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IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal ...

IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.

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  • 1. IOSR Journal of Mathematics (IOSR-JM)e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 16-19www.iosrjournals.orgwww.iosrjournals.org 16 | PageI- Function and H -Function Associated With Double IntegralAshok Singh Shekhawat*, Rakeshwar Purohit**And Jyoti Shaktawat****Department of Mathematics, Regional college for Education, Research and Technology,Jaipur, Rajasthan (India)**Department of Mathematics and Statistics, University College of Science, Mohan Lal Sukhadia University,Udaipur, Rajasthan (India)***Department of Mathematics, Kautilya Institute of Technology and Engineering,Jaipur, Rajasthan (India)Abstract: The object of this paper is to discuss certain integral properties of a I -function and H -function,proposed by Inayat-Hussain which contain a certain class of Feynman integrals, the exact partition of a Gaussianmodel in Statistical Mechanics and several other functions as its particular cases. During the course of finding,we establish certain new double integral relation pertaining to a product involving I function and H function.These double integral relations are unified in nature and act as a key formulae from which we can obtain as theirspecial case, double integral relations concerning a large number of simple special functions. For the sake ofillustration, we record here some special cases of our main results which are also new and of interest bythemselves. All the result which are established in this paper are basic in nature and are likely to find usefulapplications in several fields notably electrical network, probability theory and statistical mechanics.Key words. I function, H -function, Hermite polynomials, Laguerre polynomials.I. IntroductionThe H -function [6] is a new generalization of the well known Fox’s H-function [4]. The H -functionpertains the exact partition function of the Gaussian model in statistical mechanics, functions useful in testinghypothesis and several others as its particular cases. The conventional formulation may fail pertaining to thedomain of quantum cosmology but Feynman path integrals apply [10,11]. Feynman integral are useful in thestudy and development of simple and multiple variable hypergeometric series which in turn are useful instatistical mechanics.The I-function defined asjijiajjajijibjj(bzII[z]zI ip1,njijian1,jjaiq1,mjijibm1,jj{(bnm,iqip dszs)21 s wheres)as)b1s)a1s)bs)jijiip1njjijiiq1mj1ijjn1jjjm1j…(1.1)The H -function will be defined and represent as given in [1]   dxi21xHx]HiiP1,NjjaN1,jAjj(aQ1,MjjbM1,jj(bNM,QP,NM,QP,…(1.2)where
  • 2. I- Function And H -Function Associated With Double Integralwww.iosrjournals.org 17 | PagejjP1NjjjQ1MjjjjN1jjjM1jab1Aa1b…(1.3)which contains fractional powers of some of the gamma functions. Here aj (j = 1,…,P) and bj (j =1,…,Q) are complex parameters, aj  0 (j = 1,…,P), j  0 (j = 1,…,Q) (not all zero simultaneously and theexponents Aj (j = 1,…,N) and Bj (j = M+1,…,Q) can take on non-integer values. The contour in (1.2) isimaginary axis R() = 0. It is suitably indented in order to avoid the singularities of the gamma functions and tokeep those singularities on appropriate side. Again for Aj (j=1,…,N) not an integer, the poles of thegamma function of the numerator in (1.3) are converted to branch points. However, a long as there is nocoincidence of pole from any N)1,...,ja1andM)1,...,jb jjjj pair, thebranch cuts can be chosen so that the path of integration can be distorted in the useful manner. For the sake ofbrevity0BAT jP1NjjjQ1MjjjN1jjM1j II. Main ResultWe will obtain the following result:(A) dydxxy1yv1Hvyxy1x1Iy)x)(1(1xy1xy1y1vyxy1x1 NM,QP,1010vHvIsP1,NjjaN1,jAjja111k1Q1,MjBjjbM1,jjb1NM,1Q1,Pjijiajjajijibjj(b…(2.1)provided that  T21|Varg0b[R jjProof. We haves)as)b1sa1s)b21vxy1y)1Hvyxy1x1Ijijiip1njjijiiq1mj1ijjn1jjjm1jNM,QP, .  dvxy1y1ab1Aa1bi21dsvyxy1x1jjP1NjjjjQ1MjjjjN1jjjM1jiis…(2.2)Multiplying both sides of (2.2) by y)x)(1(1xy1xy1y1yxy1x1and integration with respect to x and y between 0 and 1 for both the variable and making a use of a known result[2, p.145],we get the required result (2.1) after a little simplification
  • 3. I- Function And H -Function Associated With Double Integralwww.iosrjournals.org 18 | Page(B) dvduv]HI(u)uvv)uNM,QP,1100 dzzz)zIs 10jijiajjajijibjj(b . dzzHP1,NjjaN1,jAjja111s1Q1,MjBjjbM1,jjb1NM,1Q1,P …(2.3)provided that  0bR( jjProof. Using (1.1) and (1.2), we havedsus)as)b1sa1s)b21v]HI(u) sjijiip1njjijiq1mj1ijjn1jjjm1jNM,QP,.  dvab1Aa1bi21jjP1NjjjjQ1MjjjjN1jjjM1jii…(2.4)Multiplying both side by11uvv)u  and integrating with respect to u and v between 0 and for both the variable and make a use of a known result [2, p.177], we get the required result. Lettingpzez)  in (2.3), we get the particular case after simplification(c) dvduv]1Hu)]I[(v(1vv)1u)(1f(uv)NM,QP,111010 1s10jijiajjajijibjj(bz)(1f(z)z)(1Is  . dzz)1HP1,NjjaN1,jAjja111s1Q1,MjBjjbM1,jjb1NM,1Q1,P…(2.5)provided that R() > 0, R() > 0.Proof. Using equation (1.1) and (1.2), we havedsu)1vs)as)b1sa1s)b21v]1Hu)]I[v(1 ssjijiip1njjijiq1mj1ijjn1jjjm1jNM,QP,
  • 4. I- Function And H -Function Associated With Double Integralwww.iosrjournals.org 19 | Page.  dv)1ab1Aa1bi21jjP1NjjjjQ1MjjjjN1jjjM1jii…(2.6)Multiplying both side of (2.6) by vv)1u)(1f(uv) 11and integrating with respect to u and vbetween 0 and 1 for both the variable and use of result [2, p.243] and by further simplification, we get the result(2.5).Letting f(z) =1z in (2.5), we get the particular result after simplification.Particular Case(i) Taking vyxy1x1Svyxy1x1I mnThe result in (2.1), (2.2) and (2.3) reduces to the known result after a slight simplification obtained by Chaurasiaand Shekhawat [2].Acknowledgement.The authors are grateful to Professor H.M. Srivastava, University of Victoria, Canada for his kind help andvaluable suggestions in the preparation of this paper.References[1]. R.G. Buschman and H.M. Srivastava, The H -function associated with a certain class of Feynman integrals, J. Phys. A: Math. Gen..23 (1990), 4707-4710.[2]. V.B.L. Chaurasia and Ashok Singh Shekhawat, Some integral properties of a general class of polynomials associated with Feynmanintegrals, Bull. Malaysian Math. Sc. Soc. (2) (28) (2) (2005), 183-189.[3]. J. Edewards, A Treatise on integral calculus, Chelsea Pub. Co., 2 (1922).[4]. C. Fox, The G and H-functions as Symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98 (1961), 395-429.[5]. C. Grosche and F. Steiner, Hand Book of Feynman Path integrals, Springer Tracts in Modern Physics Vol.145, Springer-Verlag BerlinHeidelberg, New York, 1998.[6]. A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals : I. Transformation and reductionformulae, J. Phys. A: Math. Gen. 20 (1987), 4109-4117.[7]. A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals :II. A generalization of the H-function, J. Phys. A: Math. Gen. 20 (1987), 4119-4128.[8]. Saxena, R.K., On fractional integer operators, Math. Zeitschr, 96 (1967), 288-291.[9]. Saxena, R.K. and Gupta, N., Some Abelian theorems for distributional H-function Transformation, Indian J. Pure Appl. Math. 25(8)(1994), 869-879.[10]. H.M. Srivastava, A contour integral involving Fox’s H-function, Indian J. Math. 14 (1972), 1-6.[11]. H.M. Srivastava and N.P. Singh, The integration of certain products of Multivariable H-function with a general class of Polynomials,Rend. Circ. Mat. Palermo 2(32) (1983), 157-187.[12]. C. Szego, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 23 Fourth edition, Amer. Math. Soc. Providence, Rhode Island(1975).