Anderson, W. J., Continuous-time Markov Chains: An Applications-oriented Approach, Springer series in Statistics, Springer, New York, 1991. (Cited on pp. 147, 148, 149, 150, 151, 152, 153, 155, 158, 161, 162, 178, 191, 207, 236 and 242.)Google Scholar

Akhiezer, N. I., The Classical Moment Problem and Some Related Questions in Analysis, University Mathematical Monographs, Oliver & Boyd, Edimburgh and London, 1961. (Cited on pp. 18 and 105.)Google Scholar

Andrews, G. E., Askey, R. and Roy, R., Special Functions, Encyclopedia of Mathematics, No. 71, Cambridge University Press, Cambridge, 1999. (Cited on pp. 1, 3, 16 and 17.)Google Scholar

van Assche, W., Parthasarathy, P. R. and Lenin, R. B. Spectral representation of four finite birth and death processes, Math. Sci. 24 (1999), 105–112. (Cited on pp. 194 and 196.)Google Scholar

Bhattacharya, R. N. and Waymire, E. C., Stochastic Processes with Applications, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1990. (Cited on pp. 58, 255, 257, 258, 260, 261, 262, 269, 270, 271, 274, 279 and 287.)Google Scholar

Barrett, J. F. and Lampard, D. G., An expansion for some second-order probability distributions and its application to noise problems, IRE Trans. Inf. Th. 1 (1955), 10–15. (Cited on pp. ix, 262 and 268.)Google Scholar

Bakry, D. and Mazet, O., Characterization of Markov Semigroups on Associated to Some Families of Orthogonal Polynomials, Séminaire de Probabilités XXXVII, Lecture Notes in Math., 1832, Springer, Berlin, 2003, pp. 60–80. (Cited on p. 303.)Google Scholar

Bakry, D., Orevkov, S. and Zani, M., Orthogonal Polynomials and Diffusion Operators, arXiv:1309.5632v2. (Cited on p. 303.)Google Scholar

Beals, R. and Wong, R., Special Functions: A Graduate Text, Cambridge University Press, New York, 2010. (Cited on pp. 1, 31, 45 and 55.)Google Scholar

Bochner, S., Über Sturm-Liouvillesche Polynomsysteme, Math Z. 29 (1929), 730–736. (Cited on p. 27.)Google Scholar

Brockwell, P. J., The extinction time of a general birth and death process with catastrophes, J. Appl. Prob. 23 (1986), 851–858. (Cited on p. 191.)Google Scholar

Callaert, H., On the rate of convergence in birth-and-death processes, Bull. Soc. Math. Belg. 26, (1974), 173–184. (Cited on p. 237.)Google Scholar

Castro, M. M. and Grünbaum, F. A., On a seminal paper by Karlin and McGregor, SIGMA 9 (2013), 020. (Cited on pp. 57 and 99.)Google Scholar

Cattiaux, P., Collet, P., Lambert, A., Martínez, S., Méléard, S. and San Martín, J., Quasi-stationary distributions and diffusions models in population dynamics, Ann. Prob. 37 (2009), 1926–1969. (Cited on p. 319.)Google Scholar

Chen, M. F., Exponential L2-convergence and L2- spectral gap for Markov processes, Acta Math. Sinica 7 (1991), 19–37. (Cited on p. 238.)Google Scholar

Chihara, T. S., An Introduction to Orthogonal Polynomials, Gordon and Breach, London, 1968. (Cited on pp. 1, 14, 16, 18, 69, 99 and 167.)Google Scholar

Chihara, T. S. and Ismail, M. E. H., Orthogonal polynomials suggested by a queueing model, Adv. Appl. Math. 3 (1982), 441–462. (Cited on p. 251.)Google Scholar

Clayton, A., Quasi-birth-and-death processes and matrix-valued orthogonal polynomials, SIAM J. Matrix Anal. Appl. 31 (2010), 2239–2260. (Cited on p. 142.)Google Scholar

Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. (Cited on pp. 313 and 314.)Google Scholar

Collet, P., Martínez, S. and San Martín, J., Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption, Ann. Prob. 23 (1995), 1300–1314. (Cited on p. 317.)Google Scholar

Collet, P., Martínez, S. and San Martín, J., Quasi-stationary Distributions, Probability and its Applications, Springer, New York, 2013. (Cited on pp. 236, 237, 239, 240, 241, 312, 313, 314, 315, 316, 317 and 319.)Google Scholar

Coolen-Schrijner, P. and van Doorn, E. A., Analysis of random walks using orthogonal polynomials, J. Comp. Appl. Math. 99 (1998), 387–399. (Cited on pp. 57 and 69.)Google Scholar

Coolen-Schrijner, P. and van Doorn, E. A., Quasi-stationary distributions for birth-death processes with killing, J. Appl. Math. Stoch. Anal. 2006, Art. ID 84640. (Cited on p. 241.)Google Scholar

Cox, J., Ingersoll, J. and Ross, S., An intertemporal general equilibrium model of asset prices, Econometrica 53 (1985), 363–384. (Cited on p. 291.)Google Scholar

Cox, J., Ingersoll, J. and Ross, S., A theory of the term structure of interest rates, Econometrica 53 (1985), 385–407. (Cited on p. 291.)Google Scholar

Darroch, J. N. and Seneta, E., On quasi-stationary distributions in absorbing discrete-time finite Markov chains, J. Appl. Prob. 2 (1965), 88–100. (Cited on p. 123.)Google Scholar

Deift, P. A., Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes in Mathematics, 3, American Mathematical Society, Providence, RI, 1999. (Cited on pp. 18, 21 and 24.)Google Scholar

Derman, C., A solution to a set of fundamental equations in Markov chains, Proc. Amer. Math. Soc. 5 (1954), 332–334. (Cited on p. 83.)Google Scholar

Dette, H., On a generalization of the Ehrenfest urn model, J. Appl. Prob. 31 (1994), 930–939.Google Scholar

Dette, H., On the generating functions of a random walk on the non-negative integers, J. Appl. Prob. 33 (1996), 1033–1052. (Cited on pp. 57 and 82.)Google Scholar

Dette, H., First return probabilities of birth and death chains and associated orthogonal polynomials, Proc. Amer. Math. Soc. 129 (2000), 1805–1815. (Cited on pp. 16, 77 and 83.)Google Scholar

Dette, H., Reuther, B., Studden, W. and Zygmunt, M., Matrix measures and random walks with a block tridiagonal transition matrix, SIAM J. Matrix Anal. Applic. 29 (2006), 117–142. (Cited on pp. 137, 138 and 142.)Google Scholar

Dette, H. and Reuther, B., Some comments on quasi-birth-and-death processes and matrix measures, J. Prob. Stat. 2010 (2010), https://doi.org/10.1155/2010/730543. (Cited on p. 247.)Google Scholar

Dette, H. and Studden, W. J., The Theory of Canonical Moments with Applications in Statistics, Probability and Analysis, John Wiley & Sons, Inc., New York, 1997. (Cited on pp. 16, 71, 131 and 132.)Google Scholar

van Doorn, E. A., Stochastic monotonicity of birth-death processes, Adv. Appl. Prob. 12 (1980), 59–80. (Cited on p. 242.)Google Scholar

van Doorn, E. A., Stochastic Monotonicity and Queueing Applications of Birth-Death Processes, Lecture Notes in Statistics, 4, Springer, New York, 1981. (Cited on p. 242.)Google Scholar

van Doorn, E. A., The transient state probabilities for a queueing model where potential customers are discouraged by queue length, Adv. Appl. Prob. 17 (1985) 514–530. (Cited on pp. 251 and 252.)Google Scholar

van Doorn, E. A., Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process, Adv. Appl. Prob. 17 (1985) 514–530. (Cited on pp. 238 and 240.)Google Scholar

van Doorn, E. A., Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tridiagonal matrices, J. Approx. Theory 51 (1987) 254–266.Google Scholar

van Doorn, E. A., Quasi-stationary distributions and convergence for quasi-stationarity of birth-death processes, Adv. Appl. Prob. 23 (1991), 683–700. (Cited on pp. 229, 237, 238 and 239.)Google Scholar

van Doorn, E. A., Birth-death processes and associated polynomials, J. Comp. Appl. Math. 23 (2003), 497–506. (Cited on pp. 216 and 242.)Google Scholar

van Doorn, E. A., Conditions for the existence of quasi-stationary distributions for birth-death processes with killing, Stochastic Process. Appl. 122 (2012), 2400–2410. (Cited on p. 241.)Google Scholar

van Doorn, E. A., Representations for the decay parameter of a birth-death process based on the Courant-Fischer theorem, J. Appl. Prob. 52 (2015), 278–289. (Cited on p. 238.)Google Scholar

van Doorn, E. A., On the strong ratio limit property for discrete-time birth-death processes, SIGMA 14 (2018), 047. (Cited on pp. 69, 114 and 119.)Google Scholar

van Doorn, E. A. and Pollett, P. K., Quasi-stationary distributions for discrete-state models, European J. Oper. Res. 230 (2013), 1–14. (Cited on p. 241.)Google Scholar

van Doorn, E. A. and Schrijner, P., Random walk polynomials and random walk measures, J. Comp. Appl. Math 49 (1993), 289–296. (Cited on pp. 57, 68, 117 and 128.)Google Scholar

van Doorn, E. A. and Schrijner, P., Geometric ergodicity and quasi-stationarity in discrete-time birth-death processes, J. Austral. Math. Soc. Ser. B 37 (1995), 121–144. (Cited on pp. 83, 109, 114, 123 and 130.)Google Scholar

van Doorn, E. A. and Schrijner, P., Ratio limits and limiting conditional distributions for discrete-time birth-death processes, J. Math. Anal. Appl. 190 (1995), 263–284. (Cited on pp. 114, 115, 117, 119, 120, 122, 123, 124 and 125.)Google Scholar

van Doorn, E. A. and Zeifman, A. I., Birth-death processes with killing, Statist. Prob. Lett. 72 (2005), 33–42. (Cited on pp. 188, 189, 191 and 214.)Google Scholar

van Doorn, E. A. and Zeifman, A. I., Extinction probability in a birth-death process with killing, J. Appl. Prob. 42 (2005), 185–198. (Cited on pp. 188, 190 and 191.)Google Scholar

Dunford, N. and Schwartz, J. T., Linear Operators, Part II: Spectral Theory, Wiley-Interscience, New York, 1988. (Cited on pp. 3 and 21.)Google Scholar

Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher Transcendental Functions (3 vols.), McGraw-Hill, New York, 1953. (Cited on pp. 214, 298 and 301.)Google Scholar

Feller, W., Diffusion processes in genetics, Proc. Second Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, 1951), pp. 227–246. (Cited on pp. ix, 196 and 291.)Google Scholar

Feller, W., The parabolic differential equations and the associated semi groups of transformations, Ann. Math. 55 (1952), 468–519. (Cited on p. 276.)Google Scholar

Feller, W., Diffusion processes in one dimension, Trans. Amer. Math. Soc. 77 (1954), 1–31. (Cited on p. 267.)Google Scholar

Feller, W., The birth and death processes as diffusion processes, J. Math. Pure Appl. 38 (1959), 301–345. (Cited on pp. ix and 159.)Google Scholar

Feller, W., An Introduction to Probability Theory and its Applications, vol. 1, John Wiley & Sons Inc, New York, 1968. (Cited on pp. 58, 60, 91, 143 and 202.)Google Scholar

Flajolet, P. and Guillemin, F., The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions, Adv. Appl. Prob. 32 (2000), 750–778. (Cited on p. 178.)Google Scholar

Good, P., The limiting behavior of transient birth and death processes conditioned on survival, J. Austral. Math. Soc. 8 (1968), 716–722. (Cited on p. 230.)Google Scholar

Gouriéroux, C., Renault, E. and Valéry, P., Diffusion processes with polynomial eigenfunctions, Annales d’Économie et de Statistique 85 (2007), 115–130. (Cited on pp. 267 and 268.)Google Scholar

Grosjean, C. C., The weight functions, generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the Jacobi and the Gegenbauer polynomials, J. Comp. Appl. Math. 16 (1986), 259–307. (Cited on p. 16.)Google Scholar

Grünbaum, F. A., Random walks and orthogonal polynomials: some challenges, in Pinsky, M. and Birnir, B. (eds.) Probability, Geometry and Integrable Systems, MSRI Publication, vol. 55, Cambridge University Press, Cambridge, 2008, pp. 241–260. See also arXiv: math.PR/0703375v1. (Cited on pp. 57, 91, 137 and 142.)Google Scholar

Grünbaum, F. A., QBD processes and matrix orthogonal polynomials: some new explicit examples, in Bini, D., Meini, B., Ramaswami, V., Remiche, M. A. and Taylor, P. (eds.), Numerical Methods for Structured Markov Chains, Dagstuhl Seminar Proceedings, 2008. (Cited on pp. 140 and 145.)Google Scholar

Grünbaum, F. A., The Karlin–McGregor formula for a variant of a discrete version of Walsh's spider, J. Phys. A 42 (2009), no. 45, 454010. (Cited on p. 142.)Google Scholar

Grünbaum, F. A., Gohberg, I. and Ball, J. A., A spectral weight matrix for a discrete version of Walsh's spider, in Operator Theory Advances and Applications Topics in Operator Theory, vol. 1: Operators, Matrices and Analytic Functions, Operator Theory: Advances and Applications, vol. 202, Birkhäuser Verlag, Basel, 2010 pp. 253–264. (Cited on p. 142.)Google Scholar

Grünbaum, F. A., Un urn model associated with Jacobi polynomials, Comm. App. Math. Comp. Sci. 5 (2010), 55–63. (Cited on p. 102.)Google Scholar

Grünbaum, F. A. and Haine, L., Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation, in Levi, D., Vinet, L. and Winternitz, P. (eds.), Symmetries and Integrability of Differential Equations, CRM Proc. Lecture Notes, vol. 9, American Mathematical Society, Providence, RI, 1996, pp. 143–154. (Cited on p. 74.)Google Scholar

Grünbaum, F. A. and de la Iglesia, M. D., Matrix valued orthogonal polynomials arising from group representation theory and a family of quasi-birth-and-death processes, SIAM J. Matrix Anal. Appl. 30 (2008), 741–761. (Cited on p. 142.)Google Scholar

Grünbaum, F. A. and de la Iglesia, M. D., Stochastic LU factorizations, Darboux transformations and urn models, J. Appl. Prob. 55 (2018), 862–886. (Cited on pp. 57, 74, 99, 102 and 144.)Google Scholar

Heyman, D. P., A decomposition theorem for infinite stochastic matrices, J. Appl. Prob. 32 (1995), 893–903. (Cited on p. 74.)Google Scholar

Hille, E., The abstract Cauchy problem and Cauchy's problem for parabolic differential equations, J. Anal. Math. 3 (1953), 83–196. (Cited on pp. ix and 262.)Google Scholar

Horn, R. A. and Johnson, C. R., Matrix Analysis, Cambridge University Press, Cambridge 1990. (Cited on p. 65.)Google Scholar

Ismail, M. E. H., A queueing model and a set of orthogonal polynomials, J. Math. Anal. Appl. 108 (1985), 575–594. (Cited on p. 214.)Google Scholar

Ismail, M. E. H., Classical and Quantum Orthogonal Polynomials in one Variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. (Cited on pp. 1, 15, 18, 19, 20, 30 and 214.)Google Scholar

Ismail, M. E. H., Letessier, J., Masson, D. and Valent, G., Birth and death processes and orthogonal polynomials, in Nevai, P. (ed.), Orthogonal Polynomials, Kluwer Academic Publishers, Boston, MA, 1990, pp. 229–255. (Cited on p. 248.)Google Scholar

Ismail, M. E. H., Letessier, J. and Valent, G., Linear birth and death models and associated Laguerre and Meixner polynomials, J. Approx. Theory 55 (1988), 337–348. (Cited on p. 211.)Google Scholar

Ismail, M. E. H., Letessier, J. and Valent, G., Quadratic birth and death processes and associated continuous dual Hahn polynomials, SIAM J. Math. Anal. 20 (1989), 727–737. (Cited on p. 214.)Google Scholar

Jacka, S. D. and Roberts, G. O., Weak convergence of conditioned process on a countable state space, J. Appl. Prob. 32 (1995), 902–916. (Cited on p. 237.)Google Scholar

Jones, W. B. and Magnus, A., Application of Stieltjes fractions to birth-death processes, in Saff, E. B. and Varga, R. S. (eds.), Padé and Rational Approximation, Academic Press, New York, 1977, pp. 173–179. (Cited on p. 178.)Google Scholar

Kac, M., Random walk and the theory of Brownian motion, American Math. Monthly 54 (1947), 369–391. (Cited on pp. ix and 160.)Google Scholar

Karlin, S., Total Positivity, vol. I. Stanford University Press, Stanford, CA, 1968. (Cited on p. 187.)Google Scholar

Karlin, S. and McGregor, J., Representation of a class of stochastic processes, Proc. Nat. Acad. Sci. 41 (1955), 387–391. (Cited on pp. ix, 160 and 162.)Google Scholar

Karlin, S. and McGregor, J., The differential equations of birth and death processes, and the Stieltjes moment problem, Trans. Amer. Math. Soc. 85 (1957), 489–546. (Cited on pp. 71, 126, 127, 160, 162, 164, 165, 168, 170, 172, 180, 183, 184, 186 and 187.)Google Scholar

Karlin, S. and McGregor, J., The classification of birth-and-death processes, Trans. Amer. Math. Soc. 86 (1957), 366–400. (Cited on pp. 111, 160, 216, 218, 222, 223, 227 and 252.)Google Scholar

Karlin, S. and McGregor, J., Linear growth, birth and death processes, J. Math. Mech. 7 (1958), 643–662. (Cited on pp. 207 and 251.)Google Scholar

Karlin, S. and McGregor, J., Many server queueing processes with Poisson input and exponential service times, Pacific J. Math. 8 (1958), 87–118. (Cited on pp. 79, 175, 201, 203, 207 and 250.)Google Scholar

Karlin, S. and McGregor, J., Random walks, Illinois J. Math. 3 (1959), 66–81. (Cited on pp. ix, 57, 63, 65, 67, 68, 77, 93, 95, 99, 105, 106, 114, 118, 120 and 132.)Google Scholar

Karlin, S. and McGregor, J., Coincidence probabilities, Pacific J. Math. 9 (1959), 1141–1164. (Cited on p. 267.)Google Scholar

Karlin, S. and McGregor, J., Classical diffusion processes and total positivity, J. Math. Anal. Appl. 1 (1960), 163–183. (Cited on pp. ix, 265, 267, 279, 286 and 297.)Google Scholar

Karlin, S. and McGregor, J., Total positivity of fundamental solutions of parabolic equations, Proc. Am. Math. Soc. 13 (1962), 136–139. (Cited on p. 267.)Google Scholar

Karlin, S. and McGregor, J., On a genetics model of Moran, Math. Proc. Camb. Phil. Soc. 58 (1962), 299–311. (Cited on pp. 196, 199 and 291.)Google Scholar

Karlin, S. and McGregor, J., Ehrenfest urn models, J. Appl. Prob. 2 (1965), 352–376. (Cited on p. 195.)Google Scholar

Karlin, S. and McGregor, J., Spectral theory of branching processes. I. The case of discrete spectrum, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5 (1966), 6–33. (Cited on p. 291.)Google Scholar

Karlin, S. and McGregor, J., Spectral theory of branching processes. II. Case of continuous spectrum, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5 (1966), 34–54. (Cited on p. 291.)Google Scholar

Karlin, S. and McGregor, J., On the spectral representation of branching processes with mean one, J. Math. Anal. Appl. 21 (1968), 485–495. (Cited on p. 291.)Google Scholar

Karlin, S. and Tavaré, S., The detection of a recessive visible gene in finite populations, I, Genetics Res., 37 (1981), 33–46. (Cited on p. 303.)Google Scholar

Karlin, S. and Tavaré, S., The detection of particular genotypes in finite populations, II, Theoret. Pop. Biol. 19 (1981), 215–229. (Cited on p. 303.)Google Scholar

Karlin, S. and Tavaré, S., Linear birth and death processes with killing, J. Appl. Prob 19 (1982), 477–487. (Cited on pp. 188 and 214.)Google Scholar

Karlin, S. and Tavaré, S., A class of diffusion processes with killing arising in population genetics, SIAM J. Appl. Math. 43 (1983), 31–41. (Cited on pp. 279, 286, 299 and 303.)Google Scholar

Karlin, S. and Taylor, H., A First Course in Stochastic Processes, Academic Press, New York, 1975. (Cited on p. 58.)Google Scholar

Karlin, S. and Taylor, H., A Second Course in Stochastic Processes, Academic Press, New York, 1981. (Cited on pp. 87, 97, 255, 260, 261, 262, 275, 276, 277, 279, 285, 286, 287, 291 and 297.)Google Scholar

Kendall, D. G., Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain, Ann. Math. Statist. 24 (1953), 338–354. (Cited on p. 192.)Google Scholar

Kendall, D. G., Unitary dilations of one-parameter semigroups of Markov transition operators, and the corresponding integral representation for Markov processes with a countable infinity of states, Proc. London Math. Soc. 3 (1959), 417–432. (Cited on p. ix.)Google Scholar

Kijima, M., On the existence of quasi-stationary distributions in denumerable R-transient Markov chains, J. Appl. Prob. 29 (1992), 21–36; correction 30 (1993), 496. (Cited on p. 123.)Google Scholar

Kijima, M., Nair, M. G., Pollett, P. K. and van Doorn, E. A., Limiting conditional distributions for birth-death processes, Adv. Appl. Prob. 29 (1997), 185–204. (Cited on pp. 229, 230, 232, 233 and 239.)Google Scholar

Koekoek, R., Lesky, P. A. and Swarttouw, R. F., Hypergeometric Orthogonal Polynomials and their q-Analogues, Springer Monographs in Mathematics, Springer, Berlin, 2010. (Cited on pp. 52 and 53.)Google Scholar

Kingman, J. F. C., The exponential decay of Markov transition probabilities, Proc. London Math. Soc. 13 (1963), 337–358. (Cited on p. 237.)Google Scholar

Kingman, J. F. C., Ergodic properties of continuous-time Markov processes and their discrete skeletons, Proc. London Math. Soc. 13 (1963), 593–604. (Cited on p. 237.)Google Scholar

Koelink, E., Spectral theory and special functions, Summer School, Laredo, Spain, July 24–28, 2000, http://arxiv.org/abs/math/0107036v1. (Cited on pp. 18, 21 and 24.)Google Scholar

Latouche, G. and Ramaswami, V., Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM Series on Statistics and Applied Probability, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999. (Cited on pp. 137 and 247.)Google Scholar

Ledermann, W. and Reuter, G. E., Spectral theory for the differential equations of simple birth and death processes, Phil. Trans. Roy. Soc. London, Ser. A. 246 (1954), 321–369. (Cited on pp. ix and 160.)Google Scholar

Lladser, M. and San Martín, J., Domain of attraction of the quasi-stationary distributions for the Ornstein-Uhlenbeck process, J. Appl. Prob. 37 (2000), 511–520. (Cited on p. 319.)Google Scholar

Maki, D. P., On birth-death processes with rational growth rates, SIAM J. Math. Anal. 7 (1976), 29–36. (Cited on p. 241.)Google Scholar

Mandl, P., Spectral theory of semi-groups connected with diffusion processes and its application, Czechoslovak Math. J. 11 (1961), 558–569. (Cited on p. 312.)Google Scholar

Martínez, S. and San Martín, J., Rates of decay and h-processes for one dimensional diffusions conditional on non-absorption, J. Theoret. Prob. 14 (2001), 199–212. (Cited on p. 317.)Google Scholar

McKean, H. P. Jr., Elementary solutions for certain parabolic partial differential equations, Trans. Amer. Math. Soc. 82 (1956), 519–548. (Cited on pp. ix and 262.)Google Scholar

Murphy, J. A. and O’Donohoe, J. L., Some properties of continued fractions with applications in Markov processes, J. Inst. Math. Appl. 16 (1975), 57–71. (Cited on p. 178.)Google Scholar

Nair, M. G. and Pollett, P. K., On the relationship between μ-invariant measures and quasi-stationary distributions for continuous-time Markov chains, Adv. Appl. Prob. 25 (1993), 82–102. (Cited on p. 236.)Google Scholar

Neuts, M. F., Structured Stochastic Matrices of M/G/1 Type and Their Applications, Marcel Dekker, New York, 1989. (Cited on pp. 137 and 247.)Google Scholar

Norris, J. R., Markov Chains, Cambridge University Press, Cambridge, 1997. (Cited on pp. 58 and 61.)Google Scholar

NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.20 of 2018-09-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders (eds.). (Cited on pp. 52 and 53.)Google Scholar

Ogura, Y., Spectral representation for branching processes on the real half line, Publ. Res. Inst. Math. Sci. 5 (1969/1970), 423–441. (Cited on p. 291.)Google Scholar

Parthasarathy, P. R., Lenin, R. B., Schoutens, W. and van Assche, W., A birth and death process related to the Rogers-Ramanujan continued fraction, J. Math. Anal. Appl. 224 (1998), 297–315. (Cited on p. 214.)Google Scholar

Pruitt, W., Bilateral birth and death processes, Trans. Amer. Math. Soc. 107 (1962), 508–525. (Cited on pp. 243 and 245.)Google Scholar

Reed, M. and Simon, B., Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York, 1972. (Cited on pp. 20, 21 and 24.)Google Scholar

Routh, E., On some properties of certain solutions of a differential equation of the second order, Proc. London Math. Soc. 16 (1884), 245–261. (Cited on pp. 27 and 29.)Google Scholar

Rudin, W., Functional Analysis, McGraw-Hill, New York, 1973. (Cited on pp. 20 and 21.)Google Scholar

Sasaki, R., Exactly solvable birth and death processes, J. Math. Phys. 50 (2009), 103509. (Cited on p. 214.)Google Scholar

Schoutens, W., Stochastic Processes and Orthogonal Polynomials, Lecture Notes in Statistics 146, Springer, New York, 2000. (Cited on p. ix.)Google Scholar

Seneta, E. Non-negative Matrices and Markov Chains, 2nd edition, Springer Series in Statistics XVI, Springer, New York, 1981. (Cited on p. 65.)Google Scholar

Seneta, E. and Vere-Jones, D., On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states, J. Appl. Prob. 3 (1966), 403–434. (Cited on p. 123.)Google Scholar

Serafin, G., Exit times densities of the Bessel process, Proc. Amer. Math. Soc. 145 (2017), 3165–3178. (Cited on p. 307.)Google Scholar

Simon, B., The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), 82–203. (Cited on p. 18.)Google Scholar

Sharma, O. P. and Gupta, U. C., Transient behaviour of an M/M/1/N queue, Stoch. Processes Appl. 13 (1982), 327–331. (Cited on p. 194.)Google Scholar

Shen, J., Tang, T. and Wang, L., Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics 41, Springer, Berlin, 2011. (Cited on p. 1.)Google Scholar

Sherman, Y. J., On the numerators of the convergents of the Stieltjes continued fractions, Trans. Amer. Math. Soc. 35 (1933), 64–87. (Cited on p. 16.)Google Scholar

Shohat, J. and Tamarkin, J., The Problem of Moments, AMS Mathematical Surveys, 1, American Mathematical Society, Providence, RI, 1943. (Cited on pp. 1, 18, 19, 105 and 164.)Google Scholar

Steinsaltz, D. and Evans, S.N, Quasi stationary distributions for one-dimensional diffusions with killing, Trans. Amer. Math. Soc. 359 (12007), 1285–1324. (Cited on p. 319.)Google Scholar

Stone, M. H., Linear Transformations in Hilbert Spaces and their Applications to Analysis, Colloquium Publications, vol. 15, American Mathematical Society, Providence, RI, 1932. (Cited on p. 244.)Google Scholar

Stroock, D. W., An Introduction to Markov Processes, Graduate Texts in Mathematics, 230, Springer, Berlin, 2005. (Cited on p. 58.)Google Scholar

Sumita, U. and Masuda, Y., On first passage time structure of random walks, Stoch. Processes Appl. 20 (1985), 133–147. (Cited on p. 82.)Google Scholar

Szegö, G., Orthogonal Polynomials, AMS Colloquium Publications, vol. 23, American Mathematical Society, Providence, RI, 1939. (Cited on pp. 1, 86 and 88.)Google Scholar

Vere-Jones, D., Geometric ergodicity in denumerable Markov chains, Quart. J. Math. Oxford 13 (1962), 7–28. (Cited on p. 83.)Google Scholar

Watson, G. N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922. (Cited on p. 299.)Google Scholar

Whitehurst, T., An application of orthogonal polynomials to random walks, Pacific J. Math. 99 (1982), 205–213. (Cited on p. 130.)Google Scholar

Widder, D., The Laplace Transform, Princeton University Press, Princeton, NJ, 1941. (Cited on p. 6.)Google Scholar

Wong, E. and Thomas, J. B., On polynomial expansions as second-order distributions, J. Soc. Indust. Appl. Math. 10 (1962), 507–516. (Cited on p. 303.)Google Scholar

Wong, E., The construction of a class of stationary Markoff processes, in Bellman, R. (ed.), Stochastic Processes in Mathematical Physics and Engineering, American Mathematical Society, Providence, RI, pp. 264–276, 1964. (Cited on pp. 279, 282, 308 and 310.)Google Scholar

Wright, S., The genetical structure of populations, Ann. Eugenics 15 (1951), 323–354. (Cited on pp. 196 and 291.)Google Scholar

Yoon, G. J., Darboux transforms and orthogonal polynomials, Bull. Korean Math. Soc. 39 (2002), 359–376. (Cited on p. 74.)Google Scholar

Zettl, A., Sturm–Liouville Theory, AMS Mathematical Surveys and Monographs vol. 121, American Mathematical Society, Providence, RI, 2005. (Cited on p. 26.)Google Scholar