Geodesic
Probabilistic Interpretations of q-Analogues
Séminaire lotharingien de combinatoire, Tome 30 (1993)

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In some instances, the parameter q of a q-analogue may be directly identified as the probability of tails occurring in a Bernoulli trials scheme. The simple coin-tossing game presented in the next section gives rise to a q-analogue of a standard limit formula for the exponential function and to a q-analogue of Euler's product formula for the Riemann zeta function. The context in which these q-identities arise bears some resemblance to the one Gilbert Labelle used in obtaining a q-analogue of Euler's gamma function. Slight variations of the same game also lead to probabilistic interpretations of the inversion number and of the major index.

This paper is a summary of:

Bernoulli trials and permutation statistics, Internat. J. Math. Math. Sci. 15 (1992), 291--311

Bernoulli trials and number theory, Amer. Math. Monthly 101 (1994), 948--952. 

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     author = {Don Rawlings},
     title = {Probabilistic {Interpretations} of {q-Analogues}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {30},
     year = {1993},
     url = {http://geodesic.mathdoc.fr/item/SLC_1993_30_a7/}
}
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Don Rawlings. Probabilistic Interpretations of q-Analogues. Séminaire lotharingien de combinatoire, Tome 30 (1993). http://geodesic.mathdoc.fr/item/SLC_1993_30_a7/