Hypergeometric Solutions Research Articles

Examining the Mathematica algorithm for general Heun function calculation: a comparative analysis

Abstract We investigate the numerical calculation of the general Heun equation using Wolfram Mathematica’s functions, comparing the numerical solutions with hypergeometric and explicit solutions. This exploration sheds light on the efficacy and accuracy of the numerical algorithm implemented in Mathematica for computing Heun functions.

Hypergeometric solutions to Schwarzian equations

Hypergeometric solutions to Schwarzian equations

Linear stability analysis of micropolar nanofluid flow across the accelerated surface with inclined magnetic field

PurposeThe purpose of this paper is to study the two-dimensional micropolar fluid flow with conjugate heat transfer and mass transpiration. The considered nanofluid has graphene nanoparticles.Design/methodology/approachGoverning nonlinear partial differential equations are converted to nonlinear ordinary differential equations by similarity transformation. Then, to analyze the flow, the authors derive the dual solutions to the flow problem. Biot number and radiation effect are included in the energy equation. The momentum equation was solved by using boundary conditions, and the temperature equation solved by using hypergeometric series solutions. Nusselt numbers and skin friction coefficients are calculated as functions of the Reynolds number. Further, the problem is governed by other parameters, namely, the magnetic parameter, radiation parameter, Prandtl number and mass transpiration. Graphene nanofluids have shown promising thermal conductivity enhancements due to the high thermal conductivity of graphene and have a wide range of applications affecting the thermal boundary layer and serve as coolants and thermal management systems in electronics or as heat transfer fluids in various industrial processes.FindingsResults show that increasing the magnetic field decreases the momentum and increases thermal radiation. The heat source/sink parameter increases the thermal boundary layer. Increasing the volume fraction decreases the velocity profile and increases the temperature. Increasing the Eringen parameter increases the momentum of the fluid flow. Applications are found in the extrusion of polymer sheets, films and sheets, the manufacturing of plastic wires, the fabrication of fibers and the growth of crystals, among others. Heat sources/sinks are commonly used in electronic devices to transfer the heat generated by high-power semiconductor devices such as power transistors and optoelectronics such as lasers and light-emitting diodes to a fluid medium, thermal radiation on the fluid flow used in spectroscopy to study the properties of materials and also used in thermal imaging to capture and display the infrared radiation emitted by objects.Originality/valueMicropolar fluid flow across stretching/shrinking surfaces is examined. Biot number and radiation effects are included in the energy equation. An increase in the volume fraction decreases the momentum boundary layer thickness. Nusselt numbers and skin friction coefficients are presented versus Reynolds numbers. A dual solution is obtained for a shrinking surface.

Solutions of the sl2${\mathfrak {sl}_2}$qKZ equations modulo an integer

AbstractWe study the qKZ difference equations with values in the th tensor power of the vector representation , variables , and integer step . For any integer relatively prime to the step , we construct a family of polynomials in variables with values in such that the coordinates of these polynomials with respect to the standard basis of are polynomials with integer coefficients. We show that satisfy the qKZ equations modulo . Polynomials are modulo analogs of the hypergeometric solutions of the qKZ given in the form of multidimensional Barnes integrals.

The sensitivity demonstration and propagation of hyper-geometric soliton waves in plasma physics of Kairat-II equation

This study investigates the Kairat-II equation, describing optical pulse behavior in optical fibers and plasma. To uncover new solitary wave profiles, the study employs an extended direct algebraic method. This kind of solution has never been reached in research prior to this study. This innovative approach efficiently encompasses a comprehensive set of thirty-seven solitonic wave profiles, spanning various soliton families. The investigation unveils novel solitonic wave patterns, including plane solutions, hyper-geometric solutions, mixed hyperbolic solutions, periodic and mixed periodic solutions, mixed trigonometric solutions, trigonometric solutions, shock solutions, mixed shock singular solutions, mixed singular solutions, complex solitary shock solutions, singular solutions, and shock wave solutions. To demonstrate the pulse propagation characteristics, the research presents 2-D, 3-D, and contour graphics based on parameter values, aiding in a better understanding of the phenomenon.

Identities involving special functions from hypergeometric solution of algebraic equations

Identities involving special functions from hypergeometric solution of algebraic equations

Ladder symmetries of black holes and de Sitter space: love numbers and quasinormal modes

In this note, we present a synopsis of geometric symmetries for (spin 0) perturbations around (4D) black holes and de Sitter space. For black holes, we focus on static perturbations, for which the (exact) geometric symmetries have the group structure of SO(1,3). The generators consist of three spatial rotations, and three conformal Killing vectors obeying a special melodic condition. The static perturbation solutions form a unitary (principal series) representation of the group. The recently uncovered ladder symmetries follow from this representation structure; they explain the well-known vanishing of the black hole Love numbers. For dynamical perturbations around de Sitter space, the geometric symmetries are less surprising, following from the SO(1,4) isometry. As is known, the quasinormal solutions form a non-unitary representation of the isometry group. We provide explicit expressions for the ladder operators associated with this representation. In both cases, the ladder structures help connect the boundary condition at the horizon with that at infinity (black hole) or origin (de Sitter space), and they manifest as contiguous relations of the hypergeometric solutions.

GKZ hypergeometric systems of the three-loop vacuum Feynman integrals

We present the Gel’fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems of the Feynman integrals of the three-loop vacuum diagrams with arbitrary masses, basing on Mellin-Barnes representations and Miller’s transformation. The codimension of derived GKZ hypergeometric systems equals the number of independent dimensionless ratios among the virtual masses squared. Through GKZ hypergeometric systems, the analytical hypergeometric series solutions can be obtained in neighborhoods of origin including infinity. The linear independent hypergeometric series solutions whose convergent regions have non-empty intersection can constitute a fundamental solution system in a proper subset of the whole parameter space. The analytical expression of the vacuum integral can be formulated as a linear combination of the corresponding fundamental solution system in certain convergent region.

An MHD Casson fluid flow past a porous stretching sheet with threshold Non-Fourier heat flux model

The present paper investigates the magnetohydrodynamics Casson fluid flow with porous medium. The Cu-water nanofluid is used to investigate further analytical results. Thermal radiation is considered along with the Cattaneo-Christov heat flux model and then solved analytically, then expresses the closed forms analytical solution in terms of two variables of hypergeometric function. The special form of Appell hypergeometric solutions are closed form of analytical solutions. The results of normalized shear stress at the wall and temperature profile, and rate of heat flux can be analyzed with suitable parameters, Viz., relaxation time parameter, Prandtl number, Radiation parameter, Magnetic parameter, and so on. The newness of the current work's novelty is to explain a solution to issues with stretching sheets on the basis of Appell hypergeometric form and also investigation takes place on a time relaxation parameter. The shear stress fηη0 is also calculated analytically. This problem has many industrial applications and engineering processes, namely glass-fiber production, extrusion of rubber sheets and so on. Finding that an increase in the thermal relaxation parameter and Prandtl number keeps the fluid temperature constant is the primary physical application.

Elements of spin Hurwitz theory: closed algebraic formulas, blobbed topological recursion, and a proof of the Giacchetto–Kramer–Lewański conjecture

In this paper, we discuss the properties of the generating functions of spin Hurwitz numbers. In particular, for spin Hurwitz numbers with arbitrary ramification profiles, we construct the weighed sums which are given by Orlov’s hypergeometric solutions of the 2-component BKP hierarchy. We derive the closed algebraic formulas for the correlation functions associated with these tau-functions, and under reasonable analytical assumptions we prove the loop equations (the blobbed topological recursion). Finally, we prove a version of topological recursion for the spin Hurwitz numbers with the spin completed cycles (a generalized version of the Giacchetto–Kramer–Lewański conjecture).

Generalized Brézin–Gross–Witten tau-function as a hypergeometric solution of the BKP hierarchy

In this paper, we prove that the generalized Brézin–Gross– Witten tau-function is a hypergeometric solution of the BKP hierarchy with simple weight generating function. We claim that it describes a spin version of the strictly monotone Hurwitz numbers. A family of the hypergeometric tau-functions of the BKP hierarchy, corresponding to the rational weight generating functions, is investigated. In particular, the cut-and-join operators are constructed, and the explicit description of the BKP Sato Grassmannian points is derived. Representatives of this family can be associated with interesting families of spin Hurwitz numbers including a spin version of the monotone Hurwitz numbers.

On the number of p-hypergeometric solutions of KZ equations

It is known that solutions of the KZ equations can be written in the form of multidimensional hypergeometric integrals. In 2017 in a joint paper of the author with V. Schechtman, the construction of hypergeometric solutions was modified, and solutions of the KZ equations modulo a prime number p were constructed. These solutions modulo p, called the p-hypergeometric solutions, are polynomials with integer coefficients. A general problem is to determine the number of independent p-hypergeometric solutions and understand the meaning of that number. In this paper, we consider the KZ equations associated with the space of singular vectors of weight $$n-2r$$ in the tensor power $$W^{\otimes n}$$ of the vector representation of $$\mathfrak {sl}_2$$ . In this case the hypergeometric solutions of the KZ equations are given by r-dimensional hypergeometric integrals. We consider the module of the corresponding p-hypergeometric solutions, determine its rank, and show that the rank is equal to the dimension of the space of suitable square integrable differential r-forms.

Hyperbolic and trigonometric hypergeometric solutions to the star-star equation

We construct the hyperbolic and trigonometric solutions to the star-star relation via the gauge/YBE correspondence by using the three-dimensional lens partition function and superconformal index for a certain mathcal N=2 supersymmetric gauge dual theories. This correspondence relates supersymmetric gauge theories to exactly solvable models of statistical mechanics. The equality of partition functions for the three-dimensional supersymmetric dual theories can be written as an integral identity for hyperbolic and basic hypergeometric functions.

Symmetry Reductions, Cte Method and Interaction Solutions for Sharma-Tasso-Olver-Burgers Equation

In this paper, the Sharma-Tasso-Olver-Burgers (STOB) system is analyzed by the Lie point symmetry method. The hypergeometric wave solution of the STOB equation is derived by symmetry reductions. In the meantime, the consistent tanh expansion (CTE) method is applied to the STOB equation. An nonauto-Bäcklund (BT) theorem that includes the over-determined equations and the consistent condition is obtained by the CTE method. By using the nonauto-BT theorem, the interactions between one-soliton and the cnoidal wave, and between one-soliton and the multiple resonant soliton solutions, are constructed. The dynamics of these novel interaction solutions are shown both in analytical and graphical forms. The results are potentially useful for explaining ocean phenomena.

Diagonals of Rational Functions: From Differential Algebra to Effective Algebraic Geometry

We show that the results we had previously obtained on diagonals of 9- and 10-parameter families of rational functions in three variables x, y, and z, using creative telescoping, yielding modular forms expressed as pullbacked 2F1 hypergeometric functions, can be obtained much more efficiently by calculating the j-invariant of an elliptic curve canonically associated with the denominator of the rational functions. These results can be drastically generalized by changing the parameters into arbitrary rational functions of the product p=xyz. In other cases where creative telescoping yields pullbacked 2F1 hypergeometric functions, we extend this algebraic geometry approach to other families of rational functions in three or more variables. In particular, we generalize this approach to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, or foliations in elliptic curves. We also extend this approach to rational functions in three variables when the denominator is associated with a genus-two curve such that its Jacobian is a split Jacobian, corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked 2F1 hypergeometric solutions, because the denominator corresponds to an algebraic variety that has a selected elliptic curve.

Symbolic Conversion of Holonomic Functions to Hypergeometric Type Power Series

A term $a_n$ is $m$-fold hypergeometric, for a given positive integer $m$, if the ratio $a_{n+m}/a_n$ is a rational function over a field $K$ of characteristic zero. We establish the structure of holonomic recurrence equation, i.e. linear and homogeneous recurrence equations having polynomial coefficients, that have $m$-fold hypergeometric term solutions over $K$, for any positive integer $m$. Consequently, we describe an algorithm, say $mfoldHyper$, that extends van Hoeij's algorithm (1998) which computes a basis of the subspace of hypergeometric $(m=1)$ term solutions of holonomic recurrence equations to the more general case of $m$-fold hypergeometric terms. We generalize the concept of hypergeometric type power series introduced by Koepf (1992), by considering linear combinations of Laurent-Puiseux series whose coefficients are $m$-fold hypergeometric terms. Thus thanks to $mfoldHyper$, we deduce a complete procedure to compute these power series; indeed, it turns out that every linear combination of power series with $m$-fold hypergeometric term coefficients, for finitely many values of $m$, is detected. On the other hand, we investigate an algorithm to represent power series of non-holonomic functions. The algorithm follows the same steps of Koepf's algorithm, but instead of seeking holonomic differential equations, quadratic differential equations are computed and the Cauchy product rule is used to deduce recurrence equations for the power series coefficients. This algorithm defines a normal function that yields together with enough initial values normal forms for many power series of non-holonomic functions. Therefore, non-trivial identities are automatically proved using this approach. This paper is accompanied by implementations in the Computer Algebra Systems (CAS) Maxima 5.44.0 and Maple 2019.

The Differential Equations of Gravity-free Double Pendulum: Lauricella Hypergeometric Solutions and Their Inversion

This paper solves in closed form the system of ODEs ruling the 2D motion of a gravity free double pendulum (GFDP), not subjected to any force. In such a way its movement is governed by the initial conditions only. The relevant strongly non linear ODEs have been put back to hyperelliptic quadratures which, through the Integral Representation Theorem (IRT), are driven to the Lauricella hypergeometric functions.
 We compute time laws and trajectories of both point masses forming the GFDP in explicit closed form. Suitable sample problems are carried out in order to prove the method effectiveness.

The $${{\mathbb {F}}}_p$$-Selberg Integral

We prove an $${{\mathbb {F}}}_p$$ -Selberg integral formula, in which the $${{\mathbb {F}}}_p$$ -Selberg integral is an element of the finite field $${{\mathbb {F}}}_p$$ with odd prime number p of elements. The formula is motivated by the analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo p.

The hypergeometric function, the confluent hypergeometric function and WKB solutions

Relations between the hypergeometric function with a large parameter and Borel sums of WKB solutions of the hypergeometric differential equation with the large parameter are established. The confluent hypergeometric function is also investigated from the viewpoint of exact WKB analysis. As applications, asymptotic expansion formulas for those classical special functions with respect to parameters are obtained.

Intersection numbers on {overline{M}}_{g,n} and BKP hierarchy

In their recent inspiring paper, Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar description for the Brézin-Gross-Witten tau-function. Moreover, we identify both tau-functions of the KdV hierarchy, which describe intersection numbers on the moduli spaces of punctured Riemann surfaces, with the hypergeometric solutions of the BKP hierarchy.