Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 12 (2016), 049, 15 pages      arXiv:1512.07104      https://doi.org/10.3842/SIGMA.2016.049
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Shell Polynomials and Dual Birth-Death Processes

Erik A. van Doorn
Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Received January 02, 2016, in final form May 14, 2016; Published online May 18, 2016

Abstract
This paper aims to clarify certain aspects of the relations between birth-death processes, measures solving a Stieltjes moment problem, and sets of parameters defining polynomial sequences that are orthogonal with respect to such a measure. Besides giving an overview of the basic features of these relations, revealed to a large extent by Karlin and McGregor, we investigate a duality concept for birth-death processes introduced by Karlin and McGregor and its interpretation in the context of shell polynomials and the corresponding orthogonal polynomials. This interpretation leads to increased insight in duality, while it suggests a modification of the concept of similarity for birth-death processes.

Key words: orthogonal polynomials; birth-death processes; Stieltjes moment problem; shell polynomials; dual birth-death processes; similar birth-death processes.

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References

  1. Anderson W.J., Continuous-time Markov chains. An applications-oriented approach, Springer Series in Statistics: Probability and its Applications, Springer-Verlag, New York, 1991.
  2. Berg C., Christensen J.P.R., Density questions in the classical theory of moments, Ann. Inst. Fourier (Grenoble) 31 (1981), 99-114.
  3. Berg C., Christiansen J.S., A question by T.S. Chihara about shell polynomials and indeterminate moment problems, J. Approx. Theory 163 (2011), 1449-1464, arXiv:1102.2723.
  4. Berg C., Thill M., A density index for the Stieltjes moment problem, in Orthogonal Polynomials and their Applications (Erice, 1990), IMACS Ann. Comput. Appl. Math., Vol. 9, Baltzer, Basel, 1991, 185-188.
  5. Berg C., Thill M., Rotation invariant moment problems, Acta Math. 167 (1991), 207-227.
  6. Berg C., Valent G., The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes, Methods Appl. Anal. 1 (1994), 169-209.
  7. Berg C., Valent G., Nevanlinna extremal measures for some orthogonal polynomials related to birth and death processes, J. Comput. Appl. Math. 57 (1995), 29-43.
  8. Chihara T.S., Chain sequences and orthogonal polynomials, Trans. Amer. Math. Soc. 104 (1962), 1-16.
  9. Chihara T.S., On determinate Hamburger moment problems, Pacific J. Math. 27 (1968), 475-484.
  10. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
  11. Chihara T.S., Indeterminate symmetric moment problems, J. Math. Anal. Appl. 85 (1982), 331-346.
  12. Chihara T.S., The parameters of a chain sequence, Proc. Amer. Math. Soc. 108 (1990), 775-780.
  13. Chihara T.S., Shell polynomials and indeterminate moment problems, J. Comput. Appl. Math. 133 (2001), 680-681.
  14. Coolen-Schrijner P., van Doorn E.A., Orthogonal polynomials on ${\mathbb R}^+$ and birth-death processes with killing, in Difference Equations, Special Functions and Orthogonal Polynomials, Editors S. Elaydi, J. Cushing, R. Lasser, A. Ruffing, V. Papageorgiou, W. Van Assche, World Sci. Publ., Hackensack, NJ, 2007, 726-740.
  15. Fralix B., When are two Markov chains similar?, Statist. Probab. Lett. 107 (2015), 199-203.
  16. Karlin S., McGregor J.L., The differential equations of birth-and-death processes, and the Stieltjes moment problem, Trans. Amer. Math. Soc. 85 (1957), 489-546.
  17. Karlin S., McGregor J., The classification of birth and death processes, Trans. Amer. Math. Soc. 86 (1957), 366-400.
  18. Lenin R.B., Parthasarathy P.R., Scheinhardt W.R.W., van Doorn E.A., Families of birth-death processes with similar time-dependent behaviour, J. Appl. Probab. 37 (2000), 835-849.
  19. Pedersen H.L., Stieltjes moment problems and the Friedrichs extension of a positive definite operator, J. Approx. Theory 83 (1995), 289-307.
  20. Pollett P.K., Similar Markov chains, J. Appl. Probab. 38A (2001), 53-65.
  21. Shohat J.A., Tamarkin J.D., The problem of moments, Math. Surveys Monogr., Vol. 1, Amer. Math. Soc., New York, 1943.
  22. van Doorn E.A., The indeterminate rate problem for birth-death processes, Pacific J. Math. 130 (1987), 379-393.
  23. van Doorn E.A., Representations for the decay parameter of a birth-death process based on the Courant-Fischer theorem, J. Appl. Probab. 52 (2015), 278-289.
  24. van Doorn E.A., Spectral properties of birth-death polynomials, J. Comput. Appl. Math. 284 (2015), 251-258.

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