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Abstract
We deal with monic orthogonal polynomial sequences (MOPSs), , satisfying the three-term recurrence relation
,
, with initial conditions
and
, where
and
for all
. These sequences are characterized by the relation
, where
is a MOPS. In this paper, we show that the sequence
is
-semiclassical if and only if the sequence
is
-semiclassical. Then, we express the characteristic elements of the
-semiclassical sequence
, such as the q-Pearson equation satisfied by the corresponding linear functional, the class of the linear functional, the first-order linear q-difference equation satisfied by the Stieltjes function and the coefficients of the structure relation for such a sequence of polynomials, in terms of the characteristic elements of the sequence,
. In particular, if the sequence
is
-semiclassical of class zero, then we obtain a new non-symmetric
-semiclassical sequence of polynomials
of class s = 1.
Acknowledgements
The work of the first author (BB) has been supported by Quassim University, grant SR-D-010-316. The work of the second author (FM) has been supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain, grant MTM2009-12740-C03-01. Finally, we thank the referees for their suggestions and remarks, which contributed to improve the presentation of the manuscript.
Notes
1. Email: [email protected]; [email protected]