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

DOI:
https://doi.org/10.1090/S0002-9947-98-01969-2

MathSciNet review:
1422618

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Probabilistic proofs and interpretations are given for the -binomial theorem, -binomial series, two of Euler's fundamental partition identities, and for -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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Additional Information

**Don Rawlings**

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

Email:
drawling@math.calpoly.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-01969-2

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