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

  • M. Benabdallah
  • M. J. AtiaEmail author


In this paper, we give the recursion formula (in a more general case) for the linearization coefficients \(\big (\beta ^{(N_k)}_{l}(A_k)\big )\) for Bessel polynomials \((q_n)\), mentioned by Berg and Vignat p. 21. (14) (Constr Approx 27:15–32, 2008), in the expansion \(q_{n_1}(a_1u)q_{n_2}(a_2u) \ldots q_{n_k}(a_ku)= \displaystyle \sum \nolimits _{l=\min (n_1,n_2,\ldots ,n_k)} ^{L_k}\beta ^{(N_k)}_{l}(A_k)q_l(u),\) where \(u\in \mathbb R\), \(N_k=(n_1,n_2,\ldots , n_k)\in \mathbb {N}^k,\ A_k=(a_1,a_2,\ldots , a_k) \in \mathbb {R}_+^k\) with \(\displaystyle \sum \nolimits _{1\le i\le k} a_i=1\) and \(L_k=\displaystyle \sum \nolimits _{1\le i\le k} n_i\). 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 \(\beta ^{(N_3)}_{l}(A_3)\) 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., and the second uses the one given by Boros and Moll (J Comput Appl Math 106:361–368, 1999).


Bessel polynomials Linearization coefficients Double integral Student t-density 

Mathematics Subject Classification

33C05 60E07 33C20 



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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Gabès UniversityGabèsTunisia
  2. 2.Qassim UniversityBuraydahKingdom of Saudi Arabia

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