Permutation statistics and -Fibonacci numbers.
Poset homology of Rees products, and -Eulerian polynomials.
Proof of the refined alternating sign matrix conjecture.
-analogue of a binomial coefficient congruence.
-analogues of the Hermite polynomials. (Les -analogues des polynômes d'Hermite.)
-analogues of two supercongruences of Z.-W. Sun
We give several different -analogues of the following two congruences of Z.-W. Sun: where is an odd prime, is a positive integer, and is the Jacobi symbol. The proofs of them require the use of some curious -series identities, two of which are related to Franklin’s involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012.
-Bernoulli numbers associated with -Stirling numbers.
-exponential families.
-extensions for the Apostol–Genocchi polynomials.
-Genocchi numbers and polynomials associated with -Genocchi-type -functions.
-identities related to overpartitions and divisor functions.
-Riemann zeta function.
-Selberg integrals and Macdonald polynomials
q-difference equations and Ramanujan-Selberg continued fractions
Qseries Maple tutorial: A -product tutorial for a -series Maple package.
Quintuple products and Ramanujan's partition congruence p(11n+6) ≡ 0(mod 11)
Ramanujan's method in -series congruences.
Rook theory, generalized Stirling numbers and -analogues.
Schur's determinants and partition theorems.
Secant tree calculus
A true Tree Calculus is being developed to make a joint study of the two statistics “eoc” (end of minimal chain) and “pom” (parent of maximum leaf) on the set of secant trees. Their joint distribution restricted to the set {eoc-pom ≤ 1} is shown to satisfy two partial difference equation systems, to be symmetric and to be expressed in the form of an explicit three-variable generating function.