Higher Monotonicity Properties of strongqstrong-gamma and -psi Functions.pdfVIP

Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 2, pp. 247–259 (2013) /adsa Higher Monotonicity Properties of -gamma and -psi Functions Mourad E. H. Ismail University of Central Florida Department of Mathematics Orlando, FL 32816, USA and King Saud University, Riyadh, Saudi Arabia mourad.eh.ismail@ Martin E. Muldoon York University Department of Mathematics Statistics Toronto ON M3J 1P3, Canada muldoon@yorku.ca To the memory of Panayiotis D. Siafarikas Abstract Some typical results on complete monotonicity of a function involving the gamma and -gamma functions assert that is completely monotonic for a certain range of values of a parameter, and − is completely monotonic for another range, leaving a gap–an interval of parameter values where neither nor − is com- pletely monotonic. We study this question for some examples involving -gamma functions. In all cases considered here, the width of the gap decreases to as decreases from to . AMS Subject Classifications: 33D05. Keywords: q-gamma function, q-psi function, complete monotonicity, infinite divisi- bility. Received October 31, 2012; Accepted November 6, 2012 Communicated by Andrea Laforgia 248 Mourad E. H. Ismail and Martin E. Muldoon 1 Introduction and Preliminaries Recall that a function f is completely monotonic on an interval I if − f x ≥ for n , , . . . on I