相关论文: Power sums in hyperbolic Pascal triangles
We calculate the Jacobian matrix of the dihedral angles of a generalized hyperbolic tetrahedron as functions of edge lengths and find the complete set of symmetries of this matrix.
In this note, we shall provide several properties of hypergeometric Bernoulli numbers and polynomials, including sums of products identity, differential equations and recurrence formulas.
The problem of finding a nonzero solution of a linear recurrence $Ly = 0$ with polynomial coefficients where $y$ has the form of a definite hypergeometric sum, related to the Inverse Creative Telescoping Problem of [14][Sec. 8], has now…
We present a method for obtaining congruences modulo powers of 3 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Motzkin numbers, Riordan numbers, Schr\"oder numbers,…
This paper is devoted to establishing four types of sharp capacitary inequalities within the hyperbolic space as detailed in Theorems 2.1-3.1-4.1-5.1.
Powers of Fibonacci polynomials are expressed as single sums, improving on a double sum recently seen in the literature.
Euler noted the relation $6^3=3^3+4^3+5^3$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and…
Using the $q$-Wilf--Zeilberger method and a $q$-analogue of a "divergent" Ramanujan-type supercongruence, we give several $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. One of them is a $q$-analogue of a…
In this paper, we obtain analytical solutions of Laplace transform based some generalized class of the hyperbolic integrals in terms of hypergeometric functions ${}_3F_2 (\pm1)$, ${}_4F_3 (\pm1)$, ${}_5F_4(\pm1)$, ${}_6F_5(\pm1)$,…
In parameter slices of quadratic rational functions, we identify arcs represented by matings of quadratic polynomials. These arcs are on the boundaries of hyperbolic components.
The compact hyperbolic triangle group $\Delta(p,q,r)$ admits a canonical representation to $\mathrm{PSL}_2(\mathbf{R})$ with discrete image which is unique up to conjugation. The trace field of this representation is \[K =…
The rank two Jacobi algebra $\mathcal{J}_2$ is used to provide an interpretation of the two-variable Jacobi polynomials $J_{n,k}^{(a,b,c)}(x,y)$ on the triangle, as overlaps between two representation bases. The subalgebra structure of…
Methods of summation of power series relevant to applications in quantum theory are reviewed, with particular attention to expansions in powers of the coupling constant and in inverse powers of an energy variable. Alternatives to the Borel…
We define a $q$-linear path in a hypergraph $H$ as a sequence $(e_1,\ldots,e_L)$ of edges of $H$ such that $|e_i \cap e_{i+1}| \in [\![1,q]\!]$ and $e_i \cap e_j=\varnothing$ if $|i-j|>1$. In this paper, we study the connected components…
We compute the exterior powers, with respect to the additive convolution on the general linear Lie algebra, of a parabolic Springer sheaf corresponding to a maximal parabolic subgroup of type (1, n -- 1). They turn out to be isomorphic to…
We investigate interconnected aspects of hyperderivatives of polynomials over finite fields, q-th powers of polynomials, and specializations of Vandermonde matrices. We construct formulas for Carlitz multiplication coefficients using…
Let $f(n)$ be an arithmetic function with $f(n) \ll n^\alpha$ for some $\alpha\in[0,1)$ and let $\lfloor .\rfloor $ denote the integer part function. In this paper, we evaluate asymptotically the sums $$\sum_{n_{1}n_{2}\leq x}f \left(…
The main purpose of this note is to provide an elementary discussion of some simple triangles of integer numbers in particular through their connections with representation theory of $sl_2$. The triangles under consideration are the Catalan…
The main thrust of present note is a volume formula for hyperbolic surface bundle with the fundamental group G. The novelty consists in a purely algebraic approach to the above problem. Initially, we concentrate on the Baum-Connes morphism…
This work contains different expressions for the k'th derivative of the n'th power of the trigonometric and hyperbolic sine and cosine. The first set of expressions follow from the complex definitions of the trigonometric and hyperbolic…