Related papers: New Laplace transforms for the generalized hyperge…

The Mittag-Leffler type functions arise naturally in the solution of fractional order integral and differential equations, especially in the investigations of the fractional generalization of the kinetic equation. This article introduces a…

It is shown that analytically soluble bound states of the Schr\"odinger equation for a large class of systems relevant to atomic and molecular physics can be obtained by means of the Laplace transform of the confluent hypergeometric…

After reviewing some fundamental facts from the theory of theta hypergeometric series we derive, using indefinite summation, several summation, transformation, and expansion formulas for multibasic theta hypergeometric series. Some of the…

A celebrated theorem of Klein implies that any hypergeometric differential equation with algebraic solutions is a pull-back of one of the few standard hypergeometric equations with algebraic solutions. The most interesting cases are…

We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;\lambda)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \lambda^k where p is a prime and m, s, r are positive integers. These…

In this paper, two new classes of convex functions as a generalization of convexity which is called (h-s)_{1,2}-convex functions are given. We also prove some Hadamard-type inequalities and applications to the special means are given.

In this paper, we introduce a new function, the multiple confluent hypergeometric functions, and establish a functional equation for the $r$-variable Euler--Zagier multiple zeta functions using it. In the case when $r=2$, this functional…

We propose a conjecture extending the classical construction of elliptic units to complex cubic number fields $K$. The conjecture concerns special values of the elliptic gamma function, a holomorphic function of three complex variables…

Using a ${}_7F_6$ hypergeometric transformation formula, we prove two supercongruences. In particular, one of these supercongruences confirms a recent conjecture of Guo, Liu and Schlosser, and gives an extension of a supercongruence of Long…

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 obtain new inequalities for certain hypergeometric functions. Using these inequalities, we deduce estimates for the hyperbolic metric and the induced distance function on a certain canonical hyperbolic plane domain.

We derive asymptotic expansions of the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed. For both functions we consider $b/a\le 1$ and $b/a\ge 1$, with special attention for the case…

We introduce a transformation of linear Pfaffian systems, which we call the middle Laplace transform, as a formulation of the Laplace transform from the perspective of Katz theory. While the definition of the middle Laplace transform is…

We study a class of duality transformations in generalised Z(2) gauge theories and Ising models on two- and three-dimensional compact lattices. The theories are interpreted algebraically in terms of the structure constants of a…

In this paper, we begin by applying the Laplace transform to derive closed forms for several challenging integrals that seem nearly impossible to evaluate. By utilizing the solution to the Pythagorean equation $a^2 + b^2 = c^2$, these…

We introduce new hypergeometric series expansions of the solutions to the general Heun equation. The form of the Gauss hypergeometric functions used as expansion function differs from that used before. We derive three such expansions and…

We derive new functional renormalisation group flows for quantum gravity, in any dimension. The key new achievement is that the equations apply for any theory of gravity whose underlying Lagrangian $\sim f(R_{\mu\nu\rho\sigma})$ is a…

In a recent paper, Rathie and Pogany established thirty two novel and general reductions of two and three variables generalized hypergeometric functions. In this paper we provide twenty four further novel and general reduction formulas. The…

For certain class of hypergeometric functions ${}_3F_2$ with rational parameters, we give a sufficient condition for the special value at $1$ to be expressed in terms of logarithms of algebraic numbers. We give two proofs, both of which are…

In recent work, the author, in collaboration with Allen, Long, and Tu, developed the Explicit Hypergeometric Modularity Method (EHMM), which establishes the modularity of a large class of hypergeometric Galois representations in dimensions…