Some Results Based on Generalized Mittag-Leffler Function

Transcription

1 Int. Journal of Math. Analysis, Vol. 6, 2012, no. 11, Some Results Based on Generalized Mittag-Leffler Function Pratik V. Shah Department of Mathematics C. K. Pithawalla College of Engineering & Technology Surat, India R. K. Jana, Amit D. Patel and I. A. Salehbhai Department of Mathematics S. V. National Institute of Technology, Surat, India (Corresponding Author) 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

2 504 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)

3 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 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 ,B ,B ,B ,B ,... 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 x2 + 1 x,... 2

4 506 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

5 Generalized Mittag-Leffler function 507 S B 1, S B 2, S 3 0B 3, S 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

6 508 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, W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, 3 rd Ed., John Wiley & Sons Inc., New York, H. J. Haubold, A. M. Mathai and R. K. Saxena, Mittag-Leffler Functions and Their Applications. arxiv: v2 [math.ca] 4 Oct T. R. Prabhakar, A singular integral equation with a generalized Mittag- Leffler function in the Kernel, Yokohama Math. J., 19, 7-15, A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl., 336, , A. Wiman, Uber de fundamental satz in der theorie der funktionene α (x), Acta Math. 29, , S. Zhang and J. Jin, Computations of Special Functions, John Wiley & Sons Inc., New York, Received: September, 2011