In many problems arising in physical sciences and statistics, hypergeometric functions, in logarithmic cases, are applicable, which is known from the monograph of Mathai and Haubold [4] and Mathai [3], etc. A detailed account of such applications is available from the two monographs by Mathai and Saxena [5], [6]. Expansions of the Gauss’s hypergeometric functions in logarithmic cases are given ...

Eigenvalue distributions are important dynamical quantities in matrix models, and it is a challenging problem to derive them tensor models. In this paper, we consider real symmetric order-three tensors with Gaussian as the simplest case, an explicit formula for signed of eigenvectors: Each eigenvector contributes distribution by ±1, depending on sign determinant associated Hessian matrix. The e...

Many of Ramanujan’s ideas and theorems form the seeds of questions and problems, many of which remain unresolved or even to be thoroughly examined. This survey raises questions arising from Ramanujan’s work on theta-functions and other q-series, with Gaussian hypergeometric functions making frequent appearances.

In this note our aim is to establish a Turán type inequality for Gaussian hypergeometric functions. This result completes the earlier result that G. Gasper proved for Jacobi polynomials. Moreover, at the end of this note we present some open problems.

In this paper, we derive some branched continued fraction representations for the ratios of Horn's confluent function $\mathrm{H}_6.$ The method employed is a two-dimensional generalization classical constructing Gaussian fraction. We establish estimates rate convergence expansions in region $\Omega$ (here, domain (open connected set) together with all, part or none its boundary). It also prove...

There is a two-component log-gas system with Boltzmann factor which provides an interpolation between the eigenvalue probability density function (PDF) for β=1 and β=4 invariant random matrix ensembles. Its solvability relies on construction of particular skew orthogonal polynomials, inner product linear combination products, each involving weight functions. For suitably related classical funct...

The paper classifies algebraic transformations of Gauss hypergeometric functions and pull-back transformations between hypergeometric differential equations. This classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of the Gauss hypergeometric function.