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Báo cáo toán học: "Evaluation of a Multiple Integral of Tefera via Properties of the Exponential"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Evaluation of a Multiple Integral of Tefera via Properties of the Exponential. | Evaluation of a Multiple Integral of Tefera via Properties of the Exponential Distribution Yaming Yu Department of Statistics University of California Irvine 92697 USA yamingy@uci.edu Submitted Jul 12 2008 Accepted Jul 21 2008 Published Jul 28 2008 Mathematics Subject Classihcation 26B12 05A19 60E05 Abstract An interesting integral originally obtained by Tefera A multiple integral evaluation inspired by the multi-WZ method Electron. J. Combin. 1999 N2 via the WZ method is proved using calculus and basic probability. General recursions for a class of such integrals are derived and associated combinatorial identities are mentioned. 1 Background The integral in question reads í i rni i wn -ei x _ m 2m n k - i k 2 m f 2 k - 1 m e2 x m ex x ne ei x dx --MP _ D Tk m J 0 i k 2m k - 1 k J 1 where k is a positive integer m and n are nonnegative integers x xi . xk e1 x Eti xi e2 x Ei i j k xixj y m nM y i and Tk m is defined recursively by D- k k - 2 m k - 1 2 m Tk m - Tk m-1 k _ 1 2m fc 2 Tk-i m m 1 k 2 2 and T1 m 0 m 0 Tk 0 1 k 2. Note that we are using an uncommon convention 00 0 for the case m n 0 k 1. In 1 Tefera gave a computer-aided evaluation of 1 demonstrating the power of the WZ 2 method. It was also mentioned that a non-WZ proof would be desirable especially if it is short. This note aims to provide such a proof. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N29 1 2 A short proof This is done in two steps - the first does away with the integer n using properties of the exponential distribution while the second builds a recursion using integration by parts. In this section we denote the left hand side of 1 by I n m k . Proposition 1. For n 1 we have I n m k 2m n k 1 I n 1 m k . Proof. Let Z1 . Zk be independent random variables each having a standard exponential distribution i.e. the common probability density is p z e z z 0. Denoting Z Z1 . Zk we have I n m k E e2 Z m ei Z n __ ĨT.ÍP 7 2m n e2 Z k E ei Z L1 J wi 7Z 2m n T7 I e2 Z k E ei Z Z _ 2m n k - 1 E e2 Z .