Proceedings of OPSFA12, 2013

The 12th International Symposium on Orthogonal Polynomials, Special Functions and Applications OPSFA12 was held in Sousse, Tunisia, from 25 to 29 March 2013. The conference was co-organized by The Faculty of Science of Tunis and the University of Gabès and was financially supported by the Ministry of Higher Education and Scientific Research in Tunisia, the Tunisian Mathematical Society (SMT), the Society for Industrial and Applied Mathematics (SIAM), the French Institute and the Paul Painlevé Laboratory.

The Organizing Committee included A. Fitouhi (El Manar University), M. J. Atia (Gabès University), L. Khériji (El Manar University), M. Mejri (Gabès University), I. Ben Salah (Monastir University), A. Safraoui (El Manar University), H. Elmonser (Carthage University) and S. Kouki (El Manar University).

The Scientific Committee consisted of A. Aptekarev (Russia), A. M. Finkelshtein (Spain), F. Marcellán (Spain), B. Beckermann (France), A. Fitouhi (Tunisia), P. Maroni (France), C. Berg (Denmark), M. Ismail (USA), E. B. Saff (USA), C. F. Dunkl (USA), E. Koelink (The Netherlands), W. Van Assche (Belgium) and T. H. Koornwinder (The Netherlands).

In total, there were 200 participants from 32 countries: Algeria, Austria, Belgium, Bénin, Brazil, Burundi, Cameroon, Canada, Denmark, Egypt, France, Germany, India, Iran, Italy, Mexico, Morocco, Nigeria, Poland, Portugal, Romania, Russia, Saudi Arabia, Slovak Republic, South Africa, South Korea, Spain, Sweden, The Netherlands, Tunisia, Turkey and USA. They delivered 10 plenary lectures (of 50 min), and, in 3 parallel sections, 60 communications (of 25 min). A special session honouring Professor Ahmed Fitouhi, full Professor at the Faculty of Science of Tunis, for his contribution in the large domain of Harmonic Analysis and Special Functions and q-analogues, also held. Moreover, the Gábor Szegö price was awarded to Jacob Stordal Christiansen (Sweden) and a problem session was organized during the conference.

The proceeding of OPSFA12 is published in this special issue of Integral Transforms and Special Functions. All papers have been reviewed and revised. On this occasion, we express our sincere gratitude to all referees. The articles in this special issue cover a variety of topics in special functions and orthogonal polynomials. The first paper investigates the zeros of a generalized Bessel function defined in terms of a ₀F_d -hypergeometric series. The second paper deals with the positivity of a generalized translation kernel related to the q-Hankel transform which is based on a type of q-Bessel function. The third paper presents the solution to the problem (of Chebotarev) of minimizing the logarithmic capacity of a set containing three arbitrary points or four points in special position. The key device in the proof is the use of elliptic functions. The last paper shows how Hermite, Laguerre and Jacobi polynomials can be used to construct systems of orthogonal polynomials in two variables. The techniques are illustrated in both the ordinary ℝ² -coordinate and the ℂ -coordinate systems,

The main concern of the paper ‘On the zeros of the hyper-Bessel function’ by H. Chaggara and N. Ben Rondhame is the generalized Bessel function with d parameters given by The authors prove that the zeros of the function are all of the form where x>0 and k=0, 1, 2, … , d.

In the paper ‘ Positivity of the generalized translation associated with the q-Hankel transform and applications’ by L. Dhaouadi it is shown that the kernel is nonnegative for all when 0<q<1 and ν>−1, where d_qt denotes the Jackson integral and the q-Bessel function a basic hypergeometric series. This result is then used to establish several results related to the harmonic analysis of the q-Hankel transform.

In ‘Zolotarev's conformal mapping and Chebotarev's problem’ K. Schiefermayr uses the (complex analysis) methods of conformal mapping and Jacobian elliptic functions to analyse and solve problems of finding sets containing given points and having minimal logarithmic capacity. The three-point construction relies on solving transcendental equations involving the elliptic functions . The theory is illustrated by several numerical examples. The problem is also solved for the symmetric four-point case with .

In ‘Complex versus real orthogonal polynomials of two variables’ Y. Xu illustrates the fundamental ideas of orthogonal bases, reproducing kernels, matrix recurrence relations and polynomials of classical type for the two-variable situation. The recurrence relations and the reproducing kernel (methods due to the author in papers published in the 1990s) are presented in both coordinate systems, in the general setting. The theory is illustrated for the Hermite polynomials orthogonal for on ℝ², the Zernike disk polynomials orthogonal for on the unit disk, and a family of polynomials due to Koornwinder which are orthogonal for a measure defined on the interior of Steiner's hypocycloid.