This thesis contains four papers which either prove or attempt to prove conjectures due to boyd. As applications, some well known and new generating relations are deduced. 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. The quotients of certain hypergeometric functions are presented as fractions which converge uniformly in the unit disc. If, in the future, you encounter bernoulli functions or polynomials, be sure to check what is intended. In this consent one may refer their sequel of papers 5, 6. Mathematical function, suitable for both symbolic and numerical manipulation. The parameters of the hypergeometric function are zeros and poles of this rational function. 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. This is a reasonably popular topic, with goursats original 9page contribution 18. The result of expansion are expressible in terms of nested sums or another new functions, like harmonic polylogarithms e.
Some hypergeometric generating relations motivated by the. If we replace m n by p, then we get ex np and vx n n n 1 np1 p. The rising factorial provides a direct link between the gamma and hypergeometric functions. Products of hypergeometric functions and the zeros of 4f3. Coefficient estimate of biunivalent functions based on q. The ratio m n is the proportion of ss in the population. Hypergeometric function identities in this section we summarize some facts concerning the important hypergeometric functions without giving their derivations. L inp has ve regular singularities where at least one of them is logarithmic. Pdf certain new summation formulas for the series 4f31 with. Summation and transformation formulas for hypergeometric functions at a. Hypergeometric distribution proposition the mean and variance of the hypergeometric rv x having pmf hx. 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. Transformation formulas for the generalized hypergeometric function. Generating functions, hypergeometric function, fractional derivative op.
The hierarchy of hypergeometric functions homepages of uva. The generalized hypergeometric series is sometimes just called the. International journal of mathematical analysis and applications. They were introduced by euler as power series expansions of the form 1 ab c1 z aa1. Hypergeometricpfqa 1,a 2,a 3,a 4,b 1,b 2,b 3,z 360 formulas primary definition 2 formulas specific values 189 formulas general characteristics 23 formulas series representations 31 formulas integral representations 5 formulas. We establish a simple inductive formula for the trace trnew k. Hypergeometric series, truncated hypergeometric series. Hypergeometric functions and mahler measure ubc library.
Davis goes so far as to state that of the socalled higher mathematical functions, the gamma function is undoubtedly the most fundamental cf. So, any question regarding the zoo of special functions generated by the. One can show that a function is hypergeometric if and only if in the representation fz x k. The natural generalization of the gausss hypergeometric function.
This thesis presents two algorithms to nd such solution in the following cases. Let fc q denote the group of multiplicative characters on f q. The hypergeometric functions can be generalized along the lines of basic or q number, resulting in the formation of qextensions or qanalogues. Pdf hypergeometric series, truncated hypergeometric. Trend analysis of generalized hypergeometric functions. Gaussian hypergeometric functions and traces of hecke operators sharon frechette, ken ono, and matthew papanikolas abstract. 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. Srivastava, junesang choi, in zeta and qzeta functions and associated series and integrals, 2012.
These results lead to the existence of certain hypergeometric functions in the class of either convex functions, or starlike functions 1. We will emphasize the algebraic methods of saito, sturmfels, and takayama to construct hypergeometric series and the connection with deformation techniques in commutative algebra. Applications on generalized hypergeometric and confluent 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. The hypergeometric distribution models drawing objects from a bin. Generating functions of certain hypergeometric functions by means of fractional calculus manoj singh1,sarita pundhir2, mukesh palsingh3 1department of mathematics, faculty of science,jazan university, jazan, saudi arabia. Generalized hypergeometric function, euler transformation. This chapter provides an overview of the qextensions of some special functions and polynomials. View related information in the documentation center mathworld. Watson, whipple and pfaffsaalschutz for 3f2, whipple for 4f3, dougall for.
Let be the class of analytic functions in the unit disc, normalized by and of the form. Certain properties of the generalized gauss hypergeometric. The identity between the two 4f3 hypergeometric series of unit argument is the reversal identity reverse the order of terms in the finite series expression. Trend analysis of generalized hypergeometric functions pramod joshi department of mathematics l. Hypergeometric0f1, hypergeometric1f1, and hypergeometric2f1 are special cases of hypergeometricpfq. Generating functions of certain hypergeometric functions. Hypergeometric series, truncated hypergeometric series, and gaussian hypergeometric functions article pdf available january 2015 with 222 reads how we measure reads. In many special cases, hypergeometricpfq is automatically converted to other functions. Series, expansion of the hypergeometric functions and. This constant is built into the measure dsin order to prevent. Using experimental methods, david boyd conjectured a large number of explicit relations between mahler measures of polynomials and special values of different types of lseries.
Nonterminating vwp 7f61 linear combination of two balanced 4f31s. Hypergeometric solutions of linear differential equations. 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 applications on generalized hypergeometric and. Hypergeometric function an overview sciencedirect topics. Identities for the gamma and hypergeometric functions. Pdf the main objective of this paper is to provide thirteen presumably. Some properties of generalized hypergeometric coefficients core. L inphas hypergeometric solution of degree three, i. Internal report sufpfy9601 stockholm, 11 december 1996 1st revision, 31 october 1998 last modi. Acceleration of generalized 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. Recently, a function has attracted many researchers attention due mainly to diverse applications, which are more general than the beta type function bx,y, popularly known as generalized beta type functions.
Initially this document started as an informal introduction to gauss. Pdf coefficient estimate of biunivalent functions based. Using bounded sequences, further generalizations of two main hypergeometric generating relations have also been. We consider the polynomial case and show how recent results cf. 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. In the theory of the generalized hypergeometric functions, a remarkable amount of. In section 2 we give the essential notation for the basic hypergeometric functions.
Particular cases of our results presented for the q3j and the q6j coefficients are contained in reference i and kachurik and klimyk 1990 respectiveiy. The random variate represents the number of type i. Expansion of hypergeometric functions of several variables around integer values of parameters. 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. Download all formulas for this function mathematica notebook pdf file. Generating functions of special triple hypergeometric. Campus almora, india abstract an innovative attempt to. Sgeneralized gauss hypergeometric function sanjay bhatter and richa sharma1 1department of mathematics, malaviya national institute of technology, jaipur 302017, rajasthan, india. Hypergeometric functions hypergeometricpfqa 1,a 2,a 3,a 4,b 1,b 2,b 3,z. The motivation for this study of generalised hypergeometric functions is provided by.
In this paper, we investigate the relationships among hypergeometric series, truncated hypergeometric series, and gaussian hypergeometric functions through some families of hypergeometric algebraic varieties that are higher dimensional analogues of legendre curves. Dhami head, department of mathematics kumaun university, s. Hypergeometric functions reading problems introduction the hypergeometric function fa, b. Some summation theorems for generalized hypergeometric. Reduction formulae for generalised hypergeometric functions of one variable j e gottschalk and e n maslen. 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 integral equationssee 4, 5, 7, 12. The logarithmic mahler measure of an nvariable laurent polynomial, px1. The main idea is to consider the q hypergeometric func. M is the total number of objects, n is total number of type i objects.
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