Convexity results for the largest zero and functions involving the largest zero of q-associated polynomials

Abstract

We prove the convexity of the largest zero of the q-Lommel, the associated AL-Salam Carlitz II and the q-associated Laguerre polynomials as well as the convexity of products of certain functions with the largest zero of the q-associated Laguerre polynomials and associated Al-Salam–Carlitz II polynomials. Moreover, as a consequence of our results concerning the q-associated Laguerre polynomials, we find a recent result regarding the convexity of the function (1/(α + 1))x _{n , 1}(α), where x _{n , 1}(α) is the largest zero of the classical Laguerre

polynomials. The method we use is a functional analytic one based on the three-term recurrence relations that the q-associated polynomials satisfy. By use of this method, the proofs of our results are straightforward.

Keywords:

• Convexity
• q-Associated Laguerre polynomials
• Associated Al-Salam–Carlitz II polynomials
• q-Lommel polynomials

Acknowledgements

Eugenia N. Petropoulou was supported by a Postdoc Grant from the National Foundation of Scholarships (I.K.Y.). Ioannis D. Stabolas was partially supported by the National Foundation of Scholarships (I.K.Y.).