There are several common standard forms of confluent hypergeometric functions. Gaussian hypergeometric function solving second order di erential equations in terms of bessel functions are nished by debeerst, ruben 2007 and yuan, quan 2012. Otherwise the function is called a generalized hypergeometric function. Hence, by solving the hypergeometric differential equation, one may directly compare its solutions to get the solutions of legendres differential equation, after. Hyperbolic schwarz map of the confluent hypergeometric differential equation saji, kentaro, sasaki, takeshi, and yoshida, masaaki, journal of the mathematical society of japan, 2009. Pdf the generalized hypergeometric difference equation. Three lectures on hypergeometric functions eduardo cattani abstract.
Acceleration of generalized hypergeometric functions. Every secondorder linear ode with three regular singular points can be transformed into this. If we replace m n by p, then we get ex np and vx n n n 1 np1 p. This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. 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. Secondorder linear ordinary differential equations 2. In mathematics, the gaussian or ordinary hypergeometric function 2 f 1 a,b. The ratio between and is very important so to known toward which kind of result we are going to. Hypergeometric functions reading problems introduction the hypergeometric function fa. Some properties of hypergeometric functions emine ozergin submitted to the institute of graduate studies and research in partial fulfillment of the requirements for the degree of doctor of philosophy in applied mathematics and computer science eastern mediterranean university february 2011 gazimagusa, north cyprus.
This is analogous to the derivation of the confluent hypergeometric equation. Any second order differential equation with three regular singular points can be converted to the hypergeometric differential equation by a change of variables. Solutions to the hypergeometric differential equation are built out of the hypergeometric series 2 f 1 a,b. A brief introduction to these concepts is also available in reference 3.
Cambridge university press 0521833574 basic hypergeometric series, second edition george gasper and mizan rahman excerpt more information. At each singularity, we find 8 solutions corresponding to the different cases for parameters and modified our solutions accordingly. Contiguous relations and the elliptic hypergeometric equation 266 7. In this course we will study multivariate hypergeometric functions in the sense of gelfand, kapranov, and zelevinsky gkz systems. We will emphasize the algebraic methods of saito, sturmfels, and takayama to construct hypergeometric series and the connection with deformation techniques in commutative algebra. The hypergeometric differential equation is a prototype. Equation 1 has a regular singularity at the origin and an irregular singularity at infinity. Finding all hypergeometric solutions of linear differential equations marko petkoviek department of mathematics university of ljubljana slovenia.
Kummers 24 solutions of the hypergeometric differential. The partial differential equations are related to the schrodinger equation for a system of three charged particles. We solve the secondorder linear differential equation called the hypergeometric differential equation by using frobenius method around all its regular singularities. The curves joining any two neighbouring black and white vertices are. If, and are all nonintegers, the general solution for the hypergeometric differential equation is. In this thesis we focus on a class of equations that can be solved in terms of hypergeometric functions. These functions generalize the classical hypergeometric functions of gauss, horn, appell, and lauricella. Hypergeometric differential equation article about. Hypergeometric differential equations in several variables. For instance, legendres differential equation can be shown to be a special case of the hypergeometric differential equation. It is a solution of a secondorder linear ordinary differential equation ode.
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. In this study, the discrete fractional nabla calculus operator is used to investigate the k. Hypergeometric distribution proposition the mean and variance of the hypergeometric rv x having pmf hx. Gauss hypergeometric equation ghe has 3 regular singularities at 0, 1. The ratio m n is the proportion of ss in the population. Frits beukers is a professor of mathematics at utrecht university. On some solutions of the extended confluent hypergeometric. Monodromy groups of hypergeometric functions satisfying algebraic equations kato, mitsuo and noumi, masatoshi, tohoku mathematical journal, 2003. Hypergeometric solutions of second order linear differential. Solutions to the hypergeometric differential equation are built out of the hypergeometric series. Hypergeometric systems of differential equations and. Voros coefficients of the confluent hypergeometric differential equations the kummer equation or the whittaker equation with a large parameter are defined and explicit forms of them are obtained.
The cyclic vector theorem see for example 21, theorem 5. This paper considers the solution of differential equations of hypergeometric type. Because the result of a hypergeometric series with a fixed difference are homogenous, that is the closed form are always combinations of the same kind of basic function. Asymptotics of polynomial solutions of a class of generalized lame differential equations. Regular singular point at 0, 1 and such that both at 0 and 1 one of the exponents equals 0. Cambridge university press 0521833574 basic hypergeometric. Research article solutions of hypergeometric differential. This constant is built into the measure dsin order to prevent. 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. Hypergeometric solutions of linear differential equations. Onedimensional stratonovich differential equations martin, jaime. Ordinary differential equationsfrobenius solution to the. Introduction the aim of this article is to define and to compute explicit forms of the voros co.
The main part of the paper is devoted to the confluent hypergeometric differential equation. Frobenius solution to the hypergeometric equation wikipedia. The hypergeometric distribution math 394 we detail a few features of the hypergeometric distribution that are discussed in the book by ross 1 moments let px k m k n. Hypergeometric equation article about hypergeometric. Gray, birkhouser, 2000 for the contributions of euler, pfa. The mathieu equation, for example, yields a second order. 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. Pdf second order differential equations with hypergeometric. Particular solutions of the confluent hypergeometric. Solution of differential equations of hypergeometric type.
In the following we solve the secondorder differential equation called the hypergeometric differential equation using frobenius method, named after ferdinand georg frobenius. Im hoping theres a nice way of using the series to rederive the differential equation, at least for thinking purposes. For example, atici and eloe introduced in 6 the discrete laplace. Exact solutions ordinary differential equations secondorder linear ordinary differential equations pdf version of this page.
Such surfaces have generically singularities, since any closed nonsigular. Ordinary differential equations involving power functions. Our aim is to present solutions of the form i 1 a,ipx 1. Pdf particular solutions of the confluent hypergeometric. The hypergeometric equation is a differential equation with three regular singular points cf. So it is a special case of the riemann differential equation. Journal of multivariate analysis 2, 332338 1972 partial differential equations for hypergeometric functions of two argument matrices a.
A linear differential equation of the second order with three regular. Equation 25 is secondorder differential equation that will be reduced to hypergeometric differential equation type. Finding all hypergeometric solutions of linear differential. More recently, in 24, the authors succeeded to solve the eigenvalue wave equation for an electron in the field of a molecule with an electric dipole moment by expanding the solutions of a. One can show that a function is hypergeometric if and only if in the representation fz x k. Identities for the gamma and hypergeometric functions. Hr 1 2z, we get the application of minimal length in kleingordon equation with hulthen potential using asymptotic iteration method. Extended confluent hypergeometric differential equation.
Hypergeometric equation encyclopedia of mathematics. Ordinary linear differential equations note that if we replace y by sy in the system, where s. A similar generalized coulomb problem for a class of general natanzon confluent potentials is exactly solved in 23 by reducing the corresponding system to confluent hypergeometric differential equations. The hypergeometric equation has been generalized to a system of partial differential equations with regular singularities such that the. Pdf let l be a second order linear homogeneous differential equation with rational. Euler also found the hypergeometric equation, which is the secondorder linear di. In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. Particular solutions of the confluent hypergeometric differential.
It is the startig of a book i intend to write on 1variable hypergeometric functions. A difference equation analogue of the generalized hypergeometric differential equation is defined, its contiguous relations are developed, and its relation to numerous wellknown classical special. Hypergeometric function differential equation mathematics. Fuchsian differential equations hypergeometric function references. Hypergeometric differential equations, of arbitrary order, provide some key examples of. This is usually the method we use for complicated ordinary differential equations. Combining this work with the 2descent algorithm from 6. Reduction formulae for generalised hypergeometric functions of one variable j e gottschalk and e n maslen. Pdf solutions of hypergeometric differential equations. Solutions of hypergeometric differential equations article pdf available in journal of applied mathematics april 2014 with 3,514 reads how we measure reads. Hence, by solving the hypergeometric differential equation, one may directly compare its solutions to get the solutions of legendres differential equation, after making the necessary substitutions. Hypergeometric functions in several variables, arising from connection problems for meromorphic differential equations. Partial differential equations for hypergeometric functions.
705 473 1107 142 877 541 527 1556 900 629 1435 1565 123 371 893 985 1455 1233 1059 1571 1487 1418 722 657 71 569 1275 588 734 664 1190 1178 868 517 226 797