arXiv subjects
Publications and source records attributed to Gennadiy Ilyuta.
We prove the simultaneous multiplication formulas for Apostol-Bernoulli polynomials and generalized Frobenius-Euler polynomials. These formulas contain Dedekind-Rademacher sums, Apostol-Dedekind sums and Fourier-Dedekind sums.
We connect Dedekind sums and Alexander polynomials of torus knots.
We connect Dedekind sums and some formulas for numerical semigroups.
We generalize Gassert-Shor formula for numerical semigroups.
Harer-Zagier generating functions for Euler characteristics of moduli spaces of curves contain $n$-necklace polynomials. Taylor expansions for these polynomials depend on numbers of solutions of Cohen semilinear congruences.
We prove several formulas for the distribution of positive roots.
We study Saito duality and Fourier-Ramanujan transform for power sums and multiplicities of monodromy roots.
Growth functions of Coxeter groups and the Poincare series of Kleinian and Fuchsian singularities are $2$-tangle $q$-fractions.
Twisted Markov traces and irreducible characters of the Hecke algebra are dual.
We prove some formulas relating the inverse of a Cartan matrix with algebraic and geometric invariants of finite group representations.
We use Newton divided differences for calculation of Greene sums -- the rational functions determined by linear extensions of partially ordered sets. Identities for Greene sums generate relations for Newton divided differences and Arnold differential forms. Also generalizations of the Newton interpolation series which are indexed by sequences of partially ordered sets are received.
For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued fractions, Christoffel-Darboux identity, combinatorial formula. Known results about the factorization of the Alexander-Conway polynomial permit to connect Poincare series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. One uses reconstractions of A'Campo links and Coxeter links. There is formulated a conjecture connecting obtained formulae with Poincare series of ring of functions on singularities through results of S. M. Gusein-Zade, F. Delgado, and A. Campillo.