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Session S32 - Special functions and orthogonal polynomials

Tuesday, July 13, 17:00 ~ 18:00 UTC-3

Unified construction of all hypergeometric and basic hypergeometric orthogonal polynomial sequences.

Luis Verde-Star

Universidad Autónoma Metropolitana, Iztapalapa, México   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakc7e9536c99ed52f52e98fbc2952e6a5f').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyc7e9536c99ed52f52e98fbc2952e6a5f = 'v&#101;rd&#101;' + '&#64;'; addyc7e9536c99ed52f52e98fbc2952e6a5f = addyc7e9536c99ed52f52e98fbc2952e6a5f + 'x&#97;n&#117;m' + '&#46;' + '&#117;&#97;m' + '&#46;' + 'mx'; var addy_textc7e9536c99ed52f52e98fbc2952e6a5f = 'v&#101;rd&#101;' + '&#64;' + 'x&#97;n&#117;m' + '&#46;' + '&#117;&#97;m' + '&#46;' + 'mx';document.getElementById('cloakc7e9536c99ed52f52e98fbc2952e6a5f').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyc7e9536c99ed52f52e98fbc2952e6a5f + '\'>'+addy_textc7e9536c99ed52f52e98fbc2952e6a5f+'<\/a>';

We present a construction of a class $H$ of polynomial sequences that satisfy a three-term recurrence relation and are eingenfunctions of a generalized difference equation of order one with respect to a Newton basis. The class $H$ contains all the hypergeometric and basic hypergeometric orthogonal polynomial sequences, and the sequences obtained with $q=-1$.

All the polynomial sequences in $H$ are determined by three linearly recurrent sequences of numbers that satisfy a difference equation of order three. Using the initial values of such sequences as parameters we obtain a uniform parametrization of all the families in the class. The parameters also provide alternative descriptions of the Askey and the $q$-Askey schemes.