A Concave Property of the Hypergeometric Function with Respect to a Parameter

The hypergeometric function is shown to be logarithmically concave in integer values of one of its parameters. The methods used are probabilistic. THEOREM. Let m, and g be positive integers satisfying 3 <=i + <_ g, and let z be a negative real number. Then {2Fl[-m,i’g’z]} 2 > 2Fl[-m,i + 1;g;Z]EFl[-m,i1;g;z]. We first establish the following lemma concerning the evaluation of the generating function of the negative hypergeometric distribution. () LEMMA. b+j-j 1)(k+a-j-lk_j 2Fl[-k,b;a + b;1 s] for all positive integers k, and all real s, and positive real values ofa and b. ProofofLemma. Skellam [2] has shown that if X follows a binomial distribution with parameters p and k, and if p is integrated with respect to the normalized beta function pb1(1 p)adp B(a, b) then the unconditional distribution of X is negative hypergeometric, that is, (b+j-1 {k+a-j-1)/ a+b+k1) Pr{X =j} J k-j k The left-hand side of (1), denoted below by I, is then the probability generating function of the negative hypergeometric distribution. Thus I o(sx) Op{(sXlp)} p(1 p + ps) B(a, b) [1 p(1 s)]ap-1(1 p)"-ldp 2Fl[-k,b;a+ b;1See [3, p. 20]. This proves the lemma. Proof of Theorem. The essence of the proof is to use two theorems proved elsewhere [1], one on the existence of a probability distribution with a certain property, the other giving an inequality relating to such a distribution. Received by the editors October 19, 1971, and in revised form February 22, 1972. " Department of Statistics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213. 642 THE HYPERGEOMETRIC FUNCTION 643 Lets= 1 +z,h=gandn=rn+g1. Theorem 3 of[1] states that there is a probability distribution F such that ai,,, the expected value of the ith largest of a sample of size n drawn independently from F, satisfies ai., sfor all i, 1 =< =< n. By use of a standard recurrence relation, quoted in [1, (4)], the expected value of the ith largest of some smaller sample of size h can be deduced as follows" For l<=i<=g<=n, n-h ai,h 2 j=O i+j--1 J