Abstract
As in chapter 7 we have the concept of duality introduced in definition 3.1. In chapter 11 we obtained a q-difference equation of the form (cf. (11.1.2))
with N∈{1,2,3,…} or N→∞, where
with
where e,f,g,α *,β *∈ℝ, q>0, q≠1 and ε≠0. If the regularity condition (11.2.4) holds all eigenvalues \(\lambda_{n}^{*}\) are different. This implies by using theorem 3.7 that there exists a sequence of dual polynomials. In this case we have
with ω=0 and x 0=1=q −0. Furthermore we have by using (11.2.2)
if we choose c=−1 in (11.2.1).
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Koekoek, R., Lesky, P.A., Swarttouw, R.F. (2010). Orthogonal Polynomial Solutions in q −x+uq x of Real q-Difference Equations. In: Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-05014-5_12
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DOI: https://doi.org/10.1007/978-3-642-05014-5_12
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