Pak. J. Statist. 2013 Vol. 29(2), 139-153 A GENERALIZATION OF HYPERGEOMETRIC AND GAMMA FUNCTIONS 1 2 G. Mustafa Habibullah1, Abdus Saboor1 and Munir Ahmad1 National College of Business Administration and Economics, Lahore, Pakistan. Email: mustafa1941@yahoo.com; drmunir@ncbae.edu.pk Department of Mathematics, Kohat University of Science and Technology, Kohat, Pakistan. Email: dr.abdussaboor@kust.edu.pk; saboorhangu@gmail.com ABSTRACT Al-Saqabi et al. [4] defined a gamma-type function and its probability density function involving a confluent hypergeometirc function 1 of two variables [7], where  1 a, b; c;  x  , x   a k l  b k  x     l , k0  c l  k k !l !  k    l x , x   1 and discussed some of its statistical functions. We propose extension of 1 by introducing more parameters in following form: b b+1 d d +1   a, , ,c; , ,e, f; -αx-δ , βxδ  2 2  2 2  m  b   b 1  a m  n    c n x  x      2 m  2 m    d   d 1  m, n 0  2   2   e n  f n m !n !  m  m     n , x   1 . We then define gamma-type function involving newly defined hypergeometric function of two variables and discuss its probability density function along with some of its associated statistical functions. We use inverse Mellon transform technique to derive closed form of gamma-type function and moment generating function. KEYWORDS Gamma function; inverse Mellon transform; hypergeometric function of two variables; moment generating function; moments. 1. INTRODUCTION Kobayashi [11] considered a generalized gamma function, m (u, v) . Galue et al. [8] generalized Kobayashi [11] gamma function by introducing Gauss hypergeometric function in it. Agarwal and Kalla [1] defined and studied a generalized gamma distribution. They used a modified form of the generalized gamma function of Kobayashi [11, 12]. Ghitany [9] discussed additional properties for gamma function defined by © 2013 Pakistan Journal of Statistics 139 140 A generalization of hypergeometric and gamma functions Agarwal and Kalla [1]. Al-Musallam and Kalla [2, 3] extended gamma function by involving Gauss hypergeometric function. Al-Musallam and Kalla [2, 3] and Kalla et al. [10] then discussed some of its properties. Provost et al. [14], Saboor and Ahmad [16] and Saboor et al. [17] discussed such generalizations. The remainder of this section is devoted to the inverse Mellin transform technique, which is central to the derivation of the closed form of gamma-type function and the moment generating function of the gamma-type distribution. If f  x  is a real piecewise smooth function that is defined and single valued almost everywhere for x  0 and such that 0 x k 1 f  x  dx converges for some real value k ,  then M f  s   0 x s 1 f  x  dx is the Mellin transform of f  x  . Whenever f  x  is  continuous, the corresponding the inverse Mellin transform is f  x  1 c i  s x M f  s  ds  2i c i (1.1) which together with M f  s  ; constitute a transform pair. The path of integration in the complex plane is called the Bromwich path where Bromwich path is a part of integration in the complex plane running from c  i to c  i , where c is a real positive number chosen so that the path lies to the right of all singularities of the analytic. Equation (1.1) determines f  x  uniquely if the Mellin transform is an analytic function of the complex variable s for c1    s   c  c2 where c1 and c2 are real numbers and   s  denotes the real part of s . In the case of a continuous nonnegative random variable whose density function is f  x  , the Mellin transform is its moment of order  s  1 and the inverse Mellin transform yields f  x  . Letting M f s    in1  1  ai  s  , qj m1  1  b j  s ipn1   ai  s  m j 1  bj  s (1.2) where m, n, p, q are nonnegative integers such that 0  n  p, 1  m  q, are positive number  and ai , i  1,..., p, b j , j  1,..., q , are complex number such that   b j  v  1 ai    and v,   0,1, 2,..., j  1,..., m , and i  1,..., n , the G -function can be defined as follows in terms of the inverse Mellin transform of M f  s  :  a1 ,..., a p  1 c  i  f  x   G pm,,qn  x M f  s  x  s ds ,  b1 ,..., bq  2i c i   (1.3) where M f  s  is as defined in (1.2) and the Bromwich path  c  i, c  i  separates the     points s   b j   , j  1,..., m ,   0,1, 2,... , the poles of  b j  s , j  1,..., m , from Habibullah, Saboor and Ahmad 141 the points s  1  ai    , i  1,..., n ,   0,1, 2,..., the poles of  1  ai  s  , i  1,..., n . Thus, one must have   Max1 j m R b j  c  Min1i n R 1  ai  . (1.4) The integral (1.3) converges absolutely when m  n  1  p  q  0 . 2 Moreover,  a1 ,..., a p   1 1  b1 ,...,1  bq    Gqn,,m . G pm,,qn  x p   b1 ,..., bp   x 1  a1 ,...,1  a p      (1.5) For example, when p  q , the G -function is defined for 0  x  1 , and the identity (1.5) can be used to evaluate the hypergeometric functions for x  1 . For the main properties of the G -function as well as applications to various disciplines, the reader is referred to Mathai [13]. 2. NEW  FUNCTION We introduce an extension of hypergeometric function of two variables in following form: b b+1 d d +1   a, , ,c; , ,e, f; -αx-δ , βxδ  2 2  2 2     m, n  0   a n  c n   x     e n  f n n ! m  0 n 0   n   Using Lemma 5, [15, p.22], (2.1) becomes      1   22 k     .  2 k  2 k (2.1) becomes, n  b   b 1     2  x 2  m  m  d   d 1   2   2  m!  m  m  a  n m  where  x   1 .   2 k    m b   b 1   c n x  x     2 m  2 m  d   d 1   2   2   e n  f n m !n !  n  m  a m n   m , (2.1) 142 A generalization of hypergeometric and gamma functions b b+1 d d +1   a, , ,c; , ,e, f; -αx-δ , βxδ  2 2  2 2   a n  c n x    a  n m  b 2m  x      e n  f n n ! m 0  d 2 m m ! n 0 n m  .  a n  c n x    a  n m  x       b    d  b  n  0  e n  f n n ! m  0 m! n  d  (2.2) m  1 b  2 m 1  t1 1  t1 d b 1 dt1 0 a  n m  x  t12   a n  c n x  1 b1 d b 1       t1 1  t1    b    d  b  n  0  e n  f n n ! 0 m! m 0  d  n m  dt1 . (2.3)   a n  c n x  1 b1 a n d b 1 1  x  t12  dt1  t1 1  t1     b    d  b  n  0  e n  f n n ! 0  d    n      d  1 b 1  t1  b   d  b  0 1 t1 d b1 1 x  t12   a  n 0  d   e  x   2    1  x t1  n  a n  c n   e n  f n n ! dt1 . (2.4)  f   b    d  b    a   e  a   c    f  c  111     t1b 1t2a 1t3c 1 1  t1  d b 1 000  xt2t3 exp   1  x  t 2 1  1  t2 ea 1 1  t3  f c1 1  x t12    dt1dt2 dt3 ,  a (2.5) since  x  a n  c n    1  x  t 2  1   e f n !  n  n n 0   e    n  f    a  e  a   c   f  c 11 a 1 c 1   t2 t3 1  t2  00 e  a 1  xt2t3   dt dt .  2  2 3  1  x t1  1  t3  f c1 exp  (2.6) Habibullah, Saboor and Ahmad 143 3. A GAMMA-TYPE FUNCTION We define a following gamma-type function involving newly defined hypergeometric function of two variables  .    b b+1 d d +1  H  a, b, c, d , e, f ; , , ; p,     x1e px   a, , ,c; , ,e, f;-αx-δ , βxδ  dx , 2 2 2 2   0 (3.1) where,   Re  p   0, Re     0, Re    1  0, Arg  x    . Using (2.1), one has H  a, b, c, d , e, f ; , , ; p,      a n  c n   x     e n  f n n ! n 0 n 0   n   x1e px  0  a n  c n n   n1  px e x n  0  e n  f n n ! 0     3 F2  b   b 1     2  x 2  m  m  d   d 1   2   2  m!  m  m  a  n m  x  n 1  px e 0  3 F2  a  n,  b b 1 d d 1  , ; , ;   x   dx 2 2 2 2   d   d 1     1   n 1  px 2  2   e x  b   b  1  2i 0   a  n       2  2   x  b   b 1    s    a  n  s     s     s 2 2     ds dx   d   d 1  c  i   s  s 2   2   d   d 1     2  2    b   b 1   a  n       2  2   c  i s m dx (3.2) b b 1 d d 1     a  n, 2 , 2 ; 2 , 2 ;  x  dx . (3.3)   Note that   144 A generalization of hypergeometric and gamma functions  c  i 1  2i c i   x b   b 1   s  s 2   2  d   d 1    s  s 2   2    s   s    a  n  s      n  s  1  px e dx ds 0  1 p n  /   d   d 1     2  2   b   b 1   a  n       2  2  1 c  i   2i c i b    b 1     s  sn   s  2   2    ds . d   d 1    s  s 2   2    p s   s    a  n  s    Hence,  x 0  n 1  px e  3 F2  a  n,  b b 1 d d 1  , ; , ;   x   dx 2 2 2 2   d   d 1     1 2  2    p n  /   a  n   b    b  1        2  2  b   b 1    s    a  n  s     s     s   n   /   s s  1  1 c  i 2 2         ds , 2i c i d   d 1  p    s  s 2   2    d   d 1   d d 1    1  n  ,1, ,    1  2 2  2  2  3,2  1  G . (3.4) 4,3 b b 1 p   p n  /   a  n   b    b  1  a  n, ,         2 2 2  2    where Re  p   0, Re     0, Re   n  s       0 . Equivalently, in light of (1.5), one has Habibullah, Saboor and Ahmad 145 H  a, b, c, d , e, f ; , , ; p,    1  p /    d   d 1  b b 1  n   1  a  n,1  ,1     2 2 .  2   2    c n  / p  G 2,3   p  3,4  d d 1   b   b  1  n  0  e n  f n n !   a      0, n  ,1  ,1     2 2  2  2   (3.5) Since by Slater’s theorem [13], on can express Meijer G-function as a sum of residues in terms of generalized hypergeometric functions p Fq 1    b j  bh    1  bh  a j  z  m m G pm,,qn  a1 , a2 ,....., a p  m j 1 j 1 z    b1 , b2 ,......, bq  h 1 q     1  bh  b j j  m 1    bh    a j  bh  p j  n 1  1  bh  a1 ,1  bh  a2 ,.......,1  bh  a p  p mn  p Fq 1  ;  1   1  bh  b1 ,1  bh  b2 ,.......,1  bh  b p  z   ,    (3.6) where p  q,or p  q and z  1 , and for the poles to be distinct no pair among b j , j  1, 2,....., m , may differ by an integer or zero. The asterisks in (3.6) remind us to ignore the contribution with index j  h . For m  2, n  p  3, q  4, a1  1  a  n,  b b 1 d d 1 , we have from a2  1  , a3  1  , b1  0, b2  n  , b3  1  , b4  1  2 2 2 2  (3.6)  d   d 1  n      c   / p  1 2  2  n H  a, b, c, d , e, f ; , , ; p,      p  /   a   b    b  1  n  0  e n  f n n !       2  2    b   b 1   a  n            2   2   F  a  n, b , b  1 ; n    1, d , d  1 ; p   n   3 3  2 2  2 2     d   d 1      2  2         b   b 1   n   /    n  /  2 2     d   d 1    n   /   n   /  2   2    p n /    a  2n   /     n   /     146 A generalization of hypergeometric and gamma functions  b  1 b  1  d d      3 F3  a  2n  ,  n   ,  n  ;   n  ,  n  , n   1; p    .  2  2 2  2  2 2     After some simple steps, we get H  a, b, c, d , e, f ; , , ; p,    1     /     a n   /  n  c n   / p      p /  n 0  e n  f n n ! n b b 1  d d 1    3 F3  a  n, , ; n   1, , ; p 2 2  2 2     a n  c n  n   a  2n   /     n   /     b  2  n   /       b  n  0  e n  f n n !   a  n    d  2  n   /   /    d    b  1 b  1  d d      3 F3  a  2n  ,  n   ,  n  ;   n  ,  n  , n   1; p    .  2  2 2  2  2 2     (3.7) 3.1 Particular Cases  For   0 , we get from (3.2) and (3.7) H  a, c, e, f ;0, , ; p,    x 0  1  px e  a n  c n  x     e n  f n n ! n 0   x n 1e px  2   1      p       dx  F2 a, c; e, f ;  x dx 0  n  3 F2    a,  , c; e, f ;  /   p .  For c  f ,  1   H  a, e;0, , ; p,        p       a, ; e;  / p   . (3.9)     2 F1  R.H.S of (3.9) is generalized gamma function considered by Al-Zamel [5].  For   0 , we obtain from (3.2) and (3.7) H  a, b, c, d ; ,0, ; p,   (3.8) Habibullah, Saboor and Ahmad    x1e px  0  147  3 F2  a,  b b 1 d d 1  , ; , ;   x   dx 2 2 2 2  1    /  d d 1  b b 1   ;   1, , ; p  3 F3  a, ,   p /  2 2  2 2     /    d    a   /      /     b  2   /     b   a    d  2  /   b  1 b  1  d d      3 F3  a  ,   ,  ;   ,  ,  1; p    . (3.10)  2  2 2  2  2 2     If we set b  d in (3.10) to get H  a, c; ,0, ; p,         x1e px 1 F0 a; ;   x  dx 0  1    /     F a;   1; p    / 1 1   p      /    a   /      /    a  1 F1   a    ;  1; p    .    (3.11) If we put     0 , p  1/ k and   k in (3.1) then we obtain  1  x / k dx  k    , 0 x e k which is k-gamma function [6]. 3.2 The Incomplete Gamma-Type Function The incomplete bivariate gamma-type function is defined as x u 1  pu  e 0 b b+1 d d +1   a, , ,c; , ,e, f; -αu -δ , βu δ  du 2 2 2 2    H0x  a, b, c, d , e, f ; , , ; p,   , (3.12) and the complimentary incomplete gamma-type function is defined as  u x 1  pu  e b b+1 d d +1   a, , ,c; , ,e, f; -αu -δ , βu δ  du 2 2  2 2   H x  a, b, c, d , e, f ; , , ; p,   , (3.13) 148 A generalization of hypergeometric and gamma functions 4. A GAMMA-TYPE PROBABILITY FUNCTION USING A NEWLY DEFINED HYPERGEOMETRIC FUNCTION OF TWO VARIABLES We define the following gamma-type probability density function involving the newly defined hypergeometric function of two variables specified by (2.1):  b b+1 d d +1 f  x   C x1e p x   a, , ,c; , ,e, f; -αx-δ , βxδ  , x  0 , 2 2  2 2  (4.1) where C 1  H  a, b, c, d , e, f ; , , ; p,   ,  1 0, p  0,   p,   0, Arg      .  As f  x   0 , lim f  x   0 ; lim f  x   0 and  f  x  dx  1 , f  x  defines a bona x 0 x  0 fide probability density function. 0.5 0.4 0.3 0.2 0.1 1 2 3 4 5 6 Fig. 4.1: The gamma-type pdf. a = 1.3; b = 0.6; c = 2.7;  = 0.4; d = 3.4; e = 2.4; h = 4.6; g = 3.4; f = 1.7;  = 0.3; p = 2.2;  = 1.2;   4.2(solid line);   5.2(dotted line);   6.2 (thick line) . 1.0 0.8 0.6 0.4 0.2 1 2 3 4 5 Fig. 4.2: The gamma-type pdf. a = 1.3; b = 0.6; c = 2.7;  = 4.1; d = 3.4; e = 2.4; h = 4.6; g = 3.4; f = 1.7;  = 0.3;  = 0.4;  = 1.2; p  2.2(solid line); p  3.2(dotted line); p  4.2 (thick line) . Habibullah, Saboor and Ahmad 149 1.5 1.0 0.5 1 2 3 4 5 Fig. 4.3: The gamma-type pdf. a = 1.3; b = 0.6; c = 2.7;  = 4.2; d = 3.4; e = 2.4; h = 4.6; g = 3.4; f = 1.7;  = 0.3;  = 0.4; p = 2.2;   1.2(solid line);   2.2(dotted line);   3.2 (thick line) . Figure 4.1, Figure 4.2 and Figure 4.3 illustrate how the parameters , p and  effect the gamma-type distribution. 4.1 Particular Cases of Probability Function I. If we set   0 and c  f , (4.1) becomes,   δ  p /  x1e p x 1 F1 a; e; βx f  x  ,   /  2 F1  a,  / ; e; β / p  which is modified form of pdf defined by Al-Zamel [5].  II. If we set   0 and b  d , (4.1) becomes,   f  x  where,  x1e p x 1 F0 a;-;-x-δ  , 1    /      1; p   .  1 F1  a;    p /       /    a   /      /    a 1 F1      a   ;   1; p      5. STATISTICAL FUNCTIONS Closed form representations of the moment generating function of a gamma-type random variable which is denoted by X , as well as the associated moments are provided in this section. 150 A generalization of hypergeometric and gamma functions The Moment Generating Function of X The moment generating function for density function is defined as    M X  t   E et X   et x f  x  dx 0 The inverse Mellin transform technique will be used to derive the moment generating function of the gamma-type distribution. Let X be a random variable whose pdf is specified by (4.1). The moment generating function with respect to the distribution specified by (4.1), is given by   b b+1 d d +1   M X  t   C 0 et x x1e p x   a, , ,c; , ,e, f;-αx-δ , βxδ dx . 2 2  2 2   a n  c n n   n1  p x t x  b b 1 d d 1  e e 3 F2  a  n, , ; , ;   x   dx . x e f n ! 2 2 2 2 n  0  n  n   0   C (5.1) Letting   1 and using (3.8), (5.1) becomes  d   d 1  n C     2   2    c n    M X t     b   b  1  n  0  e n  f n n !  a       2  2  1 i   2i i b   b 1   s  s    n  s  1  p t  x 2   2  e dx ds x d   d 1  0   s  s 2   2    s   s    a  n  s    n     d   d 1   c n  C       1  2  2   p t      p  t    a    b    b  1  n 0  e n  f n n ! 2  2      s    p  t     s    a  n  s    b2  s    b 2 1  s      n  s  1 i      ds ,  2i i d   d 1    s  s 2   2   where, Re  p   Re  t  , Re     Re  s  . Habibullah, Saboor and Ahmad 151 Hence,  M X t      d   d 1  C    1  2  2   p t   b   b 1   a       2  2        p t  n  d d 1  1  n  ,1, ,   1 2 2  3,2  G4,3 .  b b 1   p t  n  0  e n  f n n ! a  n , ,   2 2     c n  Equivalently, in light of (1.5), one has  d   d 1   C     1  2  2  M X t      p  t   a   b    b 1  2  2           p t  n  b b 1  1  a  n,1  ,1    2 2 . 2,3   G3,4  p t  d d 1   n 0  e n  f n n ! 0, n  ,1  ,1    2 2     c n  (5.2) The Moments The r th moment about the origen of a continuous real random variables X with density function, f  x  defined by  ur   x r f  x  dx . 0 For the density function defined in (4.1), we have r  CH  a, b, c, d , e, f ; , , ; p,   r  , (5.3) where C 1  H  a, b, c, d , e, f ; , , ; p,   . Variance The variance for the distribution of X is given by  V  X   C  H  a, b, c, d , e, f ; , , ; p,   2  CH 2  a, b, c, d , e, f ; , , ; p,   1 , (5.4) The Factorial Moments Factorial moments for probability density function defined in (4.1) are as follows 152 A generalization of hypergeometric and gamma functions E  X  X  1 X  2  ...  X  i  1    E X i  1 X i 1  2 X i 2  ...  i 1 X i 1       k  1 E X i  k , k k 0 (5.5) where  k is integer, k  0 ; which satisfy the first identity Now, E  X  X  1 X  2  ...  X  i  1  i 1   k  1 CH  a, b, c, d , e, f ; , , ; p,   i  k  . k (5.6) k 0 The Negative Moments Negative moments is defined as  1    1 E  r     r f  x  dx ,  X   0 X (5.7) using (5.5), we derive negative moments for density function defined in (4.1)  1  E  r   C H  a, b, c, d , e, f ; , , ; p,   r  ,   r  1 . X  (5.8) REFERENCES 1. Agarwal, S.K. and Kalla, S.L. 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