Rawlins, AD

SIAM Review

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A. D. Rawlins

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Two Legendre Polynomial Identities

Brunel University Research Archive

SIAM REVIEWVol. 25, No. 4, October 1983(C) 1983 Society for Industrial and Applied Mathematics0036-1445/83/2504--0005 $01.25/0PROBLEMS AND SOLUTIONSEDITED BY MURRAY S. KLAMKINCOLLABORATING EDITORS: HENRY E. FETTIS CECIL C. ROUSSEAUYUDELL L. LUKE OTTO G. RUEHRAll problems and solutions should be sent, typewritten in duplicate, to Murray S. Klamkin, Departmentof Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1. An asterisk placed beside aproblem number indicates that the problem was submitted without solution. Proposers and solvers whosesolutions are published will receive 10 reprints ofthe correspondingproblem section. Other solvers will receivejust one reprint provided a self-addressed stamped (U.S.A. or Canada) envelope is enclosed. Proposers andsolvers desiring acknowledgment of their contributions should include a self-addressed stamped postcard (nostamp necessary outside the U.S.A. and Canada). Solutions should be received by February 15, 1984.PROBLEMSTwo Legendre Polynomial IdentitiesProblem 83-16, by A. D. RAWLINS (Brunel University, Middlesex, UK).Show that(1) P,{x(x a2)-1/2}- (- 1)" d"n! (X2 O/2)(n+1)/2 dx" (x2 a2)-1/2’ n >_- 0,P" B + x B-l(n!)-2 2fi-xx (1-x2)-x -p(p- 1) +pl(2)n>__l.A System of Probabilistic InequalitiesProblem 83-17", by T. TEN RAA (Erasmus University, Rotterdam, The Netherlands).Consider n locations, n >_- 2"00...0...0...0.2 j nInitially, there is one particle at each location. Then there are two consecutive transitions,governed by real nonnegative matrices (Pu) and (x;j), respectively. Thus Pu is theprobability that the particle which is originally at location will move to j in the firstround. Similarly, xu is the probability that a particle which after the first round is atlocation will move to j in the second round.It is assumed that the system is shaky. More precisely, all particles move withcertainty in both rounds. Furthermore, in the second transition any par{icle has a positiveprobability to reach any other location. Formally,(1) Pi;=0, Pu >0 for someji, i= 1, .,n,(2) Xii 0, xu > 0 for allj i, 1, n.(It is not necessary to assume that transition probabilities add up to unity.)Consider the particle which originates from any location i. The probability that it569

Two legendre polynomial identities

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