Results and conjectures on log-concavity and zeros of hypergeometric and basic hypergeometric functions

In the talk we will discuss log-concavity for generic power series with coefficients involving gamma and q-gamma functions with respect to the variable contained in their arguments. The motivating example of such series are hypergeometric and basic hypergeometric series for which we can prove more than for the generic case. We show how log-concavity with respect to the simultaneous shift of all parameters implies Laguerre inequalities and relate it to the questions of belongingness of the generalized hypergeometric function to the Laguerre-Polya class. We further show the connection to certain classes of polynomials constructed from arbitrary real sequences in terms of the rising factorials of the argument. We prove a coefficient-wise positivity statement for one type of such polynomials and propose several conjectures about their stability and zeros under condition that the initial sequence is a Polya frequency sequence of certain order.