the service rate of a server. Explicit closed form solution is obtained by means o ..."
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Cited by 2 (0 self)
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This paper presents a novel method for the calculation of the waiting time distribution function for the M/M/m queue. It is shown that the conditional waiting time obeys an Erlang distribution with rate mµ, where µ is the service rate of a server. Explicit closed form solution is obtained by means of the probability density function of the Erlang distribution. It turns out that the derivation of the result is very simple. The significance of Khintchine's method and its close relation to our method is pointed out. It is also shown that the waiting time distribution can be obtained from Takacs's waiting time distribution for the G/M/m queue as a special case. This reveals some insight into the significance of Takacs's more general but rather complex result.
IMS Collections
, 2008
"... Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan ..."
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Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan