Birth-death processes and $q$-continued fractions
HTML articles powered by AMS MathViewer
- by Tony Feng, Rachel Kirsch, Elise Villella and Matt Wage PDF
- Trans. Amer. Math. Soc. 364 (2012), 2703-2721 Request permission
Abstract:
In the 1997 paper of Parthasarathy, Lenin, Schoutens, and Van Assche, the authors study a birth-death process related to the Rogers-Ramanujan continued fraction $r(q)$. We generalize their results to establish a correspondence between birth-death processes and a larger family of $q$-continued fractions. It turns out that many of these continued fractions, including $r(q)$, play important roles in number theory, specifically in the theory of modular forms and $q$-series. We draw upon the number-theoretic properties of modular forms to give identities between the transition probabilities of different birth-death processes.References
- Waleed A. Al-Salam and Mourad E. H. Ismail, Orthogonal polynomials associated with the Rogers-Ramanujan continued fraction, Pacific J. Math. 104 (1983), no. 2, 269–283. MR 684290
- William J. Anderson, Continuous-time Markov chains, Springer Series in Statistics: Probability and its Applications, Springer-Verlag, New York, 1991. An applications-oriented approach. MR 1118840, DOI 10.1007/978-1-4612-3038-0
- G. E. Andrews, Bruce C. Berndt, Lisa Jacobsen, and Robert L. Lamphere, The continued fractions found in the unorganized portions of Ramanujan’s notebooks, Mem. Amer. Math. Soc. 99 (1992), no. 477, vi+71. MR 1124109, DOI 10.1090/memo/0477
- Bui The Anh and D. D. X. Thanh, A Perron-Frobenius theorem for positive quasipolynomial matrices associated with homogeneous difference equations, J. Appl. Math. , posted on (2007), Art. ID 26075, 6. MR 2365983, DOI 10.1155/2007/26075
- Bruce C. Berndt, Number theory in the spirit of Ramanujan, Student Mathematical Library, vol. 34, American Mathematical Society, Providence, RI, 2006. MR 2246314, DOI 10.1090/stml/034
- T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
- W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 2, 137–162. MR 2133308, DOI 10.1090/S0273-0979-05-01047-5
- Amanda Folsom, Modular forms and Eisenstein’s continued fractions, J. Number Theory 117 (2006), no. 2, 279–291. MR 2213765, DOI 10.1016/j.jnt.2005.06.001
- Mourad E. H. Ismail, The zeros of basic Bessel functions, the functions $J_{\nu +ax}(x)$, and associated orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), no. 1, 1–19. MR 649849, DOI 10.1016/0022-247X(82)90248-7
- Mourad E. H. Ismail and Ruiming Zhang, On the Hellmann-Feynman theorem and the variation of zeros of certain special functions, Adv. in Appl. Math. 9 (1988), no. 4, 439–446. MR 968677, DOI 10.1016/0196-8858(88)90022-X
- William B. Jones and Wolfgang J. Thron, Continued fractions, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980. Analytic theory and applications; With a foreword by Felix E. Browder; With an introduction by Peter Henrici. MR 595864
- Victor Kac and Pokman Cheung, Quantum calculus, Universitext, Springer-Verlag, New York, 2002. MR 1865777, DOI 10.1007/978-1-4613-0071-7
- S. Karlin and J. L. McGregor, The differential equations of birth-and-death processes, and the Stieltjes moment problem, Trans. Amer. Math. Soc. 85 (1957), 489–546. MR 91566, DOI 10.1090/S0002-9947-1957-0091566-1
- Neal Koblitz, Introduction to elliptic curves and modular forms, 2nd ed., Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1993. MR 1216136, DOI 10.1007/978-1-4612-0909-6
- Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
- P. R. Parthasarathy, R. B. Lenin, W. Schoutens, and W. Van Assche, A birth and death process related to the Rogers–Ramanujan continued fraction, J. Math. Anal. Appl. 224 (1998), no. 2, 297–315. MR 1637462, DOI 10.1006/jmaa.1998.6005
- K. G. Ramanathan, Ramanujan’s continued fraction, Indian J. Pure Appl. Math. 16 (1985), no. 7, 695–724. MR 801801
- Sheldon M. Ross, Stochastic processes, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1996. MR 1373653
- Walter Van Assche, The ratio of $q$-like orthogonal polynomials, J. Math. Anal. Appl. 128 (1987), no. 2, 535–547. MR 917386, DOI 10.1016/0022-247X(87)90204-6
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
Additional Information
- Tony Feng
- Affiliation: 58 Plympton Street, 479 Quincy Mail Center, Cambridge, Massachusetts 02138
- Email: tfeng@college.harvard.edu
- Rachel Kirsch
- Affiliation: 7212 Longwood Drive, Bethesda, Maryland 20817
- Email: rkirsch@math.unl.edu
- Elise Villella
- Affiliation: 146 Harrison Drive, Edinboro, Pennsylvania 16412
- Email: elisemccall@gmail.com
- Matt Wage
- Affiliation: 1411 N. Briarcliff Drive, Appleton, Wisconsin 54915
- Email: mwage@princeton.edu
- Received by editor(s): July 24, 2009
- Received by editor(s) in revised form: October 26, 2010
- Published electronically: January 17, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 2703-2721
- MSC (2010): Primary 03B48; Secondary 11A55
- DOI: https://doi.org/10.1090/S0002-9947-2012-05522-X
- MathSciNet review: 2888225