On confluent hypergeometric functions and generalized bessel functions article pdf available in analysis mathematica 4371. Our principal objective in this article is to present a systematic investigation of several further. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Identities for generalized hypergeometric coefficients 1. Jhep042020121 published for sissa by springer received.
Inequalities for generalized hypergeometric functions. View enhanced pdf access article on wiley online library html. Generalized hypergeometric functions include the gaussian hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, bessel functions, and the classical orthogonal polynomials. In this paper, we apply two identities of generalized hypergeometric series in order to extend some classical summation theorems of hypergeometric functions such as gauss. In this course we will study multivariate hypergeometric functions in the sense of gelfand, kapranov, and zelevinsky gkz systems. Series of gamma functions can usually be expressed as generalised hypergeometric functions of one or more variables, providing a tool for classifying. A number of known and new results which introduced are generalization of a known results introduced by shyder and srivastava 6. Generalized hypergeometric functions, zonal polynomials, representation theory, algebraic induction, multivariate statistics, general linear group, generalized gamma functions, pochhammer symbols, laplace transforms, maximal compact subgroup, invariant poly. Differential equations for hypergeometric functions. Hanballa1 1 department of mathematics, aden university, aden, kohrmaksar, p.
Generalized hypergeometric function, euler transformation, kummer transformation, quadratic transformations, summation theorem, zeros of entire functions 1. Introduction 2f1 hypergeometric seriesfunctoins hypergeometric functions over finite fields let q ps be a prime power. This constant is built into the measure dsin order to prevent. Generalized hypergeometric series cambridge tracts in. Generalized hypergeometric functions book iopscience. Whittaker vectors generalized hypergeometric functions and.
Hypergeometric functions reading problems introduction the hypergeometric function fa, b. Pdf a new approach to the epsilon expansion of generalized. Pdf notes on generalized hypergeometric and confluent. We propose a new application of intersection theory to construct a coaction on generalized hypergeometric functions. Pdf on confluent hypergeometric functions and generalized. 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. Some properties of generalized hypergeometric thermal.
Properties of generalized univariate hypergeometric functions. Pdf a note on the asymptotic expansion of generalized. We introduce some weighted hypergeometric functions and the suitable generalization of the caputo fractional derivation. Jfca201782 new laplace transforms for the 2f2 151 it should be remarked here that whenever hypergeometric function 2f1 or gen.
Some generalizations of pochhammers symbol and their. Motivated essentially by the demonstrated potential for applications of the generalized incomplete hypergeometric functions p. Let us note that in a systematic study of the generalized. Davis goes so far as to state that of the socalled higher mathematical functions, the gamma function is undoubtedly the most fundamental cf. Let fc q denote the group of multiplicative characters on f q.
The hypergeometric function 2f1 can be written in terms of a contour inte gral involving gamma functions. Hypergeometric functions with parameter 17 duality 24 7. Generalizations of generating functions for hypergeometric and q hypergeometric orthogonal polynomials howard s. Then we obtain some generating function relations for these generalized hypergeometric functions by using each generalized fractional operators, separately. Generating functions for the generalized appell function 7 7 d. Some hypergeometric generating relations motivated by the. Integraltransforms associated with sgeneralized gauss hypergeometric function. In order to derive these transformations, two wellknown methods are used i. Askey department of mathematics, university of wisconsin. A note of generating functions involving the generalized gauss hypergeometric functions. The properties of those m functions are discussed in an appendix of the same paper. Sometimes the most general hypergeometric function pfq is called a generalized hypergeo metric function.
A note on the asymptotic expansion of generalized hypergeometric functions article pdf available in analysis and applications 1201 december 20 with 651 reads how we measure reads. A new identity for generalized hypergeometric functions. Currie a research report submitted to the faculty of science. Hypergeometric series and gaussian hypergeometric functions. For these hypergeometric functions, some linear and bilinear relations are obtained by means of the mentioned derivation operator. Hypergeometricpfqa 1,a 2,a 3,a 4,b 1,b 2,b 3,z 360 formulasprimary definition 2 formulas specific values 189 formulas. April 20, 2020 hypergeometric series representations of feynman inte. Cohlapplied and computational mathematics division, national institute of standards and echnologyt, gaithersburg, maryland spring central sectional meeting exast echt universit,y lubbock, tx special session on applications of.
Generalizations of generating functions for hypergeometric. The legendre, bessel, whittaker and other special functions, and the classical orthogonal polynomials are particular cases of the hypergeometric functions or their various combinations. There are functions which can also be evaluated analytically and expressed in form of hypergeometric function. Pad e approximations of generalized hypergeometric series tapani matalaaho tokyo 2009 march 02. Walsh on the occasion of his 75th birthday it is shown that some wellknown fade approximations for a particular form of the gaussian. Luke department of mathematics, university of missouri, kansas city, missouri 64110 received march 24, 1970 dedicated to professor j. The solution of eulers hypergeometric differential equation is called hypergeometric function or gaussian function introduced by gauss.
Identities for the gamma and hypergeometric functions. Integraltransforms associated with sgeneralized gauss hypergeometric function in this section, we have. Especially, the author 6 established a new formula between jacobi polynomials and certain fractional jacobi functions with the aid of the generalized hypergeometric functions with the nonnegative integer n. A comparative study on generating function relations for.
In mathematics, the gaussian or ordinary hypergeometric function 2f1 a, b. We establish a new identity for generalized hypergeometric functions and apply it for first and secondkind gauss summation. The more general problem of generalized generating functions with more than one free parameter requires the theory of multiple hypergeometric series and is not treated in this paper. Pdf generalized hypergeometric functions at unit argument. Reduction formulae for generalised hypergeometric functions of one. Moreover, generalized beta functions 2, 3 have played a pivotal role in the advancement of further research and have proved. A number of generalized hypergeometric functions has special names.
Explicit representation and some integral transforms of. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. Reduction formulae for generalised hypergeometric functions of one variable j e gottschalk and e n maslen department of physics, university of western australia, nedlands, western australia 6009, australia received 29 july 1987 abstract. The parameters of the hypergeometric function are zeros and poles of this rational function. Whittaker vectors, generalized hypergeometric functions, and a matrix calculus lie theory workshop university of california, santa barbara february 4, 2006 1.
It is well known that properties of these functions are revealed in the framework of intersection theory. In this paper a natural generalization of the familiar h function of fox namely the ifunction is proposed. The equation has two linearly independent solutions at each of the three regular singular points, and. A new approach to the epsilon expansion of generalized. Thus a large part of the subject of generalized hypergeometric functions became relegated to the domain of special functions, which to many people means many formulas but no mathematical depth.
Generalized hypergeometric functions and intersection theory. Abstract we shall present short proofs for type ii pad e approximations of the generalized hypergeometric and q hypergeometric series ft x1 n0 q n 1 k0 pk q n 1 k0 qk tn. Solutions to the hypergeometric differential equation are built out of the hypergeometric series. Some integrals involving generalized hypergeometric functions. 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.
We do this by looking at hypergeometric functions that are at the same time algebraic. Some integrals involving generalized hypergeometric. Transformation formulas for the generalized hypergeometric. Using bounded sequences, further generalizations of two main hypergeometric generating relations have also been. Beukers, frits 2002, gauss hypergeometric function.
This detailed monograph outlines the fundamental relationships between the hypergeometric function and special functions. Laplace transform, generalized hypergeometric functions, pochhammer symbol, gamma function. Some applications on generalized hypergeometric and. Chapter 16 generalized hypergeometric functions and meijer g function. One can show that a function is hypergeometric if and only if in the representation fz x k. In 18, gauss first outlined his studies of the hypergeometric series which has been of great significance in the mathematical modelling of physical phenomena. Hypergeometric functions are generalized from exponential functions. A note on the asymptotic expansion of generalized hypergeometric functions. Buy generalized hypergeometric series cambridge tracts in mathematics and mathematical physics no. Certain integrals of generalized hypergeometric and confluent. Appendices a and b contain some basic information about multiple polylogarithms, which are a particular class of hyperlogarithms, and the iterative solution of systems of di. A new approach to the epsilon expansion of generalized hypergeometric functions. Appendices a and b contain some basic information about multiple polylogarithms.
It is a solution of a secondorder linear ordinary differential equation ode. Contemporary mathematics 8 hypergeometric functions on domains of positivity, jack polynomials. In this paper author try to establish transformation formulas using hyper geometric functions. Generating functions for the generalized appell function. Fortunately, developments by the end of the twentieth century turned this image around somewhat, and it is. Journal of approximation theory 5, 4165 1972 inequalities for generalized hypergeometric functions yudell l. The rising factorial provides a direct link between the gamma and hypergeometric functions. A hypergeometric function is called gaussian if p 2 and q 1. Hypergeometric functions fangting tu, joint with alyson deines, jenny fuselier, ling long, holly swisher a women in numbers 3 project national center for theoretical sciences, taiwan miniworkshop at lsu on algebraic varieties, hypergeometric series, and modular forms fang ting tu ncts hypergeometric functions april 14th, 2015 1 32. Series of gamma functions can usually be expressed as generalised hypergeometric. A generalized hypergeometric function satisfying four analytic. In this paper, a unified approach to hypergeometric functions is given to derive the probability density function and. Numerical expansion of several generalized hypergeometric functions.
Generalized hypergeometric series, both qseries and. Reduction formulae for generalised hypergeometric functions. Archived from the original pdf on 20170624 cs1 maint. Then the words hypergeometric function refer to the special case. Hypergeometricpfqa 1,a 2,b 1,b 2,b 3,z 145 formulas primary definition 2 formulas specific values 24 formulas general characteristics 17 formulas series representations 20 formulas integral representations 5 formulas. Three lectures on hypergeometric functions eduardo cattani abstract. These functions generalize the classical hypergeometric functions of gauss, horn, appell, and lauricella. Hypergeometricpfqa 1,a 2,b 1,b 2,b 3,z 145 formulasprimary definition 2 formulas specific values 24 formulas. Let us note that in a systematic study of the generalized probability density, in solving problems of the theory of special functions, differential. In section 5, we will demonstrate that, for physically interesting kinematics, the twoloop sunset diagrams are expressible in terms of generalized hypergeometric functions with quarter values of parameters. The motivation for computing hypergeometric functions will be discussed, with details given of some of the practical applications of these functions. In 18, gauss first outlined his studies of the hypergeometric series which has. Confluent hypergeometric functions kummers function. The main references used in writing this chapter are erdelyi et al.
This is the most common form and is often called the hypergeometric function. Properties of generalized univariate hypergeometric functions 39 moreover we show that ruijsenaars 26 relativistic analogue r of the hypergeometric function is a degeneration of the hyperbolic hypergeometric function, and that the d4symmetry 28 of r. Apr 12, 2017 on confluent hypergeometric functions and generalized bessel functions article pdf available in analysis mathematica 4371. As applications, some well known and new generating relations are deduced. Thus, we can further obtain various integral formulas involving a number of simpler special functions. The authors are pleased to acknowledge the assistance of b. One of the purposes of the present investigation is to give a chance to the reader to compare the results corresponding to each generalized fractional operators. Generalized hypergeometric functions and laguerre polynomials in. Pdf recently, some generalizations of the generalized gamma, beta, gauss hypergeometric and confluent hypergeometric functions have been introduced in. Some summation theorems for generalized hypergeometric functions. In this paper we derive close form for the matrix elements for h.
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