We will emphasize the algebraic methods of saito, sturmfels, and takayama to construct hypergeometric series and the connection with deformation techniques in commutative algebra. Solutions of hypergeometric differential equations article pdf available in journal of applied mathematics april 2014 with 3,514 reads how we measure reads. Acceleration of generalized hypergeometric functions. Hyperbolic schwarz map of the confluent hypergeometric differential equation saji, kentaro, sasaki, takeshi, and yoshida, masaaki, journal of the mathematical society of japan, 2009. Ordinary differential equationsfrobenius solution to the. Equation 1 has a regular singularity at the origin and an irregular singularity at infinity. Pdf second order differential equations with hypergeometric. The ratio between and is very important so to known toward which kind of result we are going to. In this course we will study multivariate hypergeometric functions in the sense of gelfand, kapranov, and zelevinsky gkz systems.
Hypergeometric functions reading problems introduction the hypergeometric function fa. Solution of differential equations of hypergeometric type. Kummers 24 solutions of the hypergeometric differential. 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. Exact solutions ordinary differential equations secondorder linear ordinary differential equations pdf version of this page. These functions generalize the classical hypergeometric functions of gauss, horn, appell, and lauricella. Cambridge university press 0521833574 basic hypergeometric series, second edition george gasper and mizan rahman excerpt more information. Finding all hypergeometric solutions of linear differential equations marko petkoviek department of mathematics university of ljubljana slovenia. The curves joining any two neighbouring black and white vertices are.
Hypergeometric differential equations in several variables. 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. The cyclic vector theorem see for example 21, theorem 5. If, and are all nonintegers, the general solution for the hypergeometric differential equation is. M elements called successes l elements called failures a sample of n elements are selected at random without replacement. On some solutions of the extended confluent hypergeometric. This is analogous to the derivation of the confluent hypergeometric equation. There are several common standard forms of confluent hypergeometric functions. Im hoping theres a nice way of using the series to rederive the differential equation, at least for thinking purposes.
Onedimensional stratonovich differential equations martin, jaime. The partial differential equations are related to the schrodinger equation for a system of three charged particles. In the following we solve the secondorder differential equation called the hypergeometric differential equation using frobenius method, named after ferdinand georg frobenius. 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. Pdf particular solutions of the confluent hypergeometric. Research article solutions of hypergeometric differential. 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. Hypergeometric differential equation article about. Ordinary differential equations involving power functions. Pdf solutions of hypergeometric differential equations.
If we replace m n by p, then we get ex np and vx n n n 1 np1 p. Hypergeometric solutions of linear differential equations. Particular solutions of the confluent hypergeometric differential. The ratio m n is the proportion of ss in the population. 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. Three lectures on hypergeometric functions eduardo cattani abstract. The main part of the paper is devoted to the confluent hypergeometric differential equation. Monodromy groups of hypergeometric functions satisfying algebraic equations kato, mitsuo and noumi, masatoshi, tohoku mathematical journal, 2003. A brief introduction to these concepts is also available in reference 3. Hypergeometric function differential equation mathematics. Hence, by solving the hypergeometric differential equation, one may directly compare its solutions to get the solutions of legendres differential equation, after. At each singularity, we find 8 solutions corresponding to the different cases for parameters and modified our solutions accordingly. Gaussian hypergeometric function solving second order di erential equations in terms of bessel functions are nished by debeerst, ruben 2007 and yuan, quan 2012. The hypergeometric differential equation is a prototype.
Contiguous relations and the elliptic hypergeometric equation 266 7. Every secondorder linear ode with three regular singular points can be transformed into this. Hypergeometric equation encyclopedia of mathematics. Solutions to the hypergeometric differential equation are built out of the hypergeometric series 2 f 1 a,b. Fuchsian differential equations hypergeometric function references. 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. A linear differential equation of the second order with three regular. This constant is built into the measure dsin order to prevent. Ordinary linear differential equations note that if we replace y by sy in the system, where s. On some differential transformations of hypergeometric equations. Gauss hypergeometric equation ghe has 3 regular singularities at 0, 1. In this study, the discrete fractional nabla calculus operator is used to investigate the k.
Gray, birkhouser, 2000 for the contributions of euler, pfa. Solutions to the hypergeometric differential equation are built out of the hypergeometric series. So it is a special case of the riemann differential equation. 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 systems of differential equations and. 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.
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. Any second order differential equation with three regular singular points can be converted to the hypergeometric differential equation by a change of variables. Pdf the generalized hypergeometric difference equation. Hypergeometric distribution proposition the mean and variance of the hypergeometric rv x having pmf hx. It is a solution of a secondorder linear ordinary differential equation ode. Journal of multivariate analysis 2, 332338 1972 partial differential equations for hypergeometric functions of two argument matrices a. It is the startig of a book i intend to write on 1variable hypergeometric functions. Such surfaces have generically singularities, since any closed nonsigular. Hypergeometric equation article about hypergeometric.
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. For instance, legendres differential equation can be shown to be a special case of the hypergeometric differential equation. Frits beukers is a professor of mathematics at utrecht university. Hypergeometric differential equations, of arbitrary order, provide some key examples of. Otherwise the function is called a generalized hypergeometric function. In mathematics, the gaussian or ordinary hypergeometric function 2 f 1 a,b. Secondorder linear ordinary differential equations 2. Introduction the aim of this article is to define and to compute explicit forms of the voros co. 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. A hypergeometric function can be expressed in terms of gamma functions. Asymptotics of polynomial solutions of a class of generalized lame differential equations. Frobenius solution to the hypergeometric equation wikipedia. Equation 25 is secondorder differential equation that will be reduced to hypergeometric differential equation type. Cambridge university press 0521833574 basic hypergeometric.
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. This paper considers the solution of differential equations of hypergeometric type. In this thesis we focus on a class of equations that can be solved in terms of hypergeometric functions. One can show that a function is hypergeometric if and only if in the representation fz x k. We solve the secondorder linear differential equation called the hypergeometric differential equation by using frobenius method around all its regular singularities. 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. The mathieu equation, for example, yields a second order. Regular singular point at 0, 1 and such that both at 0 and 1 one of the exponents equals 0. Hypergeometric differential equation masayuki noro, takeshi sasaki, kotaro yamada, and masaaki yoshida abstract. Hypergeometric functions in several variables, arising from connection problems for meromorphic differential equations. This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. The hypergeometric equation has been generalized to a system of partial differential equations with regular singularities such that the.
Our aim is to present solutions of the form i 1 a,ipx 1. 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. Reduction formulae for generalised hypergeometric functions of one variable j e gottschalk and e n maslen. Finding all hypergeometric solutions of linear differential. For example, atici and eloe introduced in 6 the discrete laplace. Combining this work with the 2descent algorithm from 6.
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. Hypergeometric solutions of second order linear differential. Identities for the gamma and hypergeometric functions. Of course, the combination can be very complicated, but it is important that we. Pdf let l be a second order linear homogeneous differential equation with rational. The goal in this paper is to solve l in terms of hypergeometric function 2f1a,b. Example 1 let u1 1 0 u2 1 1 and consider the system of equations. Euler also found the hypergeometric equation, which is the secondorder linear di. The hypergeometric equation is a differential equation with three regular singular points cf.
465 798 700 946 1200 547 1164 1084 348 38 1390 376 1209 533 251 893 1169 190 1382 308 1398 361 427 1368 840 1082 74 1059 499 1116 870 983 1085 528 1445 887 1363 1030 1461 1155 867 128 196 907