Related papers: Identities between Appell's and hypergeometric fun…
This paper gives a natural extension of Frobenius-Stickelberger formula and Kiepert formula to Abelian functions for "Purely Trigonal Curves", especially, of degree four. A description on the theory of Abelian functions for general trigonal…
Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions,…
In this survey, we explore how superorthogonality amongst functions in a sequence $f_1,f_2,f_3,\ldots$ results in direct or converse inequalities for an associated square function. We distinguish between three main types of…
In this paper we determine the upper bounds of $|H_{2}(3)|$ for the inverse functions of functions of some classes of univalent functions, where $H_{2}(3)(f)=a_{3}a_{5}-a_{4}^{2}$ is the Hankel determinant of a special type.
In this paper we continue investigation of the hypergeometric function ${}_4F_3(1)$ as the function of its seven parameters. We deduce several reduction formulas for this function under additional conditions that one of the top parameters…
In solving $q$-difference equations, and in the definition of $q$-special functions, we encounter formal power series in which the $n$th coefficient is of size $q^{-\binom{n}{2}}$ with $q\in(0,1)$ fixed. To make sense of these formal…
We introduce an $\ell$-adic analogue of Gauss's hypergeometric function arising from the Galois action on the fundamental torsor of the projective line minus three points. Its definition is motivated by a relation between the KZ-equation…
The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to…
We determine closed and compact expressions for the epsilon-expansion of certain Gaussian hypergeometric functions expanded around half-integer values by explicitly solving for their recurrence relations. This epsilon-expansion is…
In this work we describe a construction that leads to an explicit solution of the problem of differentiation of hyperelliptic functions. A classical genus $g=1$ example of such a solution is a result of F.G.Frobenius and L.Stickelberger.…
The Riccati equations reducible to first-order linear equations by an appropriate change the dependent variable are singled out. All these equations are integrable by quadrature. A wide class of linear ordinary differential equations…
Ultrafunctions are a particular class of functions defined on a non-Archimedean field. They provide generalized solutions to functional equations which do not have any solutions among the real functions or the distributions. In this paper…
Although links between values of finite field hypergeometric functions and eigenvalues of elliptic modular forms are well known, we establish in this paper that there are also connections to eigenvalues of Siegel modular forms of higher…
This paper is devoted to the proof Gauss' divergence theorem in the framework of "ultrafunctions". They are a new kind of generalized functions, which have been introduced recently [2] and developed in [4], [5] and [6]. Their peculiarity is…
We consider the generalized eigenvalue problem for the classical Euler differential equation and demonstrate its intimate connection with Meijer's $G$-functions. In the course of deriving the solution of the generalized Euler eigenvalue…
This paper studies derivatives with respect to the parameters of Srivastava triple hypergeometric functions HA, HB and HC. Using basic properties of the Gamma function and Pochhammer symbols, we obtain explicit formulas for first and higher…
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…
We aim to study Mittag-Leffler type functions of two variables ${{D}_{1}}\left( x,y \right),...,{{D}_{5}}\left( x,y \right)$ by analogy with the Appell hypergeometric functions of two variables. Moreover, we targeted functions…
In investigation of boundary-value problems for certain partial differential equations arising in applied mathematics, we often need to study the solution of system of partial differential equations satisfied by hypergeometric functions and…
Recently, many researchers devoted their attention to study the extensions of the gamma and beta functions. In the present work, we focus on investigating some approximations for a class of Gauss hypergeometric functions by exploiting…