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MSIJAR VOL01 ISSUE02.Pdf ISSN No. (Online): 2581-7515 Parichay: Maharaja Surajmal Institute Journal of Applied Research Volume 1, Issue 2; July-December 2018 Contents 1. Integrals Involving Hypergeometric Function of Four Variables 1 Samta Gulia, Harish Singh 2. India’s Yes to Megatrends: Issues and Challenges with Special Emphasis on Future of Work and 4 th Industrial Revolution 7 Rajeshwari Malik 3. Impact of Demonetization: Structured and Unstructured Sector 15 Nitika Gupta, Kanika Budhiraja, Jatinder Kaur 4. Digitalization - a Key to Sustainable Development 19 Jatinder Kaur, Kanika Budhiraja 5. India’s Economic Policy: Its Bottlenecks and Implications 26 Preeti Malik, Alka Mittal, Harish Singh 6. Intrusion Detection Techniques for Mobile Cloud Computing in Heterogenous 5G Technologies 31 Harjender Singh 7. Revival of Glass Work Art of Purdilnagar, Hathras, U.P. 37 Alka Mittal, Harish Singh, Preeti Malik 8. Indian Administrative System: Some Reformative Measures 41 Kaptan Singh Vol 1. Issue 2; July-December 2018’ Parichay: Maharaja Surajmal Institute Journal of Applied Research I Parichay: Maharaja Surajmal Institute Journal of Applied Research Integrals Involving Hypergeometric Function of Four Variables Samta Gulia*, Harish Singh** Abstract: In this research paper we explain the first main case ͢! = ͢ʚ͢ − 1ʛʚ͢ − 2ʛ …………………3.2.1 of Integrals Involving Hypergeometric Function of Four We now define its generalization (read as suffix n) by ʚ∝ʛ) ∝ Variables of positive definite matrix of order m×m which has the equations many other similar sub cases, a comprehensive list of these integrals is given. Proof of all integrals are similar, therefore, ʚ∝ʛ) = ∝ ʚ∝ +1ʛʚ∝ +2ʛ ………………… ʚ∝ +͢ − 1ʛ, ͢ detailed proof is given in the case of integral (1) and so rest are quoted directly as below. ≥ 1, ) Keywords: Hypergeometric function, binomial, exponential, logarithmic and trigonometrical series = șʚ + ͢ − 1ʛ) ͕͘͢ ʚ ʛͤ = 1, ≠ 0. (Ͱͥ 1. INTRODUCTION This function is called the Factorial function. Clearly ʚ∝ʛ) First to explain this paper, we should know about sequence ʚ1ʛ) = ͢! and series as well as matrix sequence and matrix series. A matrix series is calculated by summation of all elements of For this factorial function we shall prove the result matrix sequence. Let A 0, A 1, A 2, ………., A n be the elements of matrix sequence then M(A)= Ϧ where M(A) is a )( ∝ ∝ +1 ∝ +m − 1 ∑(Ͱͤ ̻(, ʚ∝ʛ() = ͡ ʠ ʡ ƴ Ƹ … … … … … … ƴ Ƹ , matrix series. m Μ m Μ m Μ Where m is a positive integer and n is a non-negative integer. If the matrix series is a power series then we will be introduce By using above equations, We have the following formula Hypergeometric series as follows:- ( ∝ +͟ − 1 The Power series ʚ∝ʛ() Ͱ (ġĠ ș ƴ Ƹ ʚ1.1.2ʛ ͡ ) &Ͱͥ ͦ a. b aʚa + 1ʛb. ʚ͖ + 1ʛ ͮ We can define a general Hypergeometric series in real scalar 1 + ͮ + ͗ ͗ʚ͗ + 1ʛ 2! variable Z as below aʚa + 1ʛʚa + 2ʛb. ʚ͖ + 1ʛʚb + 2ʛ ͮͦ + ͗ʚ͗ + 1ʛʚ͗ + 2ʛ 3! mF n (α1, α2………. αm ; β1, β2 ……… βn; Z ) = ∞ r +⋯………….ʚ1.1.1ʛ (α1 )r ..( α 2 ) r .......( α m r Z) ∑ is one of the great generality; it includes as particular cases, r=0 (β1 r .() β 2 ) r .......( β n )r r! most of the familiar series like the binomial, exponential, logarithmic and trigonometrical series. where ( α)0 = 1 and ( α)m = α ( α+1)…………. ( α+m-1). Here α1, α2…… αm scalars and m upper parameters and β1, β2 …… βn For a=1 and b=c, the series reduces to simple geometrical are scalars and n lower parameters. Above expression is series. From the fact that the series (1.1.1) is a generalization defined for two, three and four variables for more see Exton 2, of the geometric series, it is called the Hypergeometric series. 3. Initially nineteen Hypergeometric functions of four We apply a formula in our result by using Factorial function as variables were introduced by Exton 4, then sixty four new we are familiar with the factorial of a positive integer. We Hypergeometric functions was formed. Further all forty write it as integrals of four variables has generalized by Vyas et al 11 . To *Assistant Professor, Government College, Badli, Haryana, India, [email protected] ** Professor, Maharaja Surajmal Insitute (GGSIPU), New Delhi, [email protected] Vol 1. Issue 2; July-December 2018 1 Samta Gulia, Harish Singh know more about this see Singh et al 9 and Chandel et al 8. In The notations used for quadruple hypergeometric series are this chapter we will discuss p.d.f of forty integrals. K1, K 2 ..... …… In this paper we explained first case of Integral involving K1 (a 1, a 1, a 1, a 2 ; b 1, b 1, b 1, b 1, c 1, c 2, c 3, c 4; ; x, y, z, t) Hypergeometric Function of four variable of positive definite matrix of order m×m which has many other similar cases, a ʚu,+ͮ,ͮ-ʛʚv,.ʛ ∑ ͬ+ . ͭ, . ͮ- . ͨ. comprehensive list of these integrals is given. Proof of all ʚu,+ʛʚv,,ʛʚw,-ʛʚx,.ʛ integrals are similar therefore detailed proof is given in the (1.3.1) case of integral (1) and so rest are quoted directly as below. Continuously like this we can make k 2, k3, k4, ………. We shall take p, q, r and s to be positive integers of the symbols ∆(n, a) and X stand for the sequence of parameters It is presumed that parameters and arguments in above r functions are so restricted that the series involved is α α + 1 α + n − 1 convergent. Other quadruple hypergeometric series have been , , ...., and W 1, W 2, .... W r n n n given by Sharma and Parihar. They introduced eighty three respectively. new hypergeometric series of variable four with the symbols (4) (4) (4) F1 , F 2 ...., F 83 to quote we have. 2. MULTIVARIABLE FUNCTIONS F4 (a , a , a , a , b , b , b , b , c , c , c , c ;x, y, z t) The great success of the theory of hypergeometric functions of 1 1 1 1 2 1 1 1 1 1 2 3 4 , a single variable has stimulated the development of a ∞ (a ) (a ) (b ) corresponding study of the theory in the two and more 1 p+q+r 2 s 1 p+q+r+s p q r s variables, in 1880, Appell introduced and studied = ∑ x y z t (1.3.2) p, q, r, s=0 p! q! r! s! (c 1 ) p (c 2 ) q (c 3 ) r (c 4 ) s systematically the four functions F 1, F 2, F 3 and F 4 which are the generalizations of the Gaussian hypergeometric functions 5 4 4 4 of two variables. In the year Horn defined ten hypergeometric Continuously like this we can makeF2 , F3 , F4 ……. Out function of two variables and denoted them by G 1, G 2, G 3, The of these eighty-three functions nineteen functions had already confluent forms of the four Appell functions were studied by been introduced by Exton. Humbert 6. A list of these functions is given in the work by 1 Erdely . The recent work on quadruple hypergeometric series has been done by Chandel et al 8, they introduced seven more possible In a similar manner to the generalization of the single hypergeometric functions of four variables and thus completed hypergeometric function the functions F 1 to F 4 and their the set of all possible quadruple hypergeometric series. confluent forms were further generalized by kampe de Feriet 7 who introduced a general hypergeometric function of two 4. INTEGRALS OF MATRIX VARIATE variables. The Kampe de Feriet function has further been 10 HYPERGEOMETRIC FUNCTIONS OF FOUR generalized by Srivastana et al . The explanation of Integrals VARIABLES of four variables was introduced by Singh et al 9. under Statistical Distribution associated with Hypergeometric 1 α-1 β-1 u function of matrix arguments. [1] ∫ |u| |1- u| F1(α, 1 –α ; γ; ) 0 2 3. THE QUADRUPLE HYPERGEOMETRIC 4 n n F [a , a , a , a , b , b , b , b , c , c , c , c , ; w|1-u| , x|1-u| , FUNCTIONS 1 1 1 1 2 1 1 1 1 1 2 3 4 y|1-u| n, z|1-u| n]du No specific study has been made for any hypergeometric n γ + β n 1 + γ + β series of four variables, apart from the four variables a a, a, a, b, b, b, b, c, c, c, c, ; ∆(n, β ), ∆( , ), ∆ , (4) (4) (4) (4) 1 1 1 2 1 1 1 1 1 2 3 4 1 4 2 2 2 2 Lauricella functions FA , F B , FC and FD and certain for = B F 1 n n 1 + γ + β − α their limiting cases, and the extremely generalized multiple ∆(n, γ + β ), ∆ ,γ + β + α , ∆ , ; w x y z hypergeometric functions of Srivastava et al 10 . On account of 2 2 2 the large number of such functions which arise from a (1.4.1) systematic study of all possibilities, Exton restricted here to those functions which are complete and of order second and Where which involves at least one product of type (a) p+q+r+s in the series representation : p, q, r and s are the indices of quadruple 4 F (a 1, a 1, a 1, a 2, b 1, b 1, b 1, b 1, c 1, c 2, c 3, c 4; w, x, y, z) summations. 1 Vol 1. Issue 2; July-December 2018 2 Integrals Involving Hypergeometric Function of Four Variables ∞ We assume that the series is uniformly convergent in the (a 1 ) p+q+r (a 2 ) s (b 1 ) p+q+r+s = ∑ wp xq yr zs region of integration, the inversion of integration and p, q, r, s=0 p! q! r! s! (c 1 ) p (c 2 ) q (c 3 ) r (c 4 ) s summation is infinite, then integral so that I α - 1 β − 1 u =∫ u -1 u F1(α, 1 – α; γ; ) 0 2 4 n n F1 [a 1, a 1, a 1, a 2, b 1, b 1, b 1, b 1, c 1, c 2, c 3, c 4;|1-u| w, |1-u| x, |1- n n ∞ u| y, |1-u| z] p, q p pn q qn r rm s sn × A w -1 u x -1 u y -1 u z -1 u du ∑ sr, p, q, sr, =0 4 n n n n F1 [|1-u| w, |1-u| x, |1-u| y, |1-u| z] (1.4.1.2) Then a probability density function (p.d.f.) of (1.4.1) is given by : (a ) (a ) (b ) p, q 1 p+q+r 2 s 1 p+q+r+s Where A sr, = α−1 β −1 4 p! q! r! s! (c 1 ) p (c 2 ) q (c 3 ) r (c 4 ) s |u| 1| − |u F1 (χ1 F) 1 (χ2 ) F(u) = 1 4 B F1 (χ3 ) ∞ p, q p q r s I α - 1 β − 1 = ∑ A r, s w x y z u 1 - u F1(α, 1 ∫0 = 0 else where p, q, r, s = 0 u [ 1 - u n(p + q + r + s) ] Where – α; γ; ) .
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