Integral representations 7 formulas hypergeometric2f1. Kummers confluent hypergeometric function ma, b, z, introduced by kummer 1837, is a solution to kummers differential equation. This newfound interest comes from the connections between hypergeometric functions and many areas of mathematics such as representation theory, algebraic geometry and hodge theory, combinatorics, d. Hypergeometric functions reading problems introduction the hypergeometric function fa, b. Transformation of generating function expansions and rodrigues representations. Fox hfunction, an extension of the meijer gfunction. 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. Pdf product of independent random variables involving.
The parameters of the hypergeometric function are zeros and poles of this rational function. There is a different and unrelated kummers function bearing the same name. Kummer functions, along the real line, hypergeometric function, integral representations, integrals, integrals of bessel and hankel functions, integrals of modified bessel functions, over infinite intervals notes. For the hypergeometric function 2f 1 we have an integral representation due to euler. Further integral representations can be found in,14. Each of these triple hypergeometric functions has been investigated extensively in many.
Euler hypergeometric integral, an integral representation of 2 f 1. Explicit representation and some integral transforms of sequence of functions associated with the wright type generalized hypergeometric function. Indeed, a search for the title word hypergeometric in the mathscinet database yields 3181 articles of which 1530 have been published since 1990. In this paper the analogy between the character sum expansion of a complexvalued function over gfq and the power series expansion of an analytic function is exploited in order to develop an analogue for hypergeometric series over finite fields.
The above stated three dimensional regions of convergence of the triple hypergeometric. Hypergeometric functions hypergeometric2f1a,b,c,z integral representations on the real axis. Kummer functions, along the real line, hypergeometric function, integral representations, integrals, integrals of bessel and hankel functions, integrals of modified bessel functions, over infinite intervals. In the several variables case, on the other hand, it is possible to extend each one of these approaches but one may get slightly di. One can show that a function is hypergeometric if and only if in the representation fz x k. Acknowledgements i heard and i thought that writing a dissertation is a big deal, and also a long and hard work. Some integral representations of multivariable hypergeometric.
Here we have presented only some illustrative integral representation for each of srivastavas functions. Integral representation for most of the special functions of mathematical physics and applied mathematics has been investigated in the existing literature. Similarly new integral representations of lauricella. Complete triple hypergeometric functions of the second order which were denoted by. Hypergeometric series and functions in several independent scalar variables 3. Hence mu,d is a meromorphic function which can only have poles when some e. On the computation of con uent hypergeometric functions for. Initially this document started as an informal introduction to gauss. We have the following generating function for the k hypergeometric functions given by 1. An interesting feature of hypergeometric functions is that many of them can be represented by integrals. The author thanks richard askey and simon ruijsenaars for many helpful recommendations.
Integral involving a confluent hypergeometric function. At the heart of the theory of special functions lies the hypergeometric function, in that all of the classical special functions can be expressed in terms of this powerful function. 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. This constant is built into the measure dsin order to prevent it appearing everywhere else. An integral representation of some hypergeometric functions. This chapter is based in part on chapter 15 of abramowitz and stegun by fritz oberhettinger. An application of the formulas yields some integral representations of lauricella hypergeometric seriesf a n. The present paper is devoted to establish two integral representation formulas of multivariable hypergeometric functions. Certain fractional integral operators and the generalized. Hypergeometric functions of three variables in terms of. Integral representations and further generalization 3. The generalized hypergeometric function can be written as series. The results are stated as theorems in section 3, see eqs.
This direct technique is useful for a bessel function representation exercise 14. Let us consider rstly the case of integer or halfinteger values of parameters. Foxwright function, a generalization of the generalized hypergeometric function. We recall several properties of rfrom 11 and 12 which. Our method is based on recently developed integral formulas that allow to express certain integrals through wronskians. N, where n is a nonnegative integer, the hypergeometric function is a polynomial in z see below.
The bessel and legendre functions provide examples of this approach. Of course, all 20 of extons triple hypergeometric functions x1,x20 as well as srivastavas triple hypergeometric functions ha, hb and hc are included in the set of the aforementioned 205. The hypergeometric functions are important for obtaining various properties, such as, integral representation, generating functions, solution of gauss differential equations. For onevariable hypergeometric functions this interplay has been well understood for several decades. Consequently, we derive expansions of the confluent hypergeometric function in terms of functions of the same kind. Explicit representation and some integral transforms of.
Generalized hypergeometric series have the following inductive integral representation 1, p. We will make use of the following general theorem due to rainville, see 12. Hypergeometric functions have explicit series and integral representations, and thus provide ideal tools for establishing useful summation and transformation formulae. In this paper we explore many relations and explicitly derive closed form representations of integrals of logarithmic, hypergeometric functions and the lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers. 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. Integrals of logarithmic and hypergeometric functions are intrinsically connected with euler sums. Of the direct function 3 formulas hypergeometric2f1.
Hypergeometric functions of three variables in terms of integral representations showkat ahmad. 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. 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. A few years later i had luck to met a man who turned the words long and hard into e cient and interesting. The generalized hypergeometric function was introduced by srivastava and daoust. This is also known as the confluent hypergeometric function of the first kind. To perform the expansion we use the wellknown representation. Elliptic hypergeometric series where the ratio of terms is an elliptic function of the index. An indefinite integral in the form of hypergeometric function.
On integral representations and asymptotics of some. In this paper, we derive a new contour integral representation for the confluent hypergeometric function as well as for its various special cases. Integrals of logarithmic and hypergeometric functions in. Turaev, integral representation for srivastavas hypergeometric function h, journal of korean society mathematics education,19 2012, 7145. On the computation of con uent hypergeometric functions. Pdf an integral representation of some hypergeometric. Pdf integral representations of generalized lauricella.
The euler integral representation of the 2 f 1 gauss hypergeometric function is well known and plays a prominent role in the derivation of transformation identities and in. International journal of applied physics and mathematics 121 volume 9, number 3, july 2019 which completes the proof. The main tool to derive these are eulerian and inverse laplacian integral representations, and a tauberian theorem 34, 33. Identities for the gamma and hypergeometric functions. Ruijsenaars hypergeometric function ruijsenaars 11 introduced a generalization rof the hypergeometric function as a barnes type integral. Series, expansion of the hypergeometric functions and. This report presents some of the properties of this function together with sixfigure tables and charts for the. In the present paper a new integral representation is derived.
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