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 introduce some weighted hypergeometric functions and the suitable generalization of the caputo fractional derivation. This direct technique is useful for a bessel function representation exercise 14. On a kummer type transformation for the generalized hypergeometric function 2f2 allen r. New laplace transforms of kummers confluent hypergeometric. The parameters of the hypergeometric function are zeros and poles of this rational function. The aim of this paper is to apply generalized kummers theorem and generalized dixons theorem due to lavoie et al. On a generalization of the rogers generating function. Browse other questions tagged transformation hypergeometric function or ask your own question. Transformation formulas for the generalized hypergeometric function. On the computation of con uent hypergeometric functions for. The rising factorial provides a direct link between the gamma and hypergeometric functions. Kummers confluent hypergeometric function ma, b, z, introduced by kummer 1837, is a solution to kummers differential equation.
A kummertype transformation for a 2 f 2 hypergeometric. Generating functions of special triple hypergeometric. Some interesting applications of our main results are also presented. Kummer type formulas are identities of hypergeometric series. Pdf transformations of gauss hypergeometric functions. Lauricella hypergeometric series over finite fields. A character sum evaluation and gaussian hypergeometric series. On two general integrals involving humberts and kummers. In mathematics, the gaussian or ordinary hypergeometric function 2f1 a, b. If the arguments x and parameters b are all positive and if minx hypergeometric function ra, b, x and the elementary functions defined by 2.
We believe this is a powerful technique for character sum evaluations. In the present paper, we have obtained hypergeometric generating relations associated with two hypergeometric polynomials of one variable and with their independent demonstrations via goulds identity. Kummer s function satisfies a secondorder linear differential equation called the confluent hypergeometric equation. A kummertype transformation for a 2 f2 hypergeometric function r. As a first instance of the usefulness of the generalized operator which we introduce in the next section, we give an alternative derivation of kummers theorem and weisners bilateral generating function, and discuss its numerous other applications to certain classical polynomials and appells functions. Confluent hypergeometric kummer u function matlab kummeru. We obtain a kummertype transformation for the 2f2x hypergeometric function with general parameters in the form of a sum of 2f2x functions. A derivation of two transformation formulas contiguous to.
These results lead to the existence of certain hypergeometric functions in the class of either convex functions, or starlike functions 1. A note on the appell hypergeometric matrix function f2. There is an identity, called kummers transformation, which states that. In this note i propose to discuss some transformations of a certain series of products of kummers confluent hypergeometric functions 1f1. Initially this document started as an informal introduction to gauss. Classical algebraic transformations of hypergeometric functions due to gauss,euler, kummer, pfaff and goursat. Our results also include transformation formulas such as 1 1 4x52 3 f 2 1 2. This work therefore outlines the general and basic properties of hypergeometric and confluent hypergeometric function and the kummers. Quadratic transformations of hypergeometric function and.
In a recent paper, miller derived a kummertype transformation for the generalised hypergeometric function p f p x when pairs of parameters differ by unity, by means of a reduction formula for a. Some reduction formulae associated with gauss and foxwright. Some transformation form ulas for lauricellas hypergeometric functions fd keiji matsumoto and katsuyoshi ohara june 24, 2008 abstract in this paper, we give some functional equations with a parameter c for lauricellas hypergeometric functions. In sections 2 and 3, we obtain some reduction formulae for gauss hypergeometric function and foxwright hypergeometric function associated with suitable convergence conditions by using binomial theorem, decomposition identity and pfaff kummers linear transformation. To be more precise, in each of the above cases there exists a gauge factor. In this consent one may refer their sequel of papers 5, 6. In order to nd this representation we compute its mellin transform. Olde daalhuis school of mathematics, edinburgh university, edinburgh, united kingdom acknowledgements. This work therefore outlines the general and basic properties of hypergeometric and confluent hypergeometric function and the kummers first formula. Generating kummer type formulas for hypergeometric functions. Some inequalities involving the extended gamma function and. Hypergeometric solutions of linear differential equations. Then, the second appell hypergeometric matrix function is defined in the following form.
Generalized hypergeometric function, euler transformation, kummer transformation, quadratic transformations, summation theorem, zeros of entire functions 1. The author thanks richard askey and simon ruijsenaars for many helpful recommendations. Confluent hypergeometric function kummer function file. The importance of the gauss hypergeometric function is that many elementary and special functions of mathematical physics can directly be expressed in terms of it, see e. This chapter is based in part on abramowitz and stegun 1964, chapter by l. Under the hypergeometric function, the confluent hypergeometric function is related to a countless number of different functions. Internal report sufpfy9601 stockholm, 11 december 1996 1st revision, 31 october 1998 last modi. The quotients of certain hypergeometric functions are presented as fractions which converge uniformly in the unit disc. In this paper, we establish two general transformation formulas for extons quadruple hypergeometric functions k 5 and k 12 by application of the generalized kummers summation theorem. In general case the program calculates the sum of convergent series defining the function until the next term becomes too small in comparison with the sum of all previous terms. The confluent hypergeometric function of the first kind, \y ma,b,z\, is defined to be the solution to kummers differential equation.
Transformation of a certain series of products of confluent. As applications, some well known and new generating relations are deduced. E, which depends on the energy parameter e, and a change of variables z zr, which does not, such that solutions to the corre. Solutions of hypergeometric differential equations. Hypergeometric function wikimili, the free encyclopedia. In this paper we aim to show how one can obtain so far unknown laplace transforms of three rather general cases of kummers confluent hypergeometric function 1f1a. All this shows us how the kummer transformations together with the gauss. The confluent hypergeometric function has been studied in great detail from its mathematical point of view.
Davis goes so far as to state that of the socalled higher mathematical functions, the gamma function is undoubtedly the most fundamental cf. Jun 19, 2018 in the paper, the authors present some inequalities involving the extended gamma function and the kummer confluent hypergeometric kfunction via some classical inequalities such as chebychevs inequality for synchronous or asynchronous, respectively mappings, give a new proof of the logconvexity of the extended gamma function by using the holder inequality, and introduce a turan type. The function fin i above is called the pullback function. Kummer confluent hypergeometric function, the ratio of the circumference of a circle to its diameter, e.
Oct 19, 2006 kummercomplexa,b,z is the confluent hypergeometric function 1f1 kummer function for complex parameters a, b and complex variable z. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. We also have a barnestype integral representation for the con uent hypergeometric function. Kummers confluent hypergeometric function ma, b, z, introduced by kummer, is a solution to kummers differential equation. By using kummers transformation formula 4 we obtain z 1 0 zs 1 1f 1 a c. On a kummertype transformation for the generalized hypergeometric function 2f2 allen r.
In the article 19 from 1857 he gave a complete description of the monodromy group for gausss hypergeometric function. Generalized hypergeometric function, euler transformation, kummer trans formation, quadratic transformations, summation theorem, zeros of entire. The theory and application of special functions academic, new york. In this section, we investigate some of the main properties of the second appell hypergeometric matrix function f 2 a, b 1, b 2, c 1, c 2. Transforming differential equation to a kummer s equation. Miller 1616 eighteenth street nw 210, washington, dc 200092525, usa received 29 april 2002. The bessel and legendre functions provide examples of this approach. If the real parts of z and a are positive values, then the integral representations of the kummer u function is as follows. In fact the theory of quadratic transformations of the hypergeometric function is an old topic and can be traced back to gauss,kummer, and goursat lebedev 1965. Applications of the generalized kummers summation theorem to.
For these hypergeometric functions, some linear and bilinear relations are obtained by means of the mentioned derivation operator. Kummer, dixon, watson, whipple, pfaffsaalschutz and dougall formulas and. This is also known as the confluent hypergeometric function of the first kind. Pdf a generalised kummertype transformation for the fp. This chapter is based in part on chapter 15 of abramowitz and stegun by fritz oberhettinger. Identities for the gamma and hypergeometric functions.
Methods and applications sigma 9 20, 050, pages a connection formula for the qconfluent hypergeometric function. The kummer function, as it is also named, belongs to an important class of special functions with a large number of applications in different branches of applied mathematics and physics see and references therein. For some special values of the parameter for the roots, the hypergeometric functions are radial parts of zonal spheri. Some summation theorems for generalized hypergeometric. Jan 08, 2018 in this brief note, we show how to apply kummer s and other quadratic transformation formulas for gauss and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that contain one or two continuous parameters. They send expressions in terms of sto expressions in terms of s. Transformations 6 formulas transformations and argument simplifications 1 formula products, sums, and powers of the direct function 5 formulas. A kummertype transformation for a 2f2 hypergeometric.
Some identities with generalized hypergeometric functions sep 1, 2016 series. Generalized hypergeometric function, foxwright generalized hypergeometric function, pfaff kummers linear transformation. 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. Thus the heckmanopdam hypergeometric function is determined by a root system, a parameter attached to the roots that is generic and a parameter of the eigenvalue.
This result is specialised to the case where one pair of parameters differs by unity to generalize a recent result of miller j. Further, a number of generating functions for jacobi polynomials are also derived as an applications of our main results. I can currently prove this using the integral form of kummers equation. Because of gausss work, the functions given by eulers hypergeometric series are now often called gauss hypergeometric functions. Let be the class of analytic functions in the unit disc, normalized by and of the form. The classical generalizations of the gauss hypergeometric function like p fq, the lauricella type functions, and horns hypergeometric functions all occur as special cases of the gkzsystems. This report presents some of the properties of this function together with sixfigure tables and charts for the. Analytic continuation of the first kind associated. Using bounded sequences, further generalizations of two main hypergeometric generating relations have also been. Pdf solutions of hypergeometric differential equations. 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. So any solver for nding solutions in terms of s, if it is complete, then it must be able to deal with all three transformations. We obtain a kummertype transformation for the 2f2x hypergeometric function with general parameters in the form of a sum of 2f2.
Ahmed ali atash, hussein saleh bellehaj, on two general integrals involving humberts and kummer s hypergeometric functions, international journal of advances in mathematics, volume 2018, number 2, pages 18, 2018. Hypergeometric1f1a,b,z 750 formulas primary definition 3 formulas specific values 533 formulas general characteristics formulas. Now continuing the work in connection with class of generating functions, we extend our ideas to obtain the new generating relations that involves between extons functions and hyper geometric functions. Transformation formulas for the generalized hypergeometric. Lauricella hypergeometric series over finite fields 2 it was greene who in 8 developed the theory of hypergeometric functions over nite elds and established a number of transformation and summation identities for hypergeometric series over nite elds which are analogues to those in the classical case. Kummer functions, kummers transformations, for confluent hypergeometric functions, interrelations. A symmetry by the permutations of nletters yields these formulas. There is a different and unrelated kummers function bearing the same name. A kummertype transformation for a 2f2 hypergeometric function. The motivation for computing hypergeometric functions will be discussed, with details given of some of the practical applications of these functions. Request pdf a kummertype transformation for a 2f2 hypergeometric function we obtain a kummertype transformation for the 2f2x hypergeometric.
Quadratic transformations of hypergeometric function and series with harmonic numbers martin nicholson in this brief note, we show how to apply kummers and other quadratic transformation formulas for gauss and generalized hypergeometric functions in order to obtain transfor. Thus, we shall give below two simple and transparent derivations of eq. Read a kummertype transformation for a 2 f 2 hypergeometric function, journal of computational and applied mathematics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. One can show that a function is hypergeometric if and only if in the representation fz x k. We obtain a kummertype transformation for the 2 f 2 x hypergeometric function with general parameters in the form of a sum of 2 f 2 x functions.
It is a solution of a secondorder linear ordinary differential equation ode. Hypergeometric functions reading problems introduction the hypergeometric function fa, b. Transformations of gauss hypergeometric functions rims, kyoto. A kummertype transformation formula for the generalized hypergeometric function 2f2 deduced by exton. Transformations 6 formulas identities 21 formulas differentiation 40 formulas integration 12 formulas integral transforms 1 formula operations 3 formulas representations through more general functions 71 formulas representations through equivalent functions 5 formulas theorems 0 formulas history 0 formulas. In this paper, we describe some reduction formulae for gauss hypergeometric function and foxwright hypergeometric function associated with suitable convergence conditions using series rearrangement technique. On a kummertype transformation for the generalized. Transformation of generating function expansions and rodrigues representations. Some hypergeometric generating relations motivated by the. We obtain a kummer type transformation for the 2 f 2 x hypergeometric function with general parameters in the form of a sum of 2 f 2 x functions.
Division of mathematical sciences, university of abertay dundee, dundee dd1 1hg, uk received 27 november 2003. On the computation of con uent hypergeometric functions for large imaginary part of parameters b and z guillermo navaspalencia1 and argimiro arratia2 1 numerical algorithms group ltd, uk, and dept. The gaussian analogues for evaluations and transformations of 1 f1s and 3 fs are given in 8. This integral representation can be used to prove kummers transformation formula. It seems very difficult to implement the hypergeometric module of mpmath in c.
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