New connection formulae for theq-orthogonal polynomials via a series expansion of theq-exponential

Abstract

Using a realization of the q-exponential function as an infinite multiplicative series of the ordinary exponential functions we obtain new nonlinear connection formulae of the q-orthogonal polynomials such as q-Hermite, q-Laguerre and q-Gegenbauer polynomials in terms of their respective classical analogs.

The Effects of Nonextensivity for a General Driven Dissipative Oscillator in the Coherent States

“…Furthermore, it plays a key role in the mathematical framework for nonextensive dynamical systems described by q ‐orthogonal polynomials such as q ‐Laguerre, q ‐Gegenbauer, and q ‐Hermite polynomials. [ 30 ] In fact, q ‐exponential functions emerge not only in natural systems but also in artificial and social systems with good agreement with empirical phenomena. The system is super‐extensive when and sub‐extensive in the opposite case, .…”

New connection formulae for someq-orthogonal polynomials inq-Askey scheme

“…Furthermore, in the reference [7], the multiplicative series form of the q-exponential were exploited to derive a new nonlinear connection formula between q-orthogonal polynomials and their classical versions, namely q-Hermite, q-Laguerre and q-Gegenbauer polynomials. Their results are expressed in compact form and some explicit examples are given.…”

“…Their results are expressed in compact form and some explicit examples are given. Also, the authors of [7] emphasized the possibility to extend their work for other q-orthogonal polynomials such as little q-Jacobi ones. In the present work we will take benefit of their idea to compute the connection formula between other q-orthogonal polynomials, appearing in the q-Askey scheme [5], and their classical counterparts, namely the continuous q-Laguerre, the continuous big q-Hermite and the q-Meixner-Pollaczek polynomials and we give an alternative connection formula for the q-Gegenbauer polynomials distinct from the one given in [7].…”

“…To proceed, similarly to the work of Chakrabarti et al [7], we first give the generating functions of any q-polynomials cited above and use, on one side, the series development.…”

“…In our knowledge, these cases have not been treated before. To proceed, similarly to the work of Chakrabarti et al [7], we first give the generating functions of any q-polynomials cited above and use, on the one hand, the series development. On the other hand, the Quesne formulae allow us to express the q-exponential function as a product series of the classical exponential function.…”