Related papers: Several graph sequences as solutions of a double r…
We give some results and conjectures about recurrence relations for certain sequences of binomial sums.
In this note we study a certain graph polynomial arising from a special recursion. This recursion is a member of a family of four recursions where the other three recursions belong to the chromatic polynomial, the modified matching…
Recently, Chu studied some properties of the partial sums of the sequence $P^k(F_n)$, where $P(F_n)=\big(\sum_{i=1}^nF_i\big)_{n\geq1}$ and $(F_n)_{n\geq1}$ is the Fibonacci sequence, and gave its combinatorial interpretation. We generalize…
The paper contains a combinatorial theorem (the sequence of Newton polygons of a reccurent sequence of polynomials is quasi-linear) and two applications of it in classical and quantum topology, namely in the behavior of the $A$-polynomial…
In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher-order…
For a large integer $m,$ we obtain an asymptotic formula for the number of solutions of a certain congruence modulo $m$ with four variables, where the variables belong to special sets of residue classes modulo $m.$ This formula are applied…
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
We classify combinatorial Dyson-Schwinger equations giving a Hopf subalgebra of the Hopf algebra of Feynman graphs of the considered Quantum Field Theory. We first treat single equations with an arbitrary number (eventually infinite) of…
Given a Feynman parameter integral, depending on a single discrete variable $N$ and a real parameter $\epsilon$, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in $\epsilon$. In a…
In this note we investigate the solutions of certain meta-Fibonacci recurrences of the form $f(n)=f(n-f(n-1))+f(n-2)$ for various sets of initial conditions. In the case when $f(n)=1$ for $n\leq 1$, we prove that the resulting integer…
In quantum field theory the path integral is usually formulated in the wave picture, i.e., as a sum over field evolutions. This path integral is difficult to define rigorously because of analytic problems whose resolution may ultimately…
The predictions of the standard model of particle physics are highly successful in spite of the fact that several parts of the underlying quantum field theoretical framework are analytically problematic. Indeed, it has long been suggested,…
The Feynman amplitude associated to a graph is a period of a certain motive. The sum of these motive classes over all connected graphs with no multiple edges or tadpoles and n vertices is defined in the Grothendieck ring of varieties. This…
Modifying an idea of E. Brietzke we give simple proofs for the recurrence relations of some sequences of binomial sums which have previously been obtained by other more complicated methods.
We investigate the possibility of generalizing Gopakumar's microscopic derivation [1] of Witten diagrams in large N free quantum field theory to interacting theories. For simplicity we consider a massless, matrix valued real scalar field…
Windowed recurrences are sliding window calculations where a function is applied iteratively across the window of data, and are ubiquitous throughout the natural, social, and computational sciences. In this monograph we explore the…
We discuss how basic notions of graph theory and associated graph polynomials define questions for algebraic geometry, with an emphasis given to an analysis of the structure of Feynman rules as determined by those graph polynomials as well…
We introduce a generalization of the Stirling numbers via symmetric functions involving two weight functions. The resulting extension unifies previously known Stirling-type sequences with known symmetric function forms, as well as other…
We generalize Gopakumar's microscopic derivation of Witten diagrams in large N free quantum field theory [1] to interacting theories in perturbative expansion. For simplicity we consider a matrix scalar field with $\Phi^h$ interaction in d…
We present a few combinatorial identities which were encountered in our work on the spectral theory of quantum graphs. They establish a new connection between the theory of random matrix ensembles and combinatorics.