Q-Differential Operator And Its Applications

In 1893,Rogers[122]constructed the two q-exponential operators by the q-differential operator.Then he used them to study some properties of the q-series.One hundred years later,Chen-Liu[46,47]discovered them independently.And they first gave the two identities of the q-operators.Then they used the two identities to give a system ways of studying the theory of q-series.Inspired by the two operator identities given by Chen- Liu,in this paper,we first extend the two q-exponential operators defined by Rogers.Then we give some applications of them.Using the operators extended by us,we not only obtain all of formulas proven by the old but can get some natural extensions.At the same time,applying them to study the theory of q-series,some properties are more excellent than the old.For example,we can straightforward to derive the Heine's transformation formula and the terminating and non-terminating Sears' ₃Φ₂ transformation formula from the properties of the q-exponential operators which cannot need the other q-series formulas.Applying the q-exponential operators extended by us,an extension formula of q-Pfaff-Saalsch(??)tz summation formula and an extension formula of Sears' terminating balanced ₄Φ₃ series transformation formula and an extension involving multiple sum about finite Heine's ₂Φ₁ transformation formula are given.We obtain some extensions involving multiple sum about q-Chu-Vandermonde's identities by using the extended operator identities.From these extensions,we get an extension of Dilcher's identity and an extension of Fu-Lascoux's formula.An interesting extension involving multiple sum about the finite Rogers-Fine identity is obtained by applying our operators.In addition,we also use it to give some interesting properties of the homogeneous Al-Salam and Carlitz polynomials defined by us.Such as,the generating function and the q-Mehler formula of this polynomials.As applications,we also give some formal extension formulas of the basic hypergeometric series.Such as,formal extension of Jackson's ₂Φ₂ transformation formula, formal extension of three terms of Sears' ₃Φ₂ transformation formula and formal extension of Askey-Roy integral formula and so on.At last,in this paper,we also give some other interesting applications of the q-differential operator.