Acceleration of generalized hypergeometric functions. Initially this document started as an informal introduction to gauss. Hanballa1 1 department of mathematics, aden university, aden, kohrmaksar, p. Pdf a note on the asymptotic expansion of generalized. More special cases of the generalization of classical generalized hypergeometric function are given as results. Counting points over finite fields and hypergeometric functions adriana salerno abstract. Moreover the generalized hypergeometric function includes many other functions as special cases, such as the bessel functions, the classical orthogonal polynomials and elementary functions. The code executes a while loop to calculate the generalized hypergeometric series within a specified tolerance. Kummers confluent hypergeometric function ma, b, z, introduced by kummer 1837, is a solution to kummers differential equation. The output will have the same dimensions as z and will be doubleprecision unless z is singleprecision, in which case, the output will be as well.
Some identities with generalized hypergeometric functions. Secondary 11b83, 30b10, 33b10, 33b15, 33c05, 33c20, 65b10. Several results on harmonic functions by involving hypergeometric functions have recently been studied in. The hurwitz or generalized zeta function denoted usually by, za and so for, 0 1, z n za an f. Askey department of mathematics, university of wisconsin, madison. The hypergeometric function of a matrix argument is scalarvalued,whichis a major distinction from other functions of a matrix argument e. The generalized hypergeometric function pf qis known to satisfy the following di erential equation 64. Generalized hypergeometric function is one of the most important special functions. New applications and recurrence relations for the generalized gauss hypergeometric function gghf are given. Generalized hypergeometric function, hypergeometric summation theorem, laplace transform, ramanujans integrals. Miller formerly professor of mathematics at george washington university, 1616 18th street nw, no. In this paper, we extend this result, due to igusa, to a family of mono. This is also known as the confluent hypergeometric function of the first kind.
In mathematics, the gaussian or ordinary hypergeometric function 2 f 1 a,b. This constant is built into the measure dsin order to prevent. Mathematical function, suitable for both symbolic and numerical manipulation. Some applications on generalized hypergeometric and. If we replace m n by p, then we get ex np and vx n n n 1 np1 p. The series with p 2 and q 1 is known as the gauss series or ordinary hypergeometric series. Jun, 2011 this function estimates the solution to kummers differential equation within a specified tolerance. The recursions they satisfy gives rise to a system of partial di. In this course we will study multivariate hypergeometric functions in the sense of gelfand, kapranov, and zelevinsky gkz systems. We derive extensions of the classical summation theorems of kummer and watson for the generalized hypergeometric series where r pairs of numeratorial and denominatorial parameters differ by positive integers. Paris division of complex systems, university of abertay dundee, dundee. Ordinary linear differential equations note that if we replace y by sy in the system, where s. This function is described by abramowitz and stegun 1972.
On the connection between generalized hypergeometric. One can show that a function is hypergeometric if and only if in the representation fz x k. Pdf the generalized hypergeometric difference equation. Transformation formulas for the generalized hypergeometric function with integral parameter di erences a. If you plan on executing the function more than 8,192 times in any given matlab session, you will need to either a run matlab nojvm from cygwin, or comment out all of the write statements in the fortran code. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. Hypergeometricpfqregularized is finite for all finite values of its arguments so long as. The associated parametric polynomial for the transformation 1. For certain special arguments, hypergeometricpfqregularized automatically evaluates to exact values. Agarwal and kalla 1 defined and studied a generalized ga mma. Extension of some classical summation theorems for the.
But avoid asking for help, clarification, or responding to other answers. The ratio m n is the proportion of ss in the population. Some applications on generalized hypergeometric and confluent. Reduction formulae for generalised hypergeometric functions. It is a well known result that the number of points over a nite eld on the legendre family of elliptic curves can be written in terms of a hypergeometric function modulo p.
Some integrals involving generalized hypergeometric. Three lectures on hypergeometric functions eduardo cattani abstract. Simple connections between dilogarithms and a kind of appells function are shown. Hypergeometric functions with parameter 17 duality 24 7. Pdf a generalization of hypergeometric and gamma functions.
For example, there is a notion due to horn of multivariate hypergeometric series in terms of the coe. Internal report sufpfy9601 stockholm, 11 december 1996 1st revision, 31 october 1998 last modi. Handbook on statistical distributions for experimentalists. The parameters of the hypergeometric function are zeros and poles of this rational function. The hypergeometric function of a matrix argument generalizes the classical hypergeometric function to which it reduces for n 1. As a special case, certain known or new generating relations to some classical orthogonal polynomials discussed which constitute good models for. 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. Archived from the original pdf on 20170624 cs1 maint. A note on the asymptotic expansion of generalized hypergeometric functions article pdf available in analysis and applications 1201 december 20 with 662 reads how we measure reads. Some generalizations of pochhammers symbol and their. A great number of common mathematical functions are expressible in terms of hypergeometric.
A numerical evaluator for the generalized hypergeometric function for complex arguments with large magnitudes using a direct summation of the gauss series. Moreover, we have shown a simple relationship between a kind of. Generalized hypergeometric function file exchange matlab. Beukers, frits 2002, gauss hypergeometric function.
Confluent hypergeometric function file exchange matlab. There is a different and unrelated kummers function bearing the same name. The motivation for computing hypergeometric functions will be discussed, with details given of some of the practical applications of these functions. Hypergeometric distribution proposition the mean and variance of the hypergeometric rv x having pmf hx. Gradshteyn and ryzhik 5 of utility for the evaluation of feynman loop integrals are obtained as particular cases. Several results on harmonic functions by involving hypergeometric functions have recently been studied in 1 to 4. Otherwise the function is called a generalized hypergeometric function. Hypergeometricpfqregularized can be evaluated to arbitrary numerical precision. Hypergeometricpfqa 1,a 2,b 1,b 2,b 3,z 145 formulasprimary definition 2 formulas specific values 24 formulas. Thanks for contributing an answer to mathematics stack exchange. Thus, we can further obtain various integral formulas involving a number of simpler special functions. In addition, the generalized gauss hypergeometric function i.
Several integrals involving powers and ordinary hypergeometric functions are rederived by means of a generalized hypergeometric function of two variables appells function recovering some wellknown expressions as particular cases. Special functions, generalized hypergeometric function, fractional calculus 1 introduction generalized hypergeometric function is one of the most important special functions. Motivated essentially by the demonstrated potential for applications of the generalized incomplete hypergeometric functions p. Cambridge university press 0521833574 basic hypergeometric. Transformation formulas for the generalized hypergeometric. A difference equation analogue of the generalized hypergeometric differential equation is defined, its contiguous relations are developed, and its relation to numerous wellknown classical special functions are demonstrated. Every secondorder linear ode with three regular singular points can be transformed into this. Generalized hypergeometric functions, generalized mittaglef. Hypergeometricpfqregularizedwolfram language documentation. N, d, and z can be any numeric or logical datatype and may be complex.
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