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A probabilistic approach to some of Euler's number theoretic identities

Author: Don Rawlings
Journal: Trans. Amer. Math. Soc. 350 (1998), 2939-2951
MSC (1991): Primary 60K99, 11P81, 05A30, 05A17
MathSciNet review: 1422618
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Abstract: Probabilistic proofs and interpretations are given for the $q$-binomial theorem, $q$-binomial series, two of Euler's fundamental partition identities, and for $q$-analogs of product expansions for the Riemann zeta and Euler phi functions. The underlying processes involve Bernoulli trials with variable probabilities. Also presented are several variations on the classical derangement problem inherent in the distributions considered.

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  • 1. G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976. MR 58:27738
  • 2. L. Benkherouf and J. A. Bather, Oil exploration: sequential decisions in the face of uncertainty, J. Appl. Prob. 25 (1988) 529-543. MR 89e:90068
  • 3. F. N. David and D. E. Barton, Combinatorial Chance, Hafner Publ. Co., 1962. MR 27:5305
  • 4. A. M. Garsia and J. Remmel, A combinatorial interpretation of $q$-derangement and $q$-Laguerre numbers, European J. Combin. 1 (1980) 47-59. MR 81g:05011
  • 5. I. Gessel, Counting permutations by descents, greater index, and cycle structure, unpublished manuscript, 1981.
  • 6. I. Gessel and C. Reutenauer, Counting permutations with given cycle structure and descent set, J. Combin. Theory Ser. A 64 (1993) 189-215. MR 95g:05006
  • 7. K. Griffin, The $q$-derangement problem relative to the inversion number, in preparation.
  • 8. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, John Wiley and Sons, 1968. MR 37:3604
  • 9. N. L. Johnson, S. Kotz, and A. W. Kemp, Univariate Discrete Distributions, John Wiley and Sons Inc., 1992. MR 95d:62018
  • 10. K. W. J. Kadell, A probabilistic proof of Ramanujan's $_1 \psi _1$ sum, SIAM J. Math. Anal. (18) 6 (1987) 1539-1548. MR 88k:33001
  • 11. A. W. Kemp, Heine-Euler extensions of the Poisson distribution, Comm. Statist. Theory and Methods 21 (1992) 571-588. CMP 92:15
  • 12. R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag. 61 (1988) 24-29.
  • 13. D. P. Rawlings, Bernoulli trials and number theory, Amer. Math. Monthly 101 (1994), 948-952. MR 95m:11029
  • 14. D. P. Rawlings, Absorption processes: Models for $q$-identities, Adv. Appl. Math. 18 (1997), 133-148. CMP 97:07
  • 15. D. P. Rawlings and J. A. Treadway, Bernoulli trials and Mahonian statistics: A tale of two $q$'s, Math. Mag. 67 (1994), 345-354. MR 96a:05005
  • 16. M. L. Wachs, On $q$-derangement numbers, Proc. Amer. Math. Soc. 106 (1989) 273-278. MR 89i:05032

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Additional Information

Don Rawlings
Affiliation: Department of Mathematics, California Polytechnic State University, San Luis Obispo, California 93407

Keywords: Euler's process, Euler's partition identities, $q$-binomial theorem, $q$-Poisson distribution, $q$-derangement problem, $q$-Riemann zeta function, $q$-Euler phi function
Received by editor(s): August 8, 1996
Received by editor(s) in revised form: September 16, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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