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We introduce variants of Regge finite element metrics with enhanced properties of the trace. In particular the trace operator is surjective to a finite element space of continuous functions. Multiplying these scalar functions by the…
Source identities are fundamental identities between multivariable special functions. We give a geometric derivation of rational and trigonometric source identities. We also give a systematic derivation and extension of various determinant…
Generalized trigonometric functions and generalized hyperbolic functions can be converted to each other by the duality formulas previously discovered by the authors. In this paper, we apply the duality formulas to prove dual pairs of…
Several structural properties of a universal algebra can be seen from the higher commutators of its congruences. Even on a finite algebra, the sequence of higher commutator operations is an infinite object. In the present paper, we exhibit…
The aim of this paper is to offer an algebraic definition of infinite determinants of finite potent endomorphisms using linear algebra techniques. It generalizes Grothendieck's determinant for finite rank endomorphisms and is equivalent to…
A unifying scheme of classical special functions of hypergeometric type obeying orthogonality or biorthogonality relations is described. It expands the Askey scheme of classical orthogonal polynomials and its $q$-analogue based on the…
In this work, simple exact results are presented for summations in two-particle potential with long-range interactions. Polygamma function is used to evaluate summations. Results are found when a periodic media is consider. Periodic…
For a hyperelliptic curve of genus $g$, it is well known that the symmetric products of $g$ points on the curve are expressed in terms of their Abel-Jacobi image by the hyperelliptic sigma function (Jacobi inversion formulae). Matsutani and…
New formulas for the construction of Pythagorean triples and generalizations to equations of higher powers. Application of formulas to some problems, in particular Fermat's equation with n=4.
We review "creative" strategies of closed-form evaluations of hypergeometric functions.
The paper is a survey of recent results in analysis of additive functions over function fields motivated by applications to various classes of special functions including Thakur's hypergeometric function. We consider basic notions and…
We give a survey on the theory of representation-finite and certain minimal representation-infinite algebras.The main goals are the existence of multiplicative bases and of coverings with good properties. Both are attained via…
We derive the general supertrace formula for a system with $N$ chiral superfields and one nilpotent chiral superfield in global and local supersymmetry. The nilpotent multiplet is realized by taking the scalar-decoupling limit of a chiral…
We study the set of algebraic numbers of bounded height and bounded degree where an analytic transcendental function takes algebraic values.
Following a previous article we continue our study on non-terminating hypergeometric series with one free parameter, which aims to find arithmetical constraints for a given hypergeometric series to admit a gamma product formula. In this…
In this paper we obtain explicit formulas for the traces of Hecke operators on spaces of cusp forms in certain instances related to arithmetic triangle groups. These expressions are in terms of hypergeometric character sums over finite…
We describe a new approach to the notion of general hypergeometric functions
We provide a unified method to study the adjacency matrices of regular graphs (including infinite ones) using holomorphic functional calculus. By applying this calculus on a specific ellipse that contains the spectrum, we derive an…
We present recent computer algebra methods that support the calculations of (multivariate) series solutions for (certain coupled systems of partial) linear differential equations. The summand of the series solutions may be built by…
We give an analogy of Jacobi's formula, which relates the hypergeometric function with parameters $(1/4,1/4,1)$ and theta constants. By using this analogy and twice formulas of theta constants, we obtain a transformation formula for this…