Advertisement

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

• M. Benabdallah
• M. J. Atia
Article
• 10 Downloads

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. ), 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

References

1. 1.
Berg, C., Vignat, C.: Linearization coefficients of Bessel polynomials and properties of Student $$t$$-distributions. Constr. Approx. 27, 15–32 (2008)
2. 2.
Andrews, G., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)
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)
5. 5.
Grosswald, E.: The Student t-distribution for odd degrees of freedom is infinitely divisible. Ann. Probab. 4(4), 680–683 (1976)
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).
7. 7.
Boros, G., Moll, V.H.: An integral hidden in Gradshteyn and Ryzhik. J. Comput. Appl. Math. 106, 361–368 (1999)
8. 8.
Atia, M.J., Zeng, J.: An explicit formula for the linearization coefficients of Bessel polynomials. Ramanujan J. 28, 211–221 (2012).
9. 9.
BenAbdallah, M., Atia, M.J.: Positivity and Recursion formula of the linearization coefficients of Bessel polynomials. AEJM (2018).
10. 10.
Carlitz, L.: A note on the Bessel polynomials. Duke Math. J. 24, 151–162 (1957)

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