Olvers confluent hypergeometric function, the ratio of the circumference of a circle to its diameter, e. A hypergeometric solution is a solution u for which r. The 15 gauss contiguous relations for 2f1 hypergeometric series im. Recently, wimp 5 derived explicit recursion formulae for a certain class of hypergeometric functions. The idea is to nd a recurrence relation with respect to j, for the coe cients a. How to solve recurrence relations by the generalized. In this course we will study multivariate hypergeometric functions in the sense of gelfand, kapranov, and zelevinsky gkz systems.
Applications to wave functions of certain discrete system are also given. Three lectures on hypergeometric functions eduardo cattani abstract. Sometimes, however, from the generating function you will. Some recurrence relations for the generalized basic hypergeometric functions.
Series, expansion of the hypergeometric functions and. Recursion formulae for generalized hypergeometric functions1 in. The whittaker function mkmz is defined by 1 z m kmz z e mca. Special cases of these lead to recurrence relations for the.
This report presents some of the properties of this function together with sixfigure tables and charts for the. We show how, using the constructive approach for special functions introduced by nikiforov and uvarov, one can obtain recurrence relations for the hypergeometric type functions not only for the continuous case but also for the dis. Bj, and then transform it into a di erential equation for the generating function a. The result can be extended easily to krecords statistics. In section 3, the addition theorem and three terms recurrence relation for the chebyshev matrix polynomials of the second kind are obtained and further we introduce and study the twovariable and twoindex chebyshev matrix. The following recurrence relation deducing the next approximation in. Recurrence relations for discrete hypergeometric functions article pdf available in journal of difference equations and applications 119. Construction of a summation formula allied with hyper geometric function and involving recurrence relation. A recurrence relation of hypergeometric series through record. The threeterm recurrence relation and the differentiation. Finally, section 4 deals with the study of the generalized hermite matrix polynomials by means of the hypergeometric matrix function. Browse other questions tagged recurrence relations hypergeometric function or ask your own question.
Abstractthe main aim of this paper is to create a summation formula associated to recurrence relation and hypergeometric function. Special cases of these lead to recurrence relations for the orthogonal polynomials, and many special functions. We present a general procedure for nding linear recurrence relations for the solutions of the second order di erence equation of hypergeometric type. Incomplete betafunction expansions of the solutions to the. Computation of hypergeometric functions people university of. A recurrence relation of hypergeometric series through. The secondorder linear hypergeometric differential equation and the hypergeometric function play a central role in many areas of mathematics and physics. These functions generalize the classical hypergeometric functions of gauss, horn, appell, and lauricella. Legendre polynomials let x be a real variable such that 1 x 1. Olde daalhuis school of mathematics, edinburgh university, edinburgh, united kingdom acknowledgements. In this way, the problem of summing the series would be reduced to solving a di erential equation. Pdf some recurrence relations for the generalized basic. I am reading methods of solving recurrence relation on wikipedia.
Finally, these three relationships are applied to the polynomials of hypergeometric type which form a broad subclass of functions y. Recurrence relations for hypergeometric functions of unit argument by stanislaw lewanowicz abstract. Some recurrences for generalzed hypergeometric functions. Some recurrence relations for the generalized basic. Pdf recurrence relations for discrete hypergeometric. These functions generalize the euler gauss hypergeometric function for the rank one root system and the elementary spherical functions on a real semisimple lie group for particular parameter values.
The general formula based on repeated differentiation dk dxk. Purohit abstract in the present paper, we express the generalized basic hypergeometric function r. The ab ov e mentioned technique is a q v ersion of the tec hnique used. It is a solution of a secondorder linear ordinary differential equation ode. Incomplete beta function an overview sciencedirect topics. Fourterm recurrence relations for hypergeometric functions. It was already noted by euler that many classical functions could be recognized as hypergeometric functions for special choices of the. In mathematics, the gaussian or ordinary hypergeometric function 2 f 1 a,b. On the other hand, recurrence relations for hypergeometric functions.
Computing hypergeometric solutions of linear recurrence equations. Many linear homogeneous recurrence relations may be solved by means of the generalized hypergeometric series. Most often generating functions arise from recurrence formulas. A characterization is given on the basis of this recurrence relation. In this paper, a new general recurrence relation of hypergeometric series is derived using distribution function of upper record statistics. Gpl the front end function of the package is hypergeo. In particular, recurrence relations of their solutions, their integral representations and. Using an alternative representation of the incomplete beta function through the gauss hypergeometric function, it can easily be shown that the presented finitesum solutions are always expressed in terms of elementary functions. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. This chapter is based in part on abramowitz and stegun 1964, chapter by l. Then the solution space in l of a linear equation of order n is a cvector space. For some physical applications this form is laguerre polynomials lnz result if 1.
An algorithm for computing hypergeometric solutions of linear recurrence relations with poly. The purpose of this paper is to obtain differential equations and the hypergeometric forms of the fibonacci and the lucas polynomials. Abramowitz function computed by clenshaws method, 74. Given a recurrence relation for the sequence an, we a deduce from it, an equation satis. The confluent hypergeometric function is useful in many problems in theoretical physics, in particular as the solution of the differential equation for the velocity distribution function of electrons in a high frequency gas discharge. For completeness, the explicit expressions corresponding to all classical orthogonal polynomials jacobi, laguerre, hermite, and. Generalized form of hermite matrix polynomials via the. General recurrence and ladder relations of hypergeometric. Later we shall show that this is an algebraic function over cz. In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n.
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