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

  • M. Benabdallah
  • M. J. AtiaEmail author
Article
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Abstract

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.  https://doi.org/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).

Keywords

Bessel polynomials Linearization coefficients Double integral Student t-density 

Mathematics Subject Classification

33C05 60E07 33C20 

Notes

References

  1. 1.
    Berg, C., Vignat, C.: Linearization coefficients of Bessel polynomials and properties of Student \(t\)-distributions. Constr. Approx. 27, 15–32 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrews, G., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  3. 3.
    Grosswald, E.: Bessel Polynomials. Lecture Notes in Mathematics, vol. 698. Springer, New York (1978)Google Scholar
  4. 4.
    Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products, 6th edn. Academic Press, San Diego, CA (2000)zbMATHGoogle Scholar
  5. 5.
    Grosswald, E.: The Student t-distribution for odd degrees of freedom is infinitely divisible. Ann. Probab. 4(4), 680–683 (1976)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Berg, C., Vignat, C.: Derivation of an integral of Boros and Moll via convolution of student \(t\)-densities. Ramanujan J. 27, 147–150 (2012).  https://doi.org/10.1007/s11139-010-9279-5 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boros, G., Moll, V.H.: An integral hidden in Gradshteyn and Ryzhik. J. Comput. Appl. Math. 106, 361–368 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Atia, M.J., Zeng, J.: An explicit formula for the linearization coefficients of Bessel polynomials. Ramanujan J. 28, 211–221 (2012).  https://doi.org/10.1007/s11139-011-9348-4 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    BenAbdallah, M., Atia, M.J.: Positivity and Recursion formula of the linearization coefficients of Bessel polynomials. AEJM (2018).  https://doi.org/10.1142/S1793557118500808
  10. 10.
    Carlitz, L.: A note on the Bessel polynomials. Duke Math. J. 24, 151–162 (1957)MathSciNetCrossRefGoogle Scholar

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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