Int. Journal of Math. Analysis, Vol. 6, 2012, no. 11, 503-508 Some Results Based on Generalized Mittag-Leffler Function Pratik V. Shah Department of Mathematics C. K. Pithawalla College of Engineering & Technology Surat, India pratikshah8284@yahoo.co.in R. K. Jana, Amit D. Patel and I. A. Salehbhai Department of Mathematics S. V. National Institute of Technology, Surat, India rkjana2003@yahoo.com (Corresponding Author) ptlamit83@gmail.com ibrahimmaths@gmail.com Abstract An attempt is made to obtain the result based on Generalized Mittag- Leffler function and its significance in distribution theory. Mathematics Subject Classification: 33E12, 44A10, 62E99 Keywords: Generalized Mittag-Leffler function, Laplace Transform, Statistical Distribution theory 1. Introduction In 1903, the Swedish mathematician Gosta Mittag-Leffler introduced the function E α (z), defined by E α (z) Γ(αn +1) (α C, R (α) > 0) (1) where, Γ (z) is the familiar Gamma function. The Mittag-Leffler function defined by equation (1) reduces immediately to the exponential function e z
504 Pratik V. Shah, R. K. Jana, Amit D. Patel and I. A. Salehbhai E 1 (z) when α1. For 0 <α<1, it interpolates between the pure exponential e z and a geometric function 1 ( z < 1). Its importance 1 z has been realized during the last two decades due to its involvement in the problems of Applied Sciences such as Physics, Chemistry, Biology and Engineering. Mittag-Leffler function occurs naturally in the solution of fractional order differential or integral equations [3]. In 1905, a generalization of E α (z) was studied by Wiman [6] who defined the function E (z) as follows: E (z) Γ(αn + β) The function E (z) is now known as Wiman function. (α, β C, R (α) > 0,R (β) > 0) (2) In 1971, Prabhakar [4] introduced the function E γ (z) defined by, E γ (z) (γ) n Γ(αn + β) n! Where, (λ) n is the Pochammer symbol defined by (λ) n Γ(λ + n) Γ(λ) (α, β, γ C, R (α) > 0,R (β) > 0,R (γ) > 0) { 1(n 0; λ 0) λ (λ +1)... (λ + n 1) (n N;λ C) N being the set of positive integers. In the sequel to this study, Shukla and Prajapati [5] investigated the function E γ,q (z) defined by E γ,q (z) (γ) qn Γ(αn + β) (α, β, γ C, R (α) > 0,R (β) > 0,R (γ) > 0,q (0, 1) N) The function E γ,q (z) converges absolutely for all z Cifq< R (α) + 1 (an entire function of order R (α) 1 and for z < 1ifqR (α) + 1). It is easily seen that equation (4) is an obvious generalization of equation (1), equation (2), equation (3) and the exponential function e z as follows: E 1,1 1,1 (z) e z, E 1,1 α,1 (z) E α (z), E 1,1 (z) E (z), E γ,1 α,1 (z) E γ (z) Some definitions related to this paper are given below: Distribution Function [2]: Let X be a random variable and let x 1,x 2,... be the values which it assumes; in most of what follows the x j will be integers. The aggregate of all sample points on which X assumes the fixed value x j forms the event X x j ; its n! (3) (4)
Generalized Mittag-Leffler function 505 probability is defined by P {X x j }. The function P {X x j } f (x j ) j 1, 2,... is called the (probability) distribution of the random variable. Clearly,f (x j ) 0, f (xj )1. The distribution function F (x) ofx is defined by F (x) P {X x} f (x j ), (5) x j x The last sum extending over all those x j which do not exceed x. F (x) is a non-decreasing function which tends to zero as x and to one as x. Thus the distribution function can be calculated from its probability distribution and vice versa. Bernoulli Number [7]: The Bernoulli polynomials, denoted by B n (x), are defined by the expansion te xt e t 1 t n n! B n (x), ( t < 2π) (6) The function on the left hand side of above equation is called the generating function of B n (x). When x 0, equation (6) reduces to t t n B e t 1 n! n (0) where B n (0) are called Bernoulli numbers, denoted by B n. The Bernoulli polynomials can be expressed explicitly in terms of the Bernoulli numbers as B n (x) n k0 n! k!(n k)! B kx n k (7) The Bernoulli numbers can be calculated recursively using the initial values, B 0 1, and B 1 1, and the recurrence relation given by, 2 B n ( ( ) 1 B ) n 1 k n+1 0 + B 1 j+n k B j k (n 2) k2 j2 The first several Bernoulli numbers so obtained are given by B 0 1,B 1 1 2,B 2 1 6,B 4 1 30,B 6 1 42,B 8 1 30,B 10 5 66,B 12 691 2730,... B 2n+1 0 (n 1, 2, 3,...) Substituting these into equation (7) we obtain several Bernoulli polynomial as B 0 (x) 1, B 1 (x) x 1 2, B 2 (x) x 2 x+ 1 6, B 3 (x) x 3 3 2 x2 + 1 x,... 2
506 Pratik V. Shah, R. K. Jana, Amit D. Patel and I. A. Salehbhai Digamma Function [1]: The digamma function is closely associated with the derivative of the gamma function and sometimes called logarithmic derivative function, which is defined by, ψ (x) d {ln Γ (x)} Γ (x) x 0, 1, 2,,... dx Γ(x) where, Γ (x) is the usual gamma function [1] given by, Γ(x) 0 e t t x 1 dt ;Re(x) > 0 (8) 2. Extended Bernoulli Number By Mittag-Leffler Function Let, E α (z) s (z) (9) Γ(αn +1) From equation (9), it is clear that [s (z)] z0 1. On differentiating equation (9) with respect to z, we get E α (z), which we denoted as E α (z) s 1 (z). Now for the choice of α 1, we have [s 1 (z)] z0 1. Similarly, repeating the above process we have, [s 2 (z)] z0 1 [s 3 (z)] z0 1 and so on. Hence, we can obtain extension of Bernoulli s Number for real α as, [ ] S n dn z (10) d E α (z) 1 z0 Substitute n 0 in equation (10), we get [ ] z S 0 E α (z) 1 z0 [ ] 1 E α (z) z0 Now, if we restrict the choice of α to 1, then we get, [ ] 1 S 0 1B E 0 1 (z) Similarly, substituting n 1, 2, 3,... in equation (10) and restricting α to 1, we have, z0 S 1 d [ ] z dz E α (z) 1 z0 E α (z) ze α (z) 2 { (E α (z)) 2 + E α (z) E α (z) E α (z) } z0
Generalized Mittag-Leffler function 507 S 1 1 2 B 1, S 2 1 6 B 2, S 3 0B 3, S 4 1 30 B 4,... Therefore, for the choice of α 1, extended Bernoulli Number is reduced to Bernoulli Number. 3. Mittag-Leffler function in Statistical Distribution Theorem 1. If G y (y) 1 E γ α,1 ( y α ) then, f (y) y α 1 E γ α,α ( y α );0<α 1,y >0 (11) Proof: Since G y (y) 1 E γ α,1 ( y α ) On simplifying the right hand side of equation (11), we get, G y (y) 1 k0 (γ) k ( 1) k y αk Γ(αk +1) k! G y (y) k1 Γ(γ + k) ( 1) k+1 y αk 1 Γ(γ) Γ(αk +1) k! Now differentiating both the sides with respect to y, yields the density function f (y) as: f (y) d dy [G y (y)] Special Case: k0 ( 1) k+1 Γ(γ + k) k! Γ(γ)Γ(αk +1) αkyαk 1 k1 ( 1) k Γ(γ + k +1) (αk + α) y(αk+α) 1 (k + 1)! Γ(γ)Γ((αk + α)+1) [replacing kbyk+ 1] y α 1 E γ α,α ( yα ) Hence, f (y) y α 1 E γ α,α ( yα ). On substituting γ 1 in equation (11), which reduces to [3] If G y (y) 1 E α ( y α ) then, f (y) y α 1 E α,α ( y α );0<α 1,y >0
508 Pratik V. Shah, R. K. Jana, Amit D. Patel and I. A. Salehbhai References 1. L. C. Andrews, Special Functions of Mathematics for Engineers, 2 nd Ed. McGraw Hill, New York, 1992. 2. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, 3 rd Ed., John Wiley & Sons Inc., New York, 1968. 3. H. J. Haubold, A. M. Mathai and R. K. Saxena, Mittag-Leffler Functions and Their Applications. arxiv:0909.0230v2 [math.ca] 4 Oct 2009. 4. T. R. Prabhakar, A singular integral equation with a generalized Mittag- Leffler function in the Kernel, Yokohama Math. J., 19, 7-15, 1971. 5. A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl., 336, 797-811, 2007. 6. A. Wiman, Uber de fundamental satz in der theorie der funktionene α (x), Acta Math. 29, 191-201, 1905. 7. S. Zhang and J. Jin, Computations of Special Functions, John Wiley & Sons Inc., New York, 1996. Received: September, 2011