Equation returning the kummer u function as its solution. Confluent hypergeometric kummer u function matlab kummeru. We establish a simple inductive formula for the trace trnew k. For certain special arguments, hypergeometric2f1 automatically evaluates to exact values. Hypergeometricpfqa 1,a 2,b 1,b 2,z 337 formulas primary definition 2 formulas specific values 31206 formulas general characteristics 16 formulas series representations 19 formulas integral representations 5 formulas.
Solutions to the hypergeometric differential equation are built out of the hypergeometric series. The general expression of l22z for the hypergeometric function 2f2 a, a. The equation has two linearly independent solutions at each of the three regular singular points, and. In literature the use of fractional derivatives in the theory of hypergeometric functions have wide application in the field of modeling, physics and engineering, stochastic process, probability theory, in solving ordinary and partial differential equations and. These results lead to the existence of certain hypergeometric functions in the class of either convex functions, or starlike functions 1. The quotients of certain hypergeometric functions are presented as fractions which converge uniformly in the unit disc. L inp has ve regular singularities where at least one of them is logarithmic. In this consent one may refer their sequel of papers 5, 6.
A hypergeometric function, namely the aomotosystems x2,4 and x3,6. In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. Fractional derivative, generating functions, appell function, generalized appell. A kummertype transformation for a 2f2 hypergeometric function. A kummertype transformation for a 2f2 hypergeometric. Fur thermore, linear and bilinear generating relations for these extended hypergeometric. 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. Sgeneralized gauss hypergeometric function sanjay bhatter and richa sharma1 1department of mathematics, malaviya national institute of technology, jaipur 302017, rajasthan, india. Note that if m is the value of the numerically largest a j that is a nonpositive integer, then the identity. As applications, some well known and new generating relations are deduced. In this article, we will introduce k hypergeometric functions which are extensions of the gauss hypergeometric functions including k pochhammer symbol. The main idea is to consider the q hypergeometric func. However, the corresponding k22z function is not explicitly known in general since it requires the solution of a three term recursion formula 2, 3.
The term confluent refers to the merging of singular points of families of differential equations. The solution of eulers hypergeometric differential equation is called hypergeometric function or gaussian function introduced by gauss. This is a reasonably popular topic, with goursats original 9page contribution 18 as the starting point. Amy removes three transistors at random, and inspects them. On a kummertype transformation for the generalized. Thus, we shall give below two simple and transparent derivations of eq. The solver returns the results in terms of the kummer u function and another hypergeometric function. Hypergeometric functions reading problems introduction the hypergeometric function fa, b. Introduction 2f1 hypergeometric seriesfunctoins hypergeometric functions over finite fields let q ps be a prime power. This is also known as the confluent hypergeometric function of the first kind. Reduction formulae for generalised hypergeometric functions. The qhypergeometric functions 1 are of great interest in modern mathematical physics because of the clues that they are expected to give in connection with the development of the theory of di. Let be the class of analytic functions in the unit disc, normalized by and of the. 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.
L inphas hypergeometric solution of degree three, i. Another method for a kummertype transformation for a 2f2 hypergeometric function article pdf available in communications of the korean mathematical society 223. Introduction in the world of standard functions, the hypergeometric functions take a prominent position in mathematics, both pure and applied, and in many branches of science. Integer ratios of factorials, hypergeometric functions, and related step functions by jonathan william bober a dissertation submitted in partial ful llment of the requirements for the degree of doctor of philosophy mathematics in the university of michigan 2009 doctoral committee. The parameters of the hypergeometric function are zeros and poles of this rational function. In addition to being of independent interest, this investigation was motivated by an application of gen. Some generalizations of pochhammers symbol and their. 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. The population or set to be sampled consists of n individuals, objects, or elements a nite population. Pdf another method for a kummertype transformation for. This result is specialised to the case where one pair of parameters differs by unity to generalize a recent result of miller j. 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. Pdf another method for a kummertype transformation for a.
Generating functions, hypergeometric function, fractional derivative op. This function is described by abramowitz and stegun 1972. In all these cases, the limiting behaviour is problematic. There is a different and unrelated kummers function bearing the same name. Hypergeometricpfqa 1,a 2,b 1,b 2,z 337 formulasprimary definition 2 formulas specific values 31206 formulas. Hypergeometric2f1 a, b, c, z has a branch cut discontinuity in the complex plane running from. 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 and the four askeywilson secondorder difference equa.
Some hypergeometric generating relations motivated by the. It is a solution of a secondorder linear ordinary differential equation ode. Hypergeometric2f1 automatically threads over lists. Some special cases of results were also discussed in the end.
Then the hypergeometric function pfq is defined to be the realanalytic function on s given by the series when n 1 and f is the real field, this reduces to the classical hypergeometric series. They were introduced by euler as power series expansions of the form 1 ab c1 z aa1. Hypergeometric2f1a,b,c,z 111951 formulasprimary definition 8 formulas specific values 111271 formulas. Integer ratios of factorials, hypergeometric functions. 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, in particularly ap. This report presents some of the properties of this function together with sixfigure tables and charts for the. In mathematics, the gaussian or ordinary hypergeometric function 2 f 1 a,b. Properties of generalized univariate hypergeometric functions. Hypergeometric2f1 can be evaluated to arbitrary numerical precision. The motivation for computing hypergeometric functions will be discussed, with details given of some of the practical applications of these functions. We will emphasize the algebraic methods of saito, sturmfels, and takayama to construct hypergeometric series and the connection with deformation techniques in commutative algebra. Pdf on jan 1, 20, medhat rakha and others published extension of identities due to preece and bailey involving product of generalized hypergeometric function 2f2. 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.
Department of mathematics and statistics, university of victoria, victoria, british columbia v8w 3r4, canada received. 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. Mathematical function, suitable for both symbolic and numerical manipulation. The recursions they satisfy gives rise to a system of partial di. Although many common functions can be defined as a special case of this function, it is typically better to use functions that take advantage of features of the specific function for more efficient and accurate evaluation.
Gottschalk and maslen 6 for the function 3f2 and we do so in a later section. Hypergeometric0f1, hypergeometric1f1, and hypergeometric2f1 are special cases of hypergeometricpfq. The analytic value of the hypergeometric function in these circumstances is typically of moderate modulus, but both. Pdf coefficient estimate of biunivalent functions based. Some integrals involving generalized hypergeometric functions. The series with p 2 and q 1 is known as the gauss series or ordinary hypergeometric series.
This thesis presents two algorithms to nd such solution in the following cases. Division of mathematical sciences, university of abertay dundee, dundee dd1 1hg, uk received 27 november 2003. Gaussian hypergeometric functions and traces of hecke operators sharon frechette, ken ono, and matthew papanikolas abstract. Hereei denotes the ith vector in the standardbasis of rn. Generating functions of special triple hypergeometric. In many special cases, hypergeometricpfq is automatically converted to other functions. The generalized hypergeometric series is sometimes just. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. Pdf extension of identities due to preece and bailey.
Initially this document started as an informal introduction to gauss. For some special values of the parameter for the roots, the hypergeometric functions are radial parts of zonal spheri. A kummertype transformation for a 2 f2 hypergeometric function r. Hypergeometric function identities in this section we summarize some facts concerning the important hypergeometric functions without giving their derivations. Each individual can be characterized as a success s or a failure f.
These functions generalize the euler gauss hypergeometric function for the rank one root system and the elementary spherical functions on a real semisimple lie group for particular parameter values. Algebraic ahypergeometric functions universiteit utrecht. 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. It has been ascertained that three of the transistors are faulty but it is not known which three. Coefficient estimate of biunivalent functions based on q. One can show that a function is hypergeometric if and only if in the representation fz x k. The result of expansion are expressible in terms of nested sums or another new functions, like harmonic polylogarithms e. Some generalizations of pochhammers symbol and their associated families of hypergeometric functions and hypergeometric polynomials rekha srivastava. Hypergeometric series and gaussian hypergeometric functions. Using bounded sequences, further generalizations of two main hypergeometric generating.
Series of gamma functions can usually be expressed as generalised hypergeometric. The coefficients in 7 are defined in terms of a generalization o. However, when one or more of the top parameters a j is a nonpositive integer the series terminates and the generalized hypergeometric function is a polynomial in z. On a kummertype transformation for the generalized hypergeometric function 2f2 allen r.
Let fc q denote the group of multiplicative characters on f q. This constant is built into the measure dsin order to prevent it appearing everywhere else. On a generalization of the rogers generating function. We first give an identity for kpochhammer symbol and certain linear generating functions for k hypergeometric functions. The systematic investigation of contour integrals satisfying the system of partial differential equations associated with appells hypergeometric function f 1 leads to new solutions of that system. However, it does cover aspects such as monodromy calculations for this system and a moduli interpretation of the underlying geometry. We should point out that algebraic transformations of hypergeometric functions, in particular, of modular origin, are related to the monodromy of the underlying linear di erential equations.
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