Using a general q-series expansion, we derive some nontrivial q-formulas involving many infinite products. A multitude of Hecketype series identities are derived. Some general formulas for sums of any number of squares are given. A new representation for the generating function for sums of three triangular numbers is derived, which is slightly different from that of Andrews, also implies the famous result of Gauss where every integer is the sum of three triangular numbers.

In this paper, we give an extension of the non-terminating 6 f 5 summation using the method of q-exponential operator and difference equation. This extended formula allows us to give a new generating function of the Askey-Wilson polynomials.

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In the present investigation, by using certain higher-order q-derivatives, the authors introduce and investigate several new subclasses of the family of multivalent q-starlike functions in the open unit disk. For each of these newly-defined function classes, several interesting properties and characteristics are systematically derived. These properties and characteristics include (for example) distortion theorems and radius problems. A number of coefficient inequalities and a sufficient condition for functions belonging to the subclasses studied here are also discussed. Relevant connections of the various results presented in this investigation with those in earlier works on this subject are also pointed out.

Motivated by the fact that fractional q-integrals play important roles in numerous areas of mathematical, physical and engineering sciences, it is natural to consider the corresponding iterated fractional q-integrals. The main object of this paper is to define these iterated fractional q-integrals, to build the relations between iterated fractional q-integrals and certain families of generating functions for q-polynomials and to generalize two fractional q-identities which are given in a recent work [Fract. Calc. Appl. Anal. 10 (2007), 359–373]. As applications of the main results presented here, we deduce several bilinear generating functions, Srivastava-Agarwal type generating functions, multilinear generating functions and U(n + 1) type generating functions for the Rajković-Marinković-Stanković polynomials.