Applications For An Expectation Operator In Q-series

Professor Ming-Jin Wang in order to use Lebesgue’s dominated convergence theorem and Analytic continuation theorem to get a expectation operator about q-series, constructed a new probability distribution, then proof q-binomial theorem and q-Gauss sum formula.This paper is a application research for probability in q-series fields, We use a proba-bility distribution W(x; q) to calculate the expectation of the random variable X and give the probabilistic proof of some q-series identities based on our expectation operator, for example: q-binomial theorem、q-Vandermonde sum formulas、q-Gauss sum formula、q-Kummer sum formula、q-Bailey’s 2F1(-1) sum formula、q-Gauss’s 2F1(-1) sum formula、3Φ2 transfor-mation formulas、Heine 2Φ1 transformation formula、Jackson’s 2Φ1 and 2Φ2 transformation formulas、two q-exponential functions, and so forth.