International Journal of Trends and Technology (IJMTT) – Volume 40 Number 2- December2016

Hypergeometric Functions: Application in Distribution Theory

Dr. Pooja Singh,

Maharaja Surajmal Institute of Technology (Affiliated to GGSIPU), Delhi

ABSTRACT: Hypergeometric functions are generalized from exponential functions. There are functions which can also be evaluated analytically and expressed in form of hypergeometric . In this paper, a unified approach to hypergeometric functions is given to derive the probability density function and corresponding cumulative distribution function of the noncentral F variate.

Key words and phrases: Hypergeometric functions; distribution theory; chi-square Distribution, Non- centrality Parameter.

I) Introduction extensively in Cox and Hinkley(1974), Craig(1936), Feller(1971), Hardle and The hypergeometric function is a special function Linton(1994), Muellbauer(1983). Further encountered in a variety of application. Higher-order applications in statistics/econometrics and transcendental functions are generalized from economic theory a r e suggested throughout. hypergeometric functions. These functions are Here, Hypergeometric function is applied applied in different subjects like theoretical physics, effectively to Distribution Theory. and also functional in as Maple and Mathematica. They are also broadly explored in the II) The generalized Hypergeometric area of statistical/econometric distribution theory. Before introducing the hypergeometric function, we The purpose of this paper is to understand define the Pochhammer’s symbol importance of collection of hypergeometric function in different fields especially initiating economists to the wide class of hypergeometric functions. The paper is organized as follows. In Section II, the generalized hypergeometric series is explained with some of its properties. In Sections 3 and 4, some famous special cases are discussed which will be used in our important results. In Section 5, application to distribution theory is given. It leads to the derivation of the exact cumulative distribution function of the noncentral F variate. If we substitute in eqn 1 we are left with

An appendix is attached which summarizes notational abbreviations and function names.

The paper is of an introductory nature. Much of If we multiply and divide eqn (2) by then the content is new unpublished formulae which are we get integrated with the mathematical literature. The subject was developed by the three volumes edited by Erdélyi ( 1953, 1955). The hypergeometric functions are classified in Abramowitz and stegun(1972), Carlson (1976), G a s p e r and Rehman(1990 ). For integrals involving such functions, see Chandel, Agarwal and Kumar(1992) Choi, Hasanov and Turaev(2012), So if we set Exton(1973), Joshi and Pandey(2013), Saran(1955), Seth and Sindhu(2005), Usha and Shoukat(2012).

For the theory, we referred to Whittaker and Watson (1927), Erdélyi (1953, 1955) for a more comprehensive proof refer to Anderson(1984), Slater The general Hypergeometric function is given by (1966), Luke (1969), Olver (1974), Mathai (1993). Statistics/econometric theories was explained

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International Journal of Mathematics Trends and Technology (IJMTT) – Volume 40 Number 2- December2016

An important property that we will need to use is the The and convergence criteria of the hypergeometric functions are called the numerator and denominator depending on the values of and . The radius of parameters respectively and is called the convergence of a series of variable is defined as a argument. We deal within this paper will concern value such that the series converges if eqn (4) but for the most part we will deal with and diverges if , where , in the specific cases of the general hypergeometric case , is the centre of the disc convergence. For function. For instance, hypergeometric function, provided and are not non negative integers for any , the relevant If we take in eqn (4) we get convergence criteria stated below can be derived

using the ratio test, which determines the absolute

convergence of the series using the limit of the ratio of two consective terms.

If we let and and by using eqn (1) a) If , then the ratio of coefficients of in the of the hypergeometric function

tends to as ; so the radius of

convergence is , so that the series converges for

all values of . Hence is entire. In

. particular, the radius of convergence for and

is .

b) If , the ratio of coefficients of tends to as , so the radius of convergence is ,

so that series converges only if . In particular, the radius of convergence for is

. c) If , the ratio of coefficients of which is the binomial expansion. tends to as , so the radius of convergence is Some instant result follows from eqn (4), when one , so that the series does not converge for any value of . of the parameter of is non negative integer, then We will seek approximation to the relevant

hypergeometric function for within radii of

convergence. For , as given by Luke Also, (1975), there is a restriction for convergence on the unit disc, the series only converges absolutely at if

And interchanging elements separated by commas is feasible as multiplication is commutative.

So that the selection of values for and

must reflect that.

III) The Hypergeometric Function

The Guass hypergeometric function is defined as But, interchanging of semicolons (i.e, between and ) is not allowed as division is not commutative.

Also, if , then

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International Journal of Mathematics Trends and Technology (IJMTT) – Volume 40 Number 2- December2016

Also,

where is in the radius of

convergence of the series

Integral over

where The first result is a representation of in terms of beta integral, This can also be obtained by using binomial theorem and integrating term by term. For the series is finite with terms in it, and it can also be derived by successive integration (by parts),

In terms of more familiar quantities, the hypergeometric function is

If , on rearranging the above equation in such

a way that negative term follows every two

consecutive positive term then we get for ..(14) . See Wittaker and Watson(1927) for proof. This expression can be obtained by expanding This is the expansion of log function in infinite by binomial theorem and integrating series. Series is absolutely convergent if termwise. and conditionally convergent if

If we substitute the variable this will give,

where is arbitrary.

is infinite when and or

and . The special case produces, So with these two exceptions, series expansion will give a finite value. For convergence

of the hypergeometric series, we must have . General formula for analytic continuation of

Gauss Series is given in the volumes of Erdelyi (1955)

IV) Kummer’s Confluent Hypergeometric where we applied Function

To eliminate , we put to obtain, The confluent hypergeometric function denoted by

is defined by,

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International Journal of Mathematics Trends and Technology (IJMTT) – Volume 40 Number 2- December2016

By decomposing it into an integral from 0 to ∞,

Following two formulas of Kummer are useful for our results.

where a is an arbitrary.

.…(24) This derivation shows that integrals of leads to a geometric function. The is the elementary example of the hypergeometric series. All the functions Other special case is standard normal cumulative studied here can be considered as generalization of distribution function. elementary transcendental function; Further, special cases arise when compared with Poisson process as discussed in Hardle and Linton (1994). We also have integral representation,

where is signum function,

where is the modified of the This is a case incomplete first kind with order . is used to represent cumulative distribution function (cdf) of the standard normal distribution. The incomplete gamma functions arise from Gamma distribution also have exponential pdf with Euler’s integral for the gamma function, negative value which is used in consumer theory by Delgado and Dumas(1992).

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International Journal of Mathematics Trends and Technology (IJMTT) – Volume 40 Number 2- December2016

As we know Kummer’s function satisfies a basic relation,

Equation (32) can be rewritten as,

This equation was derived by Sentana(1995) with help of Leibniz formula of fractional integral. In view this eqn (28) can be rewritten as

with the weights is from Poisson density.

where is We will get corresponding cdf by term wise standard normal density function. integration of equation (32) as,

V) Application in Distribution Theory

If follows a non central Chisquared distribution with d degree of freedom and non central parameter then

or ……(31)

In most of the following notation p is fixed and will not be explicitly stated in the notation. Also, it

is useful to set .

The formula for the pdf involves Bessel Function which can be limiting We must recall that , so behavior, if , then .

as In addition, if is independent from U, then

, with noncentral F Consider, distribution with degree of freedom and in denominator with noncentral parameter .

Cox and Minkley(1974) had given the definition of equ (32) and further follows from eqn(26). When the above distribution reduces to

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International Journal of Mathematics Trends and Technology (IJMTT) – Volume 40 Number 2- December2016

This gives the first definition and next one follows Mathematica. A major advantage they have is their from using beta function where parsimonious generality, and their ability to give

. The very last line follows from eqn (18) explicit answers to problems. It is hoped that this paper has made the case for their potential in quantitative economics. The cumulative distribution function may be found by The extension for the content of this paper is in at least three possible ways. Firstly, Meijer’s G and

Fox’s functions are of generalized hypergeometric function. These functions will help

us in analytical manipulation of functions and its argument. Secondly, in this paper we had discussed

hypergeometric function of one variable but we can extend it to more than one variable. We can rewrite

hypergeometric function for two variables, instead

of single variable functions. Thirdly, it is assumed for convenience that z is a scalar but we can pursue without it. We can define hypergeometric functions even if we have the argument as a square matrix. If we define a matrix function whose output is a scalar, we get the type of hypergeometric functions used in

multivariate distribution theory.

References:

[1] M. Abramowitz, I.A Stegun, Handbook of mathematical functions, Dover publications, New York (1972).

[2] T.W. Anderson, An introduction to multivariate statistical

analysis (2nd ed.), John Wiley & sons, New York (1984).

[3] B.C. Carlson, The need for a new classification of double hypergeometric series, Proc. Nat. Acad. Sci, 56 (1976),221- 224. ……..(38) [4] R.C.S. Chandel, R.D. Agarwal,H. Kumar, Hypergeometric functions of four variables and their integral representation,

The Mathematics Education Journal, 26 (1992),74-94. with

[5] J. Choi, A. Hasanov,M. Turaev, Integral representation for Srivastava’s Hypergeometric function Hb, Journal of Here we have applied the definition of Korean Society Mathematics Education,19 (2012), 137- Incomplete beta function and in the last step we 145. have applied the eqn (18) [6] D.R.Cox, D.V. Hinkley, Theoretical Statistics, Chapman Conclusion & Extensions and Hall, London, (1974). [7] C.C. Craig, “On the frequency function of xy”, Annals Hypergeometric functions are able to occur in of Mathematical Statistics, 7 (1936), 1-15. fractional calculus [e.g. Cox and Hinkley (1974)]. The nature of implementation of these functions is [8] A. Erdélyi, Higher transcendental functions, volumes 1-2, in data (e.g. fractionally integrated) of time series Mc.Graw-Hill, N.Y(1953). and further area of economics. Given the [9] A. Erdélyi, Higher transcendental functions, volume 3, determination of unemployment and price rises, this Mc.Graw-Hill, N.Y. (1955). relation seems to have significance for economist. We had derived the exact cumulative distribution [10] H. Exton, Some Integral representation and function by using Bessel function, incomplete transformations of hypergeometric function of four variables, Bull. Amer. Math. Soc., 14 (1973),132-140. gamma, Gauss hypergeometric function or other relevant functions. [11] W. Feller, An introduction to probability theory and its applications (2nd ed.), John Wiley & sons, New A final statement on hypergeometric functions. York(1971). They have now become so important in many areas [12] G. Gasper, M. Rahman, Basic hypergeometric series, of applied mathematics that they can be found in Cambridge University Press, Cambridge (1971). many packages, including ones allowing symbolic manipulations like Maple and

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International Journal of Mathematics Trends and Technology (IJMTT) – Volume 40 Number 2- December2016

[13] W.Härdle,O. Linton, “Applied nonparametric methods”, C , N , R, Z : the sets of complex, natural, real, and integer in R.F. Engle and D.L. McFadden (eds.), Handbook of numbers, respectively. Econometrics, Amsterdam(North-Holland) (1994). pdf: probability density function. [14] S. Joshi,R.M. Pandey, An integral involving Gauss Hypergeometric Function of the Series, International cdf: cumulative distribution function. Journal of Scientific and Innovative Mathematical Research, 1 (2013) 117-120. : the imaginary unit.

[15] Y.L.Luke, The special functions and their approximations, |z| : modulus (or absolute value) of z. volumes 1-2, Academic press, New York(1969). B(x, y) = Γ(x)Γ(y)/Γ(x + y) : Beta function. [16] A.M. Mathai, A handbook of generalized functions for statistical and physical sciences, Oxford University Press, Γ(ν) : gamma function. Oxford(1993).

Γ 1 Γ 1 j j : Binomial [17] J. Muellbauer, “Surprises in the consumption function”, Economic Journal, 93 (1983), 34-50. Coefficients.

[18] F.W.J. Olver, Asymptotics and special functions, 1 j 1 Γ

Academic Press, New York(1974). j Γ : Pochhammer’s symbol.

[19] S. Saran, Integarls associated with hypergeometric γ(ν , z), Γ(ν, z) : incomplete gamma functions. functions of these variables, National Institute of Science of India, 21,No. 2 (1955) 83-90. : generalized hypergeometric series. [20] J.P.L. Seth, B.S. Sidhu, Multivariate integral representation suggested by Laguerre and Jacobi

of matrix argument, Vihnana Prasad : Gauss Anusandhan Patrika, 48, No.2 (2005),171-219. hypergeometric series (the hypergeometric function). [21] L.J. Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge(1966). : Kummer’s function (confluent/degenerate hyperge ometric [22] B.Usha, A. Shoukat, An Integral containing function). Hypergeometric function, Advances in Computational Mathematics and its Applications, 2, No.2 (2012), 263- φ(z) , Φ(z) : standard Normal pdf and cdf 266. respectively.

[23] E.T. Whittaker, G.N.Watson, A course of modern analysis int(.) : integer part of the argument. (4th ed.), 15th printing 1988, Cambridge University Press, Cambridge(1927). : modified Bessel function of the first kind of order . APPENDIX : Special notational and functions sgn(z) : signum (sign) function of z; returning ≡ : identity; when variables or functions are equivalent for ±1 for z R±, or 0 for z = 0. all defined values of the parameters and the arguments.

= : equality; when two expressions are not equivalent, but have equal principal values or are equal for a certain range of parameter or argument values.

∼ : distributed as.

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