Q-Series from a Contemporary Perspective.

Intro -- Contents -- Preface -- The mathematical contributions of Richard Askey -- Curriculum vitae of Richard A. Askey -- Reformulations of a partition theorem of Göllnitz and q-series identities -- Schur's theorem, partitions with odd parts and the Al-Salam-Carlitz polynomials -- Singularity and monodromy of quasi-hypergeometric functions -- Incomplete elliptic integrals in Ramanujan's lost notebook -- Measure algebras associated with orthogonal polynomials -- Word straightening and q-Eulerian calculus -- Path generating transforms -- q-extensions of Erdélyi's fractional integral representations for hypergeometric functions and some summation formulas for double q-Kampé de Fériet series -- Numerical investigation of basic Fourier series -- An identity of Ramanujan, and applications -- Addition theorems for the q-exponential function -- The Schur functions for partitions with complex parts -- On Forrester's generalization of Morris constant term identity -- New combinatorial formula for modified Hall-Littlewood polynomials -- 0. Introduction -- 1. Modified Hall-Littlewood polynomials -- 1.1. Definition -- 1.2. Modified Hall-Littlewood polynomials for partition λ = (1N) -- 1.3. Hall-Littlewood polynomials and characters of the affine Lie algebra sl(n) -- 1.4. Modified Hall-Littlewood polynomials and unipotent flag varieties -- 1.5. Modified Hall-Littlewood polynomials and Demazure characters -- 1.6. Modified Hall-Littlewood polynomials and chains of subgroupsin a finite abelian p-group -- 2. Generalized mahonian statistics -- 2.1. Mahonian statistics on the set M(μ) -- 2.2. Dual mahonian statistics -- 2.3. Generalized mahonian statistics -- 3. Main results -- 3.1. Combinatorial formula for modified Hall-Littlewood polynomials -- 3.2. New combinatorial formula for the transition matrix M(e, P) -- 4. Proofs of Theorems 3.1 and 3.4.

4.1. Proof of Theorem 3.4 -- 4.2. Proof of Theorem 3.1 -- 5. Polynomials Pλμ(t) and their interpretations -- 6. Generalizations of polynomials Pλμ(t) and Kλμ(t) -- 6.1. Crystal Kostka polynomials -- 6.2. Fusion Kostka polynomials -- 6.3. Ribbon Kostka polynomials -- 6.4. Generalized Kostka polynomials -- 6.5. Summary -- 7. Fermionic formulae -- 7.1. Multinomial fermionic formulae for one dimensional sums -- 7.2. Rigged configurations polynomials -- 8. Two parameter deformation of one dimensional sums -- References -- Schur function identities and the number of perfect matchings of Holey Aztec rectangles -- A new U(n) generalization of the Jacobi triple product identity -- A new quantum algebraic interpretation of the Askey-Wilson polynomials -- Some properties of Koornwinder polynomials -- A new multidimensional matrix inversion in Ar.

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