Decomposition formulas for hypergeometric functions of. We first recall the basic euler integrals which define the beta function. Agarwal, the incomplete pochhammer symbols and their applications to hypergeometric and related functions, integral transforms spec. It is the startig of a book i intend to write on 1variable hyper geometric functions. 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. Each individual can be characterized as a success s or a failure f.
These functions generalize the classical hypergeometric functions of gauss, horn, appell, and lauricella. Multipleterm approximations for appells f1 function. The four multiparameter appell functions, appellf1, appellf2, appellf3 and appellf4 are doubly hypergeometric functions that include as particular cases the 2f1 hypergeometric and some cases of the meijerg function, and with them most of the known functions of mathematical physics. The found formulas of an analytic continuation express known hypergeometric appell function. N, where n is a nonnegative integer, the hypergeometric function is a polynomial in z see below. The population or set to be sampled consists of n individuals, objects, or elements a nite population. Appells hypergeometric function f1 which give reductions of f1. The systematic investigation of contour integrals satisfying the system of partial differential equations associated with appell s hypergeometric function f 1 leads to new solutions of that system. Wider classes of separable potentials are given by the obtained new formulae automaticaly. We first give an identity for kpochhammer symbol and certain linear generating functions for k hypergeometric functions. An indefinite integral in the form of hypergeometric function.
Beyond one loop, one also nds that feynman integrals evaluate to similar classes of hypergeometric functions. First we handle univariate specializations onto the singular lines. In this article, we will introduce k hypergeometric functions which are extensions of the gauss hypergeometric functions including k pochhammer symbol. For instance, twoloop sunrise integrals evaluate to either 2f 1, appell f 2 or appell f. One of the twovariable hypergeometric functions most commonly used in applications 16.
Starting from a secondorder fuchsian differential equation having five regular singular points, an equation obeyed by a function proportional to the first derivative of the solution of the heun equation, we construct several expansions of the solutions of the general heun equation in terms of appell generalized hypergeometric functions of two variables of the first kind. 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 appell s functions. Generating functions of special triple hypergeometric. Some reduction and transformation formulas for the appell. Initially this document started as an informal introduction to gauss hypergeometric functions for those who want to have a quick idea of some main facts on hypergeometric functions. These bounds can be viewed as new kfractional estimations of trapezoidal and midpoint type inequalities. We introduce some weighted hypergeometric functions and the suitable generalization of the caputo fractional derivation. The first appells hypergeometric function f1a, b, c, d.
By means of the obtained formulas of expansion we find the formulas of an analytic continuation of hypergeometric function of clausen. If this is the first time you use this feature, you will be asked to authorise cambridge core to connect with your account. These a hypergeometric functions or gkz hypergeometric func. 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. Fundamental sets of solutions are given for the vicinity of all singular points of the system of partial differential equations. Product and quotient of independent gauss hypergeometric. Pdf extensions of appell and lauricella hypergeometric functions. Appell functions are variations on this type of series. The following four functions of two real or complex variables x and y cannot be expressed as a product of two f 1 2 functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation 15. For properties and further results the reader is referred to luke finally, applying 10we obtain the desired result. Appell hypergeometric functions, extended lauricella. New relations and transformation formulas for the appell function f2 a, b, b c, c w. Now, using the definition of the appell s first hypergeometric function, we get the desired result.
Even when the number of degrees of freedom increases, if an ellipsoid is symmetric, the number of variables in the hypergeometric functions does not. The motivation for computing hypergeometric functions will be discussed, with details given of some of the practical applications of these functions. It is known as gauss hypergeometric function in terms of pochhammer symbol ak or general ized factorial function. Then, using these identities as an auxiliary result, we obtain some new kfractional bounds which involve k appells hypergeometric functions. Multipleterm approximations for appell s f1 function volume 39 issue 1 magdi a. Moreover, i will prove that the basic hypergeometric functions r. Every secondorder linear ode with three regular singular points can be transformed into this. A very important extension of appell type series which is just beyond the scope of this basic survey article are the multivariate hypergeometric functions considered by i. Hypergeometric functions of three variables in terms of. In this paper, the incomplete second appell hypergeometric functions are defined in terms of these symbols. Derivatives of any order of the hypergeometric function p.
Appell established the set of partial differential equations of which these functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric. Appell hypergeometric function from wolfram mathworld. These functions generalize the euler gauss hypergeometric function for the rank one root system and the elementary spherical functions on a real. Hutchinson the generalization of the hypergeometric series in one variable and three parameters to that in two variables and four parameters was first made by appell in 1880.
Hypergeometric functions of two variables project euclid. The above distribution was suggested by armero and distirbucion 6. In this course we will study multivariate hypergeometric functions in the sense of gelfand, kapranov, and zelevinsky gkz systems. The incomplete second appell hypergeometric functions. Also, they have introduced the bilateral generating function for the generalized modi ed laguerre and jacobi polynomials with the help of two linear partial di erential operators. This constant is built into the measure dsin order to prevent.
Heun equation in early papers was, based upon the immediate intuition. Soon after the publication of appells first note on the subject picard 188o. Pdf extensions of appell and lauricella hypergeometric. Numerical evaluation of appells f hypergeometric function. Generating functions of certain hypergeometric functions. Identities between appells and hypergeometric functions. On the extended appelllauricella hypergeometric functions. A formal extension of the hypergeometric function to two variables, resulting in.
Extensions of appell and lauricella hypergeometric functions article pdf available in far east journal of mathematical sciences 1026. Ordinary linear differential equations note that if we replace y by sy in the system, where s. Pdf a large number of extensions and generalizations of various special functions. Three lectures on hypergeometric functions eduardo cattani abstract. Certain class of generating functions for the incomplete hypergeometric functions article pdf available in abstract and applied analysis 20145 august 2014 with. The generalized hypergeometric function is linked to the meijer g function and the macrobert e function. On some formulas for the appell function f2 a, b, b c, c w. Lauricella 1893 further generalized the four appell functions to functions of variables and defined as lauricella 1893 10, p. It is a solution of a secondorder linear ordinary differential equation ode.
In this work, we present a simple method for computing the first appell function f 1 a,b,b. F 21 and the appell hypergeometric function of two variables of the first kind 22, are. This was achieved using the secondorder linear differential equations satis. In mathematics, the gaussian or ordinary hypergeometric function 2 f 1 a,b. In mathematics, appell series are a set of four hypergeometric series f1, f2, f3, f4 of two variables that were introduced by paul appell 1880 and that generalize gausss hypergeometric series 2f1 of one variable. The four appell functions the four multiparameter appell functions, appellf1, appellf2, appellf3 and appellf4 are doubly hypergeometric functions that include as particular cases the 2f1 hypergeometric and some cases of the meijerg function, and with them most of the known functions of mathematical physics. Fractional derivative, generating functions, appell function, generalized appell. Appell hypergeometric expansions of the solutions of the. Feynman integrals, hypergeometric functions and nested sums. Appellf1a,b 1,b 2,c,z 1,z 2 102 formulasprimary definition 1 formula specific values 21 formulas general characteristics 27 formulas. Using the same technique to continue to differentiate the first and last ones of. Then we obtain some generating function relations for these generalized hypergeometric functions by using each generalized fractional operators, separately. For these hypergeometric functions, some linear and bilinear relations are obtained by means of the mentioned derivation operator.
A comparative study on generating function relations for. These results are obtained for the functions which have the harmonic convexity property. Essentially the first book devoted entirely to ahypergeometric functions is the one by saito. Lauricella hypergeometric series over finite fields. The evaluation of twovariable hypergeometric functions is more dif. Hypergeometric functions reading problems introduction the hypergeometric function fa, b.
Primary definition 1 formula 19982020 wolfram research, inc. Schwarzs list to all irreducible appell, lauricella and horn equations. Univariate specializations of appells hypergeometric functions f1, f2, f3. Generating functions of certain hypergeometric functions by means of fractional calculus.