The generalized hypergeometric function was introduced by srivastava and daoust. Lauricellas hypergeometric function fd sciencedirect. Appelllauricella hypergeometric functions over finite. In mathematics, the gaussian or ordinary hypergeometric function 2 f 1 a,b.
Relations between lauricellas triple hypergeometric function f 3 a x, y, z and extons function x8 article pdf available in advances in difference equations 201 february 20 with. Some integral representations, transformation formulae, differentiation formulae and recurrence relations are obtained for these new generalizations. Asymptotic expansions of the lauricella hypergeometric function f. Pdf relations between lauricellas triple hypergeometric.
Similarly new integral representations of lauricella and appell function are obtained. Multiple hypergeometric functions that is, hypergeometric functions in several variables occur. Hypergeometric function of a matrix argument, the multivariate generalization of the hypergeometric series. The authors make use of some rather elementary techniques in order to derive several summation formulas associated with lauricellas hypergeometric function. Some transformation formulas for lauricellas hypergeometric. Lauricella hypergeometric series, hypergeometric series of three variables. But avoid asking for help, clarification, or responding to other answers. On the extended appelllauricella hypergeometric functions and. In this paper, we find a unique solution of the dirichlet problem for an elliptic equation with several singular coefficients in explicit form. Here, in this paper, we aim at establishing eleven new and interesting transformations between. Here, we aim to present certain extensions of the appell functions f1 and f2 of two variables and the lauricella function fd of three variables. For example, the lauricellas hypergeometric function fn.
The generalized hypergeometric function was introducedby srivastava and daoust. Since the lauricella hypergeometric series fn d has a integral representation f. Of course, all 20 of extons triple hypergeometric functions x1,x20 as well as srivastavas triple hypergeomet ric functions ha, hb and hc are included in the set of the aforementioned 205 distinct triple hypergeometric functions which were. Whittaker function, generalized whittaker function, lauricellas function, saran function, ap. Six results have been proved here one each for the functions f, fdfn a np s, two for the function f. Pdf extensions of appell and lauricella hypergeometric. If we ask, more generally, for a homogeneous function. A comparative study on generating function relations for. An integral representation of this function is obtained in the form of a generalized stieltjes transform. There are therefore a total of 14 lauricellasaran hypergeometric functions. Lauricella hypergeometric series over finite fields. Extension formulas of lauricellas functions by applications.
A lauricella hypergeometric series over finite fields. Brownian bridge, gausss hypergeometric function, lauricellas multiple hypergeometric series, uniform order statistics, appell functions. In 1893 giuseppe lauricella defined and studied four hypergeometric series fa, fb, fc, fd of three variables. Abstractlauricella hypergeometric function lhf fa is often used in the performance. The hypergeometric function for the root system of type a with a certain degenerate parameter shimeno, nobukazu and tamaoka, yuichi, tsukuba journal of mathematics, 2018. In the present paper a new integral representation is derived. Khan communicated by jose luis lopezbonilla msc 2010 classications.
Carlson institute for atomic research and department of physics, iowa state university, ames, iowa submitted by richard bellman i. In particular, the lauricella functions f a r, f d r in r real or complex variables, defined by 2 and 1, p. In the theory of hypergeometric and generalized hypergeometric series, dixons summation theorem for the sum of a 3f21 play an important role. Thegeneralized hypergeometric function wasintroduced by srivastava and daoust. Some applications of lauricella hypergeometric function fa in. Monotonicity of asymptotic relations for the lauricella hypergeometric function. In this course we will study multivariate hypergeometric functions in the sense of gelfand, kapranov, and zelevinsky gkz systems. A number of known and new transformation and reduction formulae for a hypergeometric.
In the present paper the lauricella saran functions f,f andf fm t have been. This represents the number of possible out comes in the experiment. The confluent hypergeometric function kind 1 distribution with the probability density function pdf proportional to occurs as the distribution of the ratio of independent gamma and beta variables. Relations between lauricellas triple hypergeometric function and the srivastava function. In this paper, by the extension of beta functions containing three extra parameters, we generalize appells and lauricella s hypergeometric functions. It goes beyond the classical theory of motives, but should be compatible with it. Lauricella saran functions, matrix arguments, matrix transform. Summation formulas associated with the lauricella function far. The denominator of formula 1 represents the number of ways n objects can be selected from n objects. Recent citations numerical computation of the capacity of generalized condensers mohamed m. Generating functions of special triple hypergeometric. Hypergeometric functions derived via other approaches 6.
It is perhaps interesting to note that, as an application, esther bod 6 succeeded in extending schwarzs list to all irreducible appell, lauricella and horn. When finding a solution, we use decomposition formulas and some adjacent relations for the lauricella hypergeometric function in many variables. This work was supported by the initial foundation for scienti c. Using the burchnallchaundy method, hasanov and srivastava 24, 25 found decomposition formulas for a whole class of hypergeometric functions in several three and more variables. Lauricella hypergeometric and pearsons system of partial. Carlson explains how the lauricella hypergeometric function is connected with symmetric elliptic integrals and. Abstract the lauricella saran triple hypergeometric functions of matrix arguments f,f andf kg e were introduced by us in one of our previous studies 6. Numerical calculation of oneloop integration with hypergeometric functions toshiaki kaneko 2010. Learn more about matlab, matrix, mathematics, image processing, simulation, hypergeom.
Motivated by the work of greene 8 and li et al 11, we give a nite eld analogue of the lauricella hypergeometric series. Journal of mathematical analysis and applications 7, 452470 1963 lauricella s hypergeometric function fd b. Decomposition formulas associated with the lauricella. Hypergeometric functions, generating functions, multilinear and. On the computation of lauricella functions of the fourth. Lauricella hypergeometric series over nite elds, reduction formula, transformation for mula, generating function. Decomposition formulas associated with the lauricella multivariable. Motivated by the work of greene 8 and li etal 11, we give a. Recently found all the fundamental solutions of a multidimensional singular elliptic equation are expressed in terms of the wellknown lauricella hypergeometric function in many variables. Extensions of appell and lauricella hypergeometric functions article pdf available in far east journal of mathematical sciences 1026. We will emphasize the algebraic methods of saito, sturmfels, and takayama to construct hypergeometric series and the connection with deformation techniques in commutative algebra.
Multivariate generalization of the confluent hypergeometric. Lauricellas fd this function is a generalization of f, f1 with n variables. An integral formula for the lauricella hypergeometric series and its finite field analogue are deduced. I know there are two important properties about appell hypergeometric function and lauricella functions. Monotonicity of asymptotic relations for generalized. Lauricella, appell, integral representations, mellin transformation. Relations between lauricellas triple hypergeometric. Exton 9 introduced 20 distinct triple hypergeometric functions, which he denoted by out of these exton 9 defined and gave integral representations of some hypergeometric functions of three variables which is denoted. We used integral form of extonns functions in the form of laplace integral to obtain the new generating relations among extons and hypergeometric functions.
An identity is proved which simplifies the computation of lauricella functions. Riemannliouville, caputo and kobererdelyi fractional operators by using confluent hypergeometric function with six parameters. Asymptotic expansions of the lauricella hypergeometric function f d chelo ferreira1 and jos. Integral representations for lauricella functions for a given multivariable hypergeometric function, it is useful to. Journal of computational and applied mathematics 21 1988 369375 369 northholland on the computation of lauricella functions of the fourth kind peter j. Asymptotic expansions of the lauricella hypergeometric.
These functions generalize the classical hypergeometric functions of gauss, horn, appell, and lauricella. The dirichlet problem for elliptic equation with several. Hypergeometric transformations of linear forms in one logarithm viola, carlo and zudilin, wadim, functiones et approximatio commentarii mathematici, 2008. Some special cases of the main results here are also considered. In this paper, we find a unique solution of the dirichlet problem for an. Every secondorder linear ode with three regular singular points can be transformed into this. Hypergeometric functions of three variables in terms of. Pdf extensions of appell and lauricella hypergeometric functions. The hypergeometric function for the root system of type a with a certain degenerate parameter shimeno, nobukazu and tamaoka, yuichi, tsukuba journal of mathematics, 2018 transformation relations of matrix functions associated to the hypergeometric function of gauss under modular transformations watanabe, humihiko, journal of the mathematical. The lauricella hypergeometric function f d r a,b 1,b r.
Notes on ahypergeometric functions 3 in 2006 the present author found a combinatorial characterisation for algebraic ahypergeometric functions in the irreducible case in 9. Integral representations for srivastavas triple hypergeometric functions ha, hb and hc 2753. In this article, a multivariate generalization of this distribution is defined and derived. Such a theory would explain a recent and surprising conjecture arising in the context of scattering amplitudes for a. Such a theory would explain a recent and surprising conjecture arising in the context of scattering. Integral representations of generalized lauricella. Lauricella hypergeometric functions, unipotent fundamental. Thanks for contributing an answer to mathematics stack exchange. Furthermore, linear and bilinear generating relations for these extended hypergeometric. This report presents some of the properties of this function together with sixfigure tables and charts for the. Three lectures on hypergeometric functions eduardo cattani abstract. In 1, a family of generalized elliptictype integrals is introduced, which admits an explicit representation in terms of the lauricella function. Some generating relations for extended appells and. Since the lauricella hypergeometric series fn d has a integral representation fn.
The lauricella saran functions f, fdfn a np s have been defined for the matrix arguments case. Confluence of hypergeometric functions and integrable. It is a solution of a secondorder linear ordinary differential equation ode. Lauricella also indicated the existence of ten other hypergeometric functions of three variables. 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. Relations between lauricellas triple hypergeometric function.
About the integral forms of appell hypergeometric function. On the extended appelllauricella hypergeometric functions. Generating functions for khypergeometric functions. Old and new relations between hypergeometric functions managed by computer 6. The lauricella hypergeometric function, the riemann hilbert problem, and some applications to cite this article. In this paper, we present further generalizations of the beta function. Hypergeometric functions reading problems introduction the hypergeometric function fa, b. Some transformation of a multiple hypergeometric series of. Pdf very recently, the incomplete pochhammer ratios were defined in terms of the incomplete beta function b y x, z. About the confluent versions of appell hypergeometric. Algebraic evaluations of some euler integrals, duplication formulae. Introduction in 1880 appell defined four hypergeometric series in two variables, which were generalized to n variables in a straightforward way by. For the sake of convenience the symbol a m denotes the product j m a j a 1 3.
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