English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The two linearly independent solutions of the three-term recurrence relationof the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22],are slightly modified so as to make it transparent that these functions satisfya beautiful symmetry property. It essentially means that the geometric and thespectral parameters are interchangeable in these functions. We call theresulting functions the Askey-Wilson functions. Then, we show that by addingbound states (with arbitrary weights) at specific points outside of thecontinuous spectrum of some instances of the Askey-Wilson difference operator,we can generate functions that satisfy a doubly infinite three-term recursionrelation and are also eigenfunctions of $q$-difference operators of arbitraryorders. Our result provides a discrete analogue of the solutions of the purelydifferential version of the bispectral problem that were discovered in thepioneering work [8] of Duistermaat and Gr\"unbaum.Comment: 42 pages, Section 3 moved to the end, minor correction

Askey-Wilson type functions with bound states