Related papers: Generating Functions for Multi-j-Symbols
A generating function is derived that counts the number of diagrams in an arbitrary scalar field theory. The number of graphs containing any number $n_j$ of $j$-point vertices is given. The count is also used to obtain the number of…
In the paper, 2 explicit formulas for the Euler numbers of the second kind are obtained. Based on those formulas a exponential generating function is deduced. Using the generating function some well-known and new identities for the Euler…
In the present paper, we prove an identity for the generating function of the quadruple zeta values. Taking homogeneous parts on both sides of the identity and substituting appropriate values for the variables, we obtain the sum formula for…
The purpose of this paper is to provide definitions for, and proofs of, the asymptotic formulae given by Edmonds, which relate the 3j and 6j symbols to rotation matrices.
We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences some of which are known to have an alternative interpretation. We…
The hypergeometric distribution is a popular distribution, whose properties have been extensively investigated. Generating functions of this distribution, such as the probability-generating function, the moment-generating function, and the…
We enumerate weighted graphs with a certain upper bound condition. We also compute the generating function of the numbers of these graphs, and prove that it is a rational function. In particular, we show that if the given graph is a…
In this paper, we consider the generating functions of the complete and elementary symmetric functions and provide a new generalization of these classical symmetric functions. Some classical relationships involving the complete and…
We propose a symbolic generative task description language and a corresponding inference engine capable of representing arbitrary multimodal tasks as structured symbolic flows. Unlike conventional generative models that rely on large-scale…
The questions when two Morse function on closed manifolds are conjugated is investigated. Using the handle decompositions of manifolds the condition of conjugation is formulated. For each Morse function on 3-manifold the ordered generalized…
Explicit expressions are found for the $6j$ symbols in symmetric representations of quantum $\mathfrak{su}_N$ through appropriate hypergeometric Askey-Wilson (q-Racah) polynomials. This generalizes the well-known classical formulas for…
We give a generating function for the number of unimodal permutations with a given cycle structure.
The concern of this paper is a famous combinatorial formula known under the name "exponential formula". It occurs quite naturally in many contexts (physics, mathematics, computer science). Roughly speaking, it expresses that the exponential…
Quantum 3D R-matrix in the classical (i.e. functional) limit gives a symplectic map of dynamical variables. The corresponding 3D evolution model is considered. An auxiliary problem for it is a system of linear equations playing the role of…
We first present some identities involving the Pochhammer symbol (rising factorial). We also recall and present some new properties of the Jacobi polynomials. We use them to expand a general hypergeometric function in an orthogonal series…
Using generalized hypergeometric functions to perform symbolic manipulation of equations is of great importance to pure and applied scientists. There are in the literature a great number of identities for the Meijer-G function. On the other…
The article studies a class of generalized factorial functions and symbolic product sequences through Jacobi type continued fractions (J-fractions) that formally enumerate the divergent ordinary generating functions of these sequences. The…
We develop a generating-function formulation for the symbolic reduction of multi-loop Feynman integrals. In this framework, integration-by-parts identities are rewritten as differential equations for sector-wise generating functions, so the…
In a recent article a generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal expressions was a key-point to allow to give them a statistical…
3D objects (artefacts) are made to fulfill functions. Designing an object often starts with defining a list of functionalities that it should provide, also known as functional requirements. Today, the design of 3D object models is still a…