Derivation of a double integral via convolution of Student t-densities.

In this paper, we give the recursion formula (in a more general case) for the linearization coefficients (β l (N k) (A k)) for Bessel polynomials (q n) , mentioned by Berg and Vignat p. 21. (14) (Constr Approx 27:15–32, 2008), in the expansion q n 1 (a 1 u) q n 2 (a 2 u) ... q n k (a k u) = ∑ l = min (n 1 , n 2 , ... , n k) L k β l (N k) (A k) q l (u) , where u ∈ R , N k = (n 1 , n 2 , ... , n k) ∈ N k , A k = (a 1 , a 2 , ... , a k) ∈ R + k with ∑ 1 ≤ i ≤ k a i = 1 and L k = ∑ 1 ≤ i ≤ k n i . This recursion formula yields, again, the positivity of the coefficients. In addition, in the case when k = 3 , we give either an explicit formula for β l (N 3) (A 3) and we derive a double integral formula via convolution of Student t-densities. As a bonus, two reductions for this double integral are given; the first reduction uses an integral formula given by Atia and Zeng (Ramanujan J 28:211–221, 2012. 10.1007/s11139-011-9348-4), and the second uses the one given by Boros and Moll (J Comput Appl Math 106:361–368, 1999). [ABSTRACT FROM AUTHOR]