Riemann surface, 29 addition theorem for sn(u), 28 additive number theory, 15 algebraic function, 29 balanced hypergeometric series, 18 Bell number, 17, 107 Bernoulli counting scheme, 111 Bernoulli number, 17 Bernoulli polynomial, 17 beta integral, 19 binomial series, 18 branched covering map, 28 branchpoint, 29 Catalan number, 16 Chu-Vandermonde summation formula, 19 commutative ordinary differential operators, 117 complete symmetric polynomial, 12 completely multiplicative, 15 complex structure, 29 conformally equivalent, 29 conjugate partition, 5 contiguous relation, 20 Dedekind sum, 107 digamma function, 21 Dirichlet region, 26 double gamma function, 27 Eisenstein series, 26 elementary symmetric polynomial, 12 elliptic function, 26 elliptic integral, 25 Euler number, 22 Euler product, 15 Euler’s dilogarithm, 19 Euler’s transformation formula, 19 Euler-Maclaurin summation formula, 15, 21 Eulerian number, 17, 107 Fermat measure, 53 Fermat’s last theorem, 27 fractional differentiation, 18 function element, 28 fundamental region, 26 Galois field, 65, 121 Gauss second summation theorem, 21 Gauss summation formula, 20 Gegenbauer polynomial, 51 generalised Stirling number, 16 generalised Stirling polynomial, 16 generalized hypergeometric series, 18 generalized Laguerre polynomial, 22 generalized Vandermonde determinant, 125 Genocchi number, 17

The history of q-calculus and a new method

On characterizations of classical polynomials

On the q -polynomials: a distributed study

q -Classical polynomials and the q -Askey and Nikiforov-Uvarov tableaus

Modified Clebsch-Gordan-type expansions for products of discrete hypergeometric polynomials

For the most general case of hypergeometric type difference equations on uniform and nonuniform lattices the explicit expression of polynomials solutions in terms of generalized q-hypergeometric series is obtained from the difference analog of the Rodrigues formula. From this expression, the formulas for all particular cases are derived by an appropriate choice of parameters and by taking various limits. Basic hypergeometric series are also used [14]. The consideration of these polynomials gives us the classification of corresponding q-polynomials in accordance with the values of parameters q, μ of the lattice , and with the number of zeroes of the function σ(s) that is the coefficient before the senior difference derivative in the equation of hypergeometric type. All earlier introduced q-polynomials are included in our scheme of classification. This review is an expanded version of the report which was made by one of the authors on the VII simposium sobre polinomios ortogonales y aplicationes (23-27 Sept...

Vasily B. Uvarov

Papers

Polynomial solutions of hypergeometric type difference equations and their classification