Abstract
We discuss a probability distribution I q depending on a parameter 0 < q < 1 and determined by its moments n!/(q; q)n. The treatment is purely analytical. The distribution has been discussed recently by Bertoin, Biane and Yor in connection with a study of exponential functionals of Lévy processes.
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References
Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver and Boyd, Edinburgh.
Askey, R. (1986). Limits of some q-laguerre polynomials. J. Approx. Theory, 46:213–216.
Askey, R. (1989). Orthogonal polynomials and theta functions. In Ehrenpreis, L. and Gunning, R. C., editors, Theta functions—Bowdoin 1987, Part 2 (Brunswick, ME, 1987), volume 49 of Proc. Symp. Pure Math., pages 299–321. Amer. Math. Soc., Providence, RI.
Askey, R. and Roy, R. (1986). More q-beta integrals. Rocky Mountain J. Math., 16:365–372.
Berg, C. (1980). Quelques remarques sur le cône de Stieltjes. In Seminar on Potential Theory, Paris, No. 5. (French), volume 814 of Lecture Notes in Mathematics, pages 70–79. Springer, Berlin.
Berg, C. (1981). The Pareto distribution is a generalized Γ-convolution. A new proof. Scand. Actuarial J., pages 117–119.
Berg, C. (1985). On the preservation of determinacy under convolution. Proc. Amer. Math. Soc., 93:351–357.
Berg, C. (1995). Indeterminate moment problems and the theory of entire functions. J. Comp. Appl. Math., 65:27–55.
Berg, C. (2000). On infinitely divisible solutions to indeterminate moment problems. In Dunkl, C., Ismail, M., and Wong, R., editors, Special Functions (Hong Kong, 1999), pages 31–41. World Scientific, Singapore.
Berg, C. and Duran, A. J. (2004). A transformation from Hausdorff to Stieltjes moment sequences. Ark. Mat. To appear.
Berg, C. and Forst, G. (1975). Potential theory on locally compact abelian groups. Springer-Verlag, New York.
Bertoin, J. (1996). Lévy processes. Cambridge University Press, Cambridge.
Bertoin, J., Biane, P., and Yor, M. (2004). Poissonian exponential functionals, q-series, q-integrals, and the moment problem for the log-normal distribution. Birkhäuser. To appear.
Bertoin, J. and Yor, M. (2001). On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Elect. Comm. in Probab., 6:95–106.
Biane, P., Pitman, J., and Yor, M. (2001). Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc., 38:435–465.
Bondesson, L. (1992). Generalized gamma convolutions and related classes of distributions and densities, volume 76 of Lecture Notes in Statistics. Springer, New York.
Carmona, P., Petit, F., and Yor, M. (1994). Sur les fonctionelles exponentielles de certain processus de Lévy. Stochastics and Stochastics Reports, 47:71–101.
Carmona, P., Petit, F., and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Yor, M., editor, Exponential functionals and principal values related to Brownian motion, Biblioteca de la Revista Matemitica Iberoamericana, pages 73–121. Revista Matemática Iberoamericana, Madrid.
Cowan, R. and Chiu, S.-N. (1994). A stochastic model of fragment formation when DNA replicates. J. Appl. Probab., 31:301–308.
Dumas, V., Guillemin, F., and Robert, P. (2002). A Markovian analysis of additive-increase multiplicative-decrease algorithms. Adv. in Appl. Probab., 34:85–111.
Gasper, G. and Rahman, M. (1990). Basic Hypergeometric Series. Cambridge University Press, Cambridge.
Itô, M. (1974). Sur les cônes convexes de Riesz et les noyaux complètement sousharmoniques. Nagoya Math. J., 55:111–144.
Lukacs, E. (1960). Characteristic functions. Griffin, London.
Pakes, A. G. (1996). Length biasing and laws equivalent to the log-normal. J. Math. Anal. Appl., 197:825–854.
Reuter, G. E. H. (1956). Über eine Volterrasche Integralgleichung mit totalmonotonem Kern. Arch. Math., 7:59–66.
Stoyanov, J. (2000). Krein condition in probabilistic moment problems. Bernoulli, 6:939–949.
Thorin, O. (1977a). On the infinite divisibility of the lognormal distribution. Scand. Actuarial. J., (3):121–148.
Thorin, O. (1977b). On the infinite divisibility of the Pareto distribution. Scand. Actuarial. J., (1):31–40.
Widder, D. V. (1941). The Laplace Transform, volume 6 of Princeton Mathematical Series. Princeton University Press, Princeton, N. J.
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Berg, C. (2005). On a Generalized Gamma Convolution Related to the q-Calculus. In: Ismail, M.E., Koelink, E. (eds) Theory and Applications of Special Functions. Developments in Mathematics, vol 13. Springer, Boston, MA. https://doi.org/10.1007/0-387-24233-3_4
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DOI: https://doi.org/10.1007/0-387-24233-3_4
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