The Bessel Polynomials and the StudenttDistribution

Abstract

The quotient 2Fo(-n + 1, n;-;-1/2x) P,_,(x),,/'-x2Fo(-n,n+ 1;-;-1/2x)-P.(x) arose in connection with the problem of the infinite divisibility of the Student distribution. It is shown that P,_,(x)/P,(,,Ux) is completely monotonic in [0, oo) for n 4, 5 and 6. This implies that the Student distribution is infinitely divisible for 9, 11 and 13 degrees of freedom. We show that certain power sums of the zeros of the simple Bessel polynomials are zero. This is then used to show that for every n =0, 1, 2,. ., there exists a 0,>0 such that the inverse Laplace transform of P,_,(qx)/P,,(4x) is nonnegative in [0,, oo). This supports our conjecture that P,_,(,,/-x)/P,(x) is completely monotonic in (0, oo) for all n, and that the Student distribution is infinitely divisible for odd degrees of freedom.

Some open problems of generalised Bessel polynomials

“…From (4) one can easily see that, for ordinary Bessel polynomials, the sum S2k+l = 0 for ks 1 as was pointed out by Ismail and Kelker (1976). Notice from the same equation that the quantities s,/ n for n + c o and r = 1,2, .…”

Rate of approximation of zf′(z) by special sums associated with the zeros of the Bessel polynomials

“…(The right side in (11) tends to infinity as m → ∞, and the example A 1 m = (3 m −1)/2 m shows that the quantities m − A n m are indeed unbounded as m → ∞.) Thus (7) immediately follows from (6), |f m | ≤ 1, Proposition 1 and (11):…”

“…To prove (8), put x = x n = √ n/(1 + √ n) in (7) and apply the Stirling formula n! = √ 2πn(n/e) n exp c n , 0 < c n < 1/(12n), and also the inequalities x n < 1, (1+1/ √ n) √ n < e, c n + √ n + n −1/2 < 0.3n (n ≥ 13) and e 1.3 /4 < 0.92:…”

Singularities of bessel‐zeta functions and Hawkins' polynomials

“…For the definition of K v (z) see [13, p. 78] The Bessel polynomials y n (z,2) = y n (z) = 2 F 0 (-n, 1 + n; --) (3.4) are related to the 6 n {z) by /1\ (3.5) Here 6> 0 (z) = l, 0,(z) = z+l, 0 2 (z) = z 2 + 3z + 3, etc., (3.6) [6, p. 7]. As in [6, p. 76] we let jS, -= )3 jn denote the j-th zero of 6 n (z), 1 ^; ^ n. Then by formula (14) of [6, p. 95] (see also [9]) we have From (3.2) and the expansion of the right-hand side of (3.7) into a geometric series we find that applies to negative powers of the zeros).…”

“…This reduces to a multiple of the Riemann zeta function when v = ±\. The Bessel-theta functions corresponding to (1.3) arise in probability theory; see [10] and also [8,9,11]. John Hawkins [7] has established THEOREM 2.…”