It is of course interesting to derive these solutions for their intrinsic value. So it is a special case of the riemann differential equation. There are three possible ways in which one can characterize hypergeometric functions. This equation has a regular singularity at the origin with indices 0 and 1b, and an irregular singularity at infinity of rank one. Suppose lhas ve regular singularities and at least one of them is logarithmic. Every ordinary differential equation of secondorder with at most three regular singular points can be brought to the hypergeometric differential equation by means of a suitable change of variables. On generalized hypergeometric solutions of firstorder linear di. The equation has two linearly independent solutions at each of the three regular singular points, and. Rational transformations of confluent hypergeometric. Secondorder linear ordinary differential equations. We classify pullback transformations of the conuent hypergeometric equations, because the linear equations. Home page exact solutions methods software education about this site math forums. Exact solutions ordinary differential equations secondorder linear ordinary differential.
On modular forms arising from a differential equation of hypergeometric type masanobu kaneko and masao koike x1. If a differential equation or a system of equations with a singular point at z 0 has a basis of solutions with components in c. The q hypergeometric equation 7 can be rewritten as c abqxu q2x. Ordinary differential equationsfrobenius solution to the. The outcome of the above threepart recipe is a system of four equations in four unknowns that emerge from the method, which, when solved and combined with the rest of the analysis, leads to the general solution of 1 which has, as a special case, a rodrigues formula format. Solution of differential equations of hypergeometric type 3 the riemannliouville fractional derivative, d. Although there is no complete algorithm which can nd closed form solution of every second order di erential equation, there are algorithms to treat some classes of di erential equations. Algebraic aspects of hypergeometric differential equations 9 the series approach to solving di. Hypergeometric solutions of second order linear differential. In mathematics, the gaussian or ordinary hypergeometric function 2 f 1 a,b. Solving the confluent hypergeometric differential equation using the method of frobenius. This equation was found by euler 1 and was studied extensively by gauss 2, kummer 3, 4, and riemann 5. Finding all hypergeometric solutions of linear differential equations marko petkoviek department of mathematics university of ljubljana slovenia. Monodromy for the hypergeometric function hypergeometric differential equation in general.
Spectral properties of solutions of hypergeometric type differential equations. Decomposition formulas for hypergeometric functions of. The irregularity of qdifference equations are studied using the newton polygons by j. Hypergeometric solutions of second order linear di erential equations with five singularities. A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it wolfram koepf.
Solution of some integral equations involving confluent k. We note that riemanns functions and the fundamental solutions of the degenerate second order partial differential equations are expressible by means of hypergeometric functions of several variables 6. Monodromy for the hypergeometric function solution of this problem was the primary goal of this paper. In 4, we applied the arguments in 1 to the solution of the homogeneous hypergeometric equation. Finding all hypergeometric solutions of linear differential. Frobenius solution to the hypergeometric equation wikipedia. Research article solutions of hypergeometric differential equations. The hypergeometric equation is a secondorder differential equation with three regular singular points. The pochhammer symbol is defined as and, for, where. 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. Solution of some integral equations involving confluent k hypergeometric functions author. It can be regarded as the limiting form of the hypergeometric differential equation 15. The principle aim of this research article is to investigate the properties of kfractional integration introduced and defined by mubeen and habibullah 1,and secondly to solve the integral equation of the form, for k 0.
When talking about differential equations, the term order is commonly used for the degree of. We first investigate the solution, singular at z 1, and in w 2 and w 4 we give integral representations of it. On the rodrigues formula solution of the hypergeometric. Gauss hypergeometric equation is ubiquitous in mathematical physics as many wellknown partial di. Solutions of hypergeometric differential equations shahid mubeen, mammona naz, abdur rehman, and gauhar rahman d e p a r t m e n to fm a t h e m a t i c s,u n i v e r s i t yo fs a r g o d h. The confluent hypergeometric equation, which is close to equation 1. Hypergeometric series and differential equations 1. Then hence, hypergeometric differential equation 7 takes the form since, the solution of the hypergeometric differential equation at is the same as the solution for this equation at.
Identities for the gamma and hypergeometric functions. Hypergeometric functions with special arguments reduce to elementary functions, for example. If a fundamental solution is known for 7r2, the relationship may permit the construction of a fundamental solution for rrx. Derivatives of are given by magnus and oberhettinger 1949, p. The method is characterized by using the mellin transform to convert the original differential equation into a complex difference equation solving the differential. This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. But if the order of the differential equation is higher than two this is not the case. What about equations of orders 3, 4, what about systems of equations. The complete solution to the hypergeometric differential equation is the hypergeometric series is convergent for arbitrary, and for real, and for if. A generic polynomial solution for the differential equation. If we can get a short list which contains all solutions, we can then test out each one and throw out the invalid ones. However, it turned out that without too much effort one could also describe the differential galois group of the hypergeometric differential equation in general. We assume that lhas no liouvillian solutions, otherwise. Pdf spectral properties of solutions of hypergeometric.
The main part of the paper is devoted to the confluent hypergeometric differential equation. Series solution of legendres differential equation in hindi. Hypergeometric equation encyclopedia of mathematics. The solutions of hypergeometric differential equation include many of the most interesting. The hypergeometric differential equation is a prototype. At each singularity, we find 8 solutions corresponding to the different cases for parameters and modified our solutions accordingly. The hypergeometric function is a solution of eulers hypergeometric differential equation. We solve the secondorder linear differential equation called the hypergeometric differential equation by using frobenius method around all its.
He showed that the linear equation can be transformed to the gauss hypergeometric equation for three, four and six divided points of picard s solutions. Solution of inhomogeneous differential equations with. Pdf we solve the secondorder linear differential equation called the ii. Solution of differential equations of hypergeometric type. We solve the secondorder linear differential equation called the ii hypergeometric differential equation by using frobenius method around all its regular singularities. 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. Solutions to the hypergeometric differential equation are built out of the hypergeometric series.
The solution of eulers hypergeometric differential equation is called hypergeometric function or gaussian function introduced by gauss. Hypergeometric solutions of linear differential equations. Series solutions to ordinary differential equations. Series solutions to odes revised and extended by m. 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 functions of several variables are used in physical and quantum chemical applications as well cf. Introduction hardly there is a necessity to speak about importance of properties of hypergeometric functions for any scientist and the engineer dealing with practical application of differential equations. Using a linear fractional transformation, we can place the three singularities at 0, 1, and.
At the end we present some glimpses into the general hypergeometric differential equation as well. The hypergeometric equation has been generalized to a system of partial differential equations with regular singularities such that the appell or lauricella hypergeometric function in several variables is a solution, cf. This equation was found by euler and was studied extensively by gauss, kummer 3, 4, and riemann. Initially this document started as an informal introduction to gauss. Hypergeometric function differential equation mathematics. An important element in the proof of the above results is a theorem of levelt, which gives a simple algebraic characterisation of the monodromy group of a hypergeometric differential equation le. Research article solutions of k hypergeometric differential equations article pdf available in journal of applied mathematics january 2014 with 74 reads how we measure reads. Im hoping theres a nice way of using the series to rederive the differential equation, at least for thinking purposes.
Pdf research article solutions of k hypergeometric. Kummers 24 solutions of the hypergeometric differential. The solutions of hypergeometric differential equation include many of the most interesting special functions of mathematical physics. In 1, we discussed it in terms of the aclaplace transform. Frobenius series solution of fuchs secondorder ordinary. The reader who is not familiar with algorithm hyper is encouraged to read this section focusing on the case m 1,in order to get a feeling of how this algorithm works. Degenerate hypergeometric equation exact solutions keywords. Exact solutions ordinary differential equations secondorder linear ordinary differential equations. Differential equations i department of mathematics. Second order differential equations with hypergeometric. Then the solution space in l of a linear equation of order n is a cvector space of dimension at most n.
In that paper, kummers 8 solutions of kummers differential equation are obtained by using the method which is adopted in the present paper to obtain the solutions of the hypergeometric differential equation. Gauss hypergeometric function frits beukers october 10, 2009 abstract we give a basic introduction to the properties of gauss hypergeometric functions, with an emphasis on the determination of the monodromy group of the gaussian hyperegeometric equation. We study connection problems for linear qdifference equations with irregular singular points. Notes on differential equations and hypergeometric functions not. Solution of some integral equations involving confluent khypergeometric functions author.
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. The general solution of a wide class of differential equations can be written in terms of hypergeometric functions f. Jul 08, 2003 the hypergeometric equation is a secondorder differential equation with three regular singular points. But the solution at z 0 is identical to the solution we obtained for the point x 0, if we replace each. Series solution of some special differential equations 1 hypergeometric.
Hypergeometric solutions of second order linear di. In fact, meijer himself studied the hypergeometric differential equation more general than 1. Apparently, equation 19, with 20 in mind, is of the same class as equation 1, so that equation 19 comes within the compass of the rodrigues formula solution methodology and we can write down a rodrigues formula solution to equation 19. In section2, we present the formulas in distribution theory, which are given in the book of zemanian 5, section 6. Solutions of hypergeometric differential equations hindawi. Hypergeometric solutions of second order linear di erential. On the rodrigues formula solution of the hypergeometrictype. It is a solution of a secondorder linear ordinary differential equation ode. Although there is no complete algorithm which can nd closed form solution of every second.
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