New indefinite integrals of Heun functions

We present a conspicuous number of indefinite integrals involving Heun functions and their products obtained by means of the Lagrangian formulation of a general homogeneous linear ordinary differential equation. As a by-product we also derive new indefinite integrals involving the Gauss hypergeometric function and products of hypergeometric functions with elliptic functions of the first kind. All integrals we obtained cannot be computed using Maple and Mathematica.     

Analytic continuation of the Appell function <Emphasis Type="Italic">F</Emphasis><Subscript>1</Subscript> and integration of the associated system of equations in the logarithmic case

The Appell function F ₁ (i.e., a generalized hypergeometric function of two complex variables) and a corresponding system of partial differential equations are considered in the logarithmic case when the parameters of F ₁ are related in a special way. Formulas for the analytic continuation of F ₁ beyond the unit bicircle are constructed in which F ₁ is determined by a double hypergeometric series. For the indicated system of equations, a collection of canonical solutions are presented that are two-dimensional analogues of Kummer solutions well known in the theory of the classical Gauss hypergeometric equation. In the logarithmic case, the canonical solutions are written as generalized hypergeometric series of new form. The continuation formulas are derived using representations of F ₁ in the form of Barnes contour integrals. The resulting formulas make it possible to efficiently calculate the Appell function in the entire range of its variables. The results of this work find a number of applications, including the problem of parameters of the Schwarz–Christoffel integral.     

Generalized Abel type integral equations with localized fractional integrals and derivatives

Generalized Abel type integral equations with Gauss, Kummer's and Humbert's confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.     

广义分数积分的复合与Gauss超几何函数积分的估计

In this paper, composition formulas for generalized fractional integral oper-ators involving Gauss hypergeometric function are applied to evaluating of de finite integrals involving two Gauss hypergeometric functions.     

Elliptic hypergeometric functions and Calogero-Sutherland-type models

We consider an elliptic analogue of the Gauss hypergeometric function and two of its multivariate generalizations. We describe their relation to elliptic beta integrals, the exceptional Weyl group E₇, the elliptic hypergeometric equation, and Calogero-Sutherland-type models. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 311–324, February, 2007.     

A generalization of Clausen’s identity

The paper gives an extension of Clausen’s identity to the square of any Gauss hypergeometric function. Accordingly, solutions of the related third-order linear differential equation are found in terms of certain bivariate series that can reduce to ₃F₂ series similar to those in Clausen’s identity. The general contiguous variation of Clausen’s identity is found as well. The related Chaundy’s identity is generalized without any restriction on the parameters of the Gauss hypergeometric function. The special case of dihedral Gauss hypergeometric functions is underscored.     

Interior-boundary value problem with Saigo operators for the gellerstedt equation

For an equation of mixed elliptic-parabolic type, we consider an interior-boundary value problem in which the Dirichlet condition is posed on the elliptic part of the boundary and a point condition relating generalized derivatives and fractional integrals with the Gauss hypergeometric function of the values of the solution on the characteristics to the values of the solution and its derivative on the parabolic degeneration line is posed on the hyperbolic part.     

Erdélyi-type integrals for generalized hypergeometric functions with integral parameter differences

In this paper, we extend one of Erdélyi's classical integrals for the Gauss hypergeometric functions to a special class of generalized hypergeometric functions in which certain pairs of numerator and denomiator parameters differ by positive integers. Our main results are achieved by applying the fractional integration by parts and series manipulation technique. Some special cases are also pointed out which includes a new extension of a Thomae-type transformation.     

Some Examples of RS 2 3(3)-Transformations of Ranks 5 and 6 as the Higher Order Transformations for the Hypergeometric Function

A combination of rational mappings and Schlesinger transformations for a matrix form of the hypergeometric equation is used to construct higher order transformations for the Gauss hypergeometric function.     

On Relationships Between Classical Pearson Distributions and Gauss Hypergeometric Function

In this paper, it is shown that the classical Pearson distributions and Gauss hypergeometric function satisfy a unique differential equation of hypergeometric type. Hence, they are directly related to each other. This connection leads to some new integral relations between them. For instance, two special cases of Pearson distributions, namely the generalized T distribution and Beta distribution, are considered and their direct relationships with Gauss hypergeometric function are obtained.     

Summation formulae involving the Laguerre polynomial

A general result involving the generalized hypergeometric function is deduced by the elementary manipulation of series. Kummer's first theorem for the confluent hypergeometric function and two summation formulae for the Gauss hypergeometric function are then applied and new summation formulae involving the Laguerre polynomial are deduced.     

Some properties of generalized hypergeometric coefficients

A generalization to several variables of the Gauss hypergeometric series has been given in [13]. Defining generalized hypergeometric coefficients as Schur function transforms of this series, we develop here new properties and relations possessed by these coefficients. An integral representation of the generalized hypergeometric series is developed and application to q-analog series indicated.     

Numerical methods for the computation of the confluent and Gauss hypergeometric functions

The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss–Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide “roadmaps” with our recommendation for which methods should be used in each situation.     

Solution of a second-order linear differential equation with polynomial coefficients and Fuchsian point at zero

We specify the structure of the power series determining a solution of a Fuchsian second-order differential equation with polynomial coefficients in a neighborhood of zero. The power series is represented via hypergeometric functions of fractional order. The structure of the coefficients of the series is clarified.     

A new approach to hypergeometric transformation formulas

The Ramanujan Journal - We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi’s canonical form of the...     

Logarithmic concavity of series in gamma ratios

We find two-sided bounds and prove non-negativeness of Taylor coefficients for the Turán determinants of power series with coefficients involving the ratio of gamma-functions. We consider these series as functions of simultaneous shifts of the arguments of the gamma-functions located in the numerator and the denominator. The results are then applied to derive new inequalities for the Gauss hypergeometric function, the incomplete normalized beta-function and the generalized hypergeometric series. This communication continues the research of various authors who investigated logarithmic convexity and concavity of hypergeometric functions in parameters.     

A New Integral Involving the Product of Bessel Functions and Associated Laguerre Polynomials

A new result for integrals involving the product of Bessel functions and Associated Laguerre polynomials is obtained in terms of the hypergeometric function. Some special cases of the general integral lead to interesting finite and infinite series representations of hypergeometric functions.     

New fractional integral formulas and kinetic model associated with the hypergeometric superhyperbolic sine function

In the paper, we consider some fractional integral formulas in terms of the Riemann–Liouville, Erdélyi–Kober type, and Weyl fractional integral operators and present the general fractional kinetic model involving the hypergeometric superhyperbolic sine function via the Gauss hypergeometric series.     

Boundary-Value Problem With Saigo Operators for Mixed Type Equation With Fractional Derivative

We set up and solve a non-local problem for a differential equation, which contains the diffusion equation of fractional order. The boundary condition contains a linear combination of generalized operators with the Gauss hypergeometric function in the kernel. For various values of parameters of these operators we write a solution in explicit form.