Journal of Applied Mathematics and Stochastic Analysis, 15:4 (2002), 371-384. ON A GENERALIZED JACOBI TRANSFORM JOSI SARABIA UNEXPO SecciDn de Matem[ttica, Apartado Postal 352 Barquisimeto, Venezuela E-mail: jsarabia196@hotmail.com.ve S.L. KALLA Kuwait University Department of Mathematics and Computer Science P.O. Box 5959, Safat 13050, Kuwait E-mail: kalla@mcs.sci.kuniv.edu.kw (Received May, 2001; Revised November, 2001) In this paper, we study a generalized Jacobi transform and obtain images of certain functions under this transform. Furthermore, we define a J acobi random variable and derive its moments, distribution function, and charac- teristic function. Key words: Jacobi Transform, Random Variable, Integrals. AMS subject classifications: 33C45, 44A15, 60El0. 1. Introduction Kalla et al. [4] have studied the following integral, ,f j (1 x) (1 -4- x)bp(c’)(x)dx (1) -1 (c ft) is the Jacobi function, where with Re(a) > 1, Re(b) > 1 and Pu (c, (c + 1) (-u,u+A l_x) Pu )(x) r(t + 1)" 2F1 (2) c+l and A-c +/ + 1. The authors considered its partial derivatives with respect to a and b. Some more general results were obtained in [6]. These results were extended by Sarabia [10] using the following integral" i,,b,c,p= j b (o c,p) c,f (1 x)a(1 + x) Pu (x)dx, (3) -1 (a 1) is a generalized Jacobi function defined as where Pu Printed in the U.S.A. ()2002 by North Atlantic Science Publishing Company 371