A Frog’s Random Jump and the Pólya Identity

A frog jumps along the lattice points on the x-axis. Starting from x = x0, he jumps ` steps to the left with probability p, or he jumps r steps to the right with probability 1− p at each time. What is the probability that he ever lands on the origin? We answer this question by using a closed formula for ∑k≥0 (ck+s dk+t ) zk, which is an extension of the Pólya identity. We also include a combinatorial proof of the Pólya identity. 1. A FROG PROBLEM AND THE PÓLYA IDENTITY In this paper we consider the following problem (cf. section 10.6 of [3]). Problem 1. A frog lives on the line Z. Starting from x = x0, he jumps ` steps to the left (from x to x− `) with probability p, or he jumps r steps to the right (from x to x + r) with probability q = 1− p at each time t = 1,2, . . .. What is the probability that we can catch him by setting a trap at the origin? More formally we consider random variables X1,X2, . . . with Prob(Xi =−`) = p and Prob(Xi = r) = q for all i≥ 1. Let fk be the probability that the frog lands on the origin after k jumps for the first time, i.e., fk = Prob(∑i=1 Xi =−x0 and ∑i=1 Xi 6=−x0 for all ` < k). Then what is ∑i=1 fk? This definition of the probability is valid for all starting position x0 ∈ Z, but we exceptionally define the probability for the case x0 = 0 (the case starting from the origin) to be 1 just for a technical reason. Another way to state the problem is as follows. Problem 2. A frog lives in Z2. Starting from the origin, he jumps one unit up with probability p, or he jumps one unit right with probability q = 1− p at each time. Then what is the probability that the frog ever lands on the line rx− `y+ x0 = 0? An l steps jump to the left (resp. r steps jump to the right) in Problem 1 is corresponding to one unit jump upwards (resp. one unit jump to the right) in Problem 2. Problem 2 is naturally arisen when one deals with multiply intersecting families in extremal set theory, which was one of the motivations of this paper. In fact, the answer to the problem (and its variations) plays an important role in [1] and [8]. Also the problem is related to some interesting identities appeared in enumerative combinatorics. Among others, we give a closed formula for ∑k≥0 (ck+s dk+t ) , which is an extension of the Pólya identity. The author was supported by MEXT Grant-in-Aid for Scientific Research (B) 16340027.