Abstract
A power series expression in closed form for the transient probabilities of a state-dependent birth-death process is presented with suitable illustrations.
A power series expression in closed form for the transient probabilities of a state-dependent birth-death process is presented with suitable illustrations.
G. E. Andrews, The Theory of Partitions, vol. 2 of Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Massachusetts, 1976.
View at: Zentralblatt MATH | MathSciNetB. C. Berndt, Ramanujan's Notebooks. Part II, Springer, New York, 1989.
View at: Zentralblatt MATH | MathSciNetP. Flajolet, “Combinatorial aspects of continued fractions,” Discrete Mathematics, vol. 32, no. 2, pp. 125–161, 1980.
View at: Google Scholar | Zentralblatt MATH | MathSciNetJ. Gill, “A note on extending Euler's connection between continued fractions and power series,” Journal of Computational and Applied Mathematics, vol. 106, no. 2, pp. 299–305, 1999.
View at: Google Scholar | Zentralblatt MATH | MathSciNetI. P. Goulden and D. M. Jackson, Combinatorial Enumeration, A Wiley-Interscience Publication. Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, New York, 1983.
View at: Zentralblatt MATH | MathSciNetF. Guillemin and D. Pinchon, “Excursions of birth and death processes, orthogonal polynomials, and continued fractions,” Journal of Applied Probability, vol. 36, no. 3, pp. 752–770, 1999.
View at: Google Scholar | Zentralblatt MATH | MathSciNetW. B. Jones and W. J. Thron, Continued Fractions. Analytic Theory and Applications, vol. 11 of Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Massachusetts, 1980.
View at: Zentralblatt MATH | MathSciNetR. B. Lenin, P. R. Parthasarathy, W. R. W. Scheinhardt, and E. A. van Doorn, “Families of birth-death processes with similar time-dependent behaviour,” Journal of Applied Probability, vol. 37, no. 3, pp. 835–849, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNetJ. A. Murphy and M. R. O'Donohoe, “Some properties of continued fractions with applications in Markov processes,” Journal of the Institute of Mathematics and its Applications, vol. 16, no. 1, pp. 57–71, 1975.
View at: Google Scholar | Zentralblatt MATH | MathSciNetP. R. Parthasarathy, “A transient solution to an queue: a simple approach,” Advances in Applied Probability, vol. 19, no. 4, pp. 997–998, 1987.
View at: Google Scholar | Zentralblatt MATH | MathSciNetP. R. Parthasarathy, “Some unusual birth-and-death processes,” The Mathematical Scientist, vol. 28, no. 2, pp. 79–90, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNetP. R. Parthasarathy and R. B. Lenin, Birth and Death Process (BDP) Models with Applications—Queueing, Communication Systems, Chemical Models, Biological Models: The State-of-the-Art with a Time-Dependent Perspective, American Science Press, New York, 2004.
View at: Zentralblatt MATHP. R. Parthasarathy, R. B. Lenin, W. Schoutens, and W. Van Assche, “A birth and death process related to the Rogers-Ramanujan continued fraction,” Journal of Mathematical Analysis and Applications, vol. 224, no. 2, pp. 297–315, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNetP. R. Parthasarathy and R. Sudhesh, “A formula for the coefficients of orthogonal polynomials from the three-term recurrence relations,” to appear in Applied Mathematics Letters.
View at: Google ScholarL. J. Rogers, “On the representation of certain asymptotic series as convergent continued fractions,” Proceedings of the London Mathematical Society. Second Series, vol. 4, no. 2, pp. 72–89, 1907.
View at: Google ScholarR. P. Stanley, Enumerative Combinatorics. Vol. 2, vol. 62 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999.
View at: Zentralblatt MATH | MathSciNetR. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, vol. 134 of Applied Mathematical Sciences, Springer, New York, 1999.
View at: Zentralblatt MATH | MathSciNetH. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand, New York, 1948.
View at: Zentralblatt MATH | MathSciNetA. J. Zajta and W. Pandikow, “Conversion of continued fractions into power series,” Mathematics of Computation, vol. 29, no. 130, pp. 566–572, 1975.
View at: Google Scholar | Zentralblatt MATH | MathSciNet