Hypergeometric equation encyclopedia of mathematics. Differential equations department of mathematics, hkust. Im hoping theres a nice way of using the series to rederive the differential equation, at least for thinking purposes. We solve the secondorder linear differential equation called the ii hypergeometric differential equation by. The purpose of this paper is to obtain differential equations and the hypergeometric forms of the fibonacci and the lucas polynomials. On the rodrigues formula solution of the hypergeometric. Hypergeometric equation an overview sciencedirect topics. A linear differential equation of the second order with three regular. In mathematics, the gaussian or ordinary hypergeometric function 2 f 1 a,b. In the following we solve the secondorder differential equation called the hypergeometric differential equation using frobenius method, named after ferdinand georg frobenius. Similar algorithms concerning differential equations are considered. Hypergeometric series and differential equations 1. We solve the secondorder linear differential equation called the hypergeometric differential equation by using frobenius method.
This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. Specifically, the transformations considered apply to hypergeometric solutions of hypergeometric differential equations with the local exponent differences 1 k, 1. Generalized euler integrals and a hypergeometric functions. All web surfers are welcome to download these notes, watch the youtube videos, and to use the notes and videos freely for teaching and learning. This paper presents explicit algebraic transformations of some gauss hypergeometric functions. The secondorder linear hypergeometric differential equation and the hypergeometric function play a central role in many areas of mathematics and physics. Shahid mubeen, mammona naz, abdur rehman, and gauhar. It is well known that asymptotic expansions for a function can in many cases be derived using a differential equation satisfied by the function see e. An equivalent theory of hyperexponential integration due to almkvist and zeilberger completes the book. Kummers 24 solutions of the hypergeometric differential. At each singularity, we find 8 solutions corresponding to the different cases for parameters and modified our solutions accordingly.
Pdf solutions of hypergeometric differential equations. Solutions to the hypergeometric differential equation are built out of the hypergeometric series. We present a method for solving the classical linear ordinary differential equations of hypergeometric type 8, including bessels equation, legendres equation, and others with polynomial coefficients of a certain type. A difference equation analogue of the generalized hypergeometric differential equation is defined, its contiguous. The solution of eulers hypergeometric differential equation is called. Ordinary linear differential equations note that if we replace y by sy in the system, where s. More precisely, properties of the hypergeometric and gegenbauer equation can be derived from generalized symmetries of the laplace equation in 4, resp.
The theory is presented in an entirely selfcontained way, starting with an introduction of the solution of the. Solution of differential equations of hypergeometric type. Transformations of some gauss hypergeometric functions. The author thanks richard askey and simon ruijsenaars for many helpful recommendations. Hyperbolic schwarz map of the confluent hypergeometric differential equation saji, kentaro, sasaki, takeshi, and yoshida, masaaki, journal of the mathematical society of japan, 2009 on the stability of the linear delay differential and difference equations ashyralyev, a. However, since, ja kj b k, so it follows that the radius of convergence of wz p a kzk cannot be less than r. Solutions of hypergeometric differential equations. On some solutions of the extended confluent hypergeometric. In this paper we discuss its generalization to the case. We show that the schr\odinger and kleingordon equations can both be derived from an hypergeometric differential equation. This shows that the radius of convergence of p b krk is r.
Erdelyi 4, and so, with this in mind, a study of differential equations satisfied by certain hypergeometric functions certainly seems justified. In this thesis we focus on a class of equations that can be solved in terms of hypergeometric functions. There are many omissions, some of which are rectified elsewhere in the literature. Regular singular point at 0, 1 and such that both at 0 and 1 one of the exponents equals 0. The method is characterized by using the mellin transform to convert the original differential equation into a complex difference equation solving the differential.
The same applies to non linear generalizations of these equations. Solutions of hypergeometric differential equations hindawi. The systematic investigation of contour integrals satisfying the system of partial differential equations associated with appells hypergeometric function f 1. Second order differential equations special functions. Solving second order di erential equations in terms of bessel functions are nished by debeerst, ruben 2007 and yuan, quan 2012. The combination of these results gives orthogonal polynomials and hypergeometric and q hypergeometric special functions a solid algorithmic foundation. In ggz, gzk 1, gzk 21 we have constructed a basis in the. He shows that these may be written in the form h y s tz t dt, 3 g t being a solution of a hypergeometric differential equation of the order n1. The hypergeometric function is a solution of the hypergeometric differential equation, and is known to be expressed in terms of the riemannliouville fractional derivative fd 1, p. This chapter is based in part on chapter 15 of abramowitz and stegun by fritz oberhettinger. The generalized hypergeometric difference equation. Hypergeometric differential equations semantic scholar. In the following we solve the secondorder differential equation called the hypergeometric.
In my joint papers with iwaki and koike ikot1, ikot2 we found an intriguing relation between the voros coefficients in the exact wkb analysis and the free energy in the topological recursion introduced by eynard and orantin in the case of the confluent family of the gauss hypergeometric differential equations. A hypergeometric function can be expressed in terms of gamma functions. It is a solution of a secondorder linear ordinary differential equation ode. Equation 1 has a regular singularity at the origin and an irregular singularity at infinity. Hypergeometric functions reading problems introduction the hypergeometric function fa, b. Frobenius solution to the hypergeometric equation wikipedia. The systematic investigation of contour integrals satisfying the system of partial differential equations associated with appells hypergeometric function f 1 leads to new solutions of that system. Hypergeometric functions of two variables project euclid. Unfortunately lebedev plugs in a series solution to the given hypergeometric differential equation, whereas id like to use the hypergeometric series as a means of deriving the differential equation. Second order differential equations presents a classical piece of theory concerning hypergeometric special functions as solutions of secondorder linear differential equations. Pdf particular solutions of the confluent hypergeometric.
Pdf the generalized hypergeometric difference equation. Solutions of khypergeometric differential equations. The book contains several classical and modern methods for the. This is usually the method we use for complicated ordinary differential equations. We call it the a hypergeometric system, and its solutions the a hypergeometric functions. Every secondorder linear ode with three regular singular points can be transformed into this. Ordinary and partial differential equation by md raisinghania pdf download. Differential equations play a relevant role in many disciplines and provide powerful tools for analysis and modeling in applied sciences. Differential equations for engineers click to view a promotional video.
We present a method for solving the classical linear ordinary differential equations of hypergeometric type 8, including bessels equation, legendres equation, and others with polynomial coe. The mathieu equation, for example, yields a second order difference equation, which is not solvable by the proposed method. In this work we trace a brief history of the development of the gamma and hypergeometric functions, illustrate the close relationship between them and present a range of their most useful properties and identities, from the earliest ones to those developed in more recent years. Distributional solutions of the hypergeometric differential equation. Extended confluent hypergeometric differential equation. Research article solutions of hypergeometric differential. The hypergeometric equation has been generalized to a system of partial differential equations with regular singularities such that the. It is the startig of a book i intend to write on 1variable hypergeometric functions. The hypergeometric equation is a differential equation with three regular singular points cf. We solve the secondorder linear differential equation called the hypergeometric differential equation by using frobenius method around all its regular singularities. Hypergeometric functions and their applications james b. Initially this document started as an informal introduction to gauss hypergeometric functions for those who want to have a quick idea of some main facts on hypergeometric functions.
The procedure followed in most texts on these topics e. The method produces, tout court, the general solution of these equations in the form of a combination of a standard rodrigues formula and a generalized rodrigues. The hypergeometrictype differential equation is a secondorder homogeneous differential. Hypergeometric solutions of linear differential equations. Solutions to the hypergeometric differential equation are built out of the. Identities for the gamma and hypergeometric functions. We show that properties of hypergeometric class equations and functions become transparent if we derive them from appropriate 2nd order differential equations with constant coefficients. The leading example, in the theory of linear ordinary differential equations with regular singular points of one complex variable, is doubtless the hypergeometrie differential equation. So it is a special case of the riemann differential equation.
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