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Lévy–Khintchine Formula and Dual Convolution Semigroups Associated with Laguerre and Bessel Functions

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Abstract

In this work we consider a system of partial differential operators D 1,D 2 on K=[0,+∞[×R, whose eigenfunctions are the functions φγ(x,t), (x,t)∋K, γ∋Γ=((R∖0)×N)∪(0×[0,+∞[), which are related to the Laguerre functions for γ∋((R∖ 0)×N)∪(0,0) and which are the Bessel functions for γ∋(0×[0,+∞[). We provide K and Γ with a convolution structure. We prove a Lévy–Khintchine formula on K, which permits us to characterize dual convolution semigroups on Γ.

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References

  1. Ben Salem, N.: 'Convolution semigroups and central limit theorem associated with a dual convolution structure', Journal of Theoretical Probability 7(2) (1994), 417-436.

    Google Scholar 

  2. Bloom, W.R. and Heyer, H.: 'Harmonic analysis of probability measures on hypergroups', in H. Bauer-J. L. and Kazdan-E. Zehnder (eds.), de Gruyter studies in Mathematics, 20, Walter de Gruyter. Berlin-New York, 1994.

    Google Scholar 

  3. Koornwinder, T.H.: 'Positivity proofs for linearisation and connection coefficients of orthogonal polynomials satisfying an addition formula', J. London. Math. Soc. 2,18(1978), 101-114.

    Google Scholar 

  4. Koornwinder, T.H. and Kortas, H.: Dual Product Formula for Laguerre Functions. Preprint. Faculty of Sciences of Tunis, 1997.

  5. Kortas, H.: 'Convolution généralisée et formules de produit duales pour les fonctions de Laguerre', Thèse de3ème cycle, Faculté des Sciences de Tunis, 1993.

  6. Nessibi, M.M. and Sifi, M.: 'Laguerre hypergroup and limit theorem', in B.P. Komrakov, I.S. Krasil'shchik, G.L. Litvinov and A.B. Sossinsky (eds.), Lie Groups and Lie Algebras - Their Representations, Generalisations and Applications, Kluwer Acad. Publ., Dordrecht/ Boston/London, 1998.

    Google Scholar 

  7. Sifi, M.: 'Central limit theorem and infinitely divisible probabilities associated with partial differential operators', Journal of Theoretical Probability 8(3) (1995), 475-499.

    Google Scholar 

  8. Sifi, M.: 'Analyse harmonique associée à des opérateurs aux dérivées partielles', Thèse d'Etat, Faculté des Sciences de Tunis, 1996.

  9. Szegö, G.: 'Orthogonal polynomials', Amer. Math. Soc. Colloquium publications Vol. XXIII.

  10. Trimèche, K.: 'Probabilités indéfiniment divisibles et théorème de la limite centrale pour une convolution généralisée sur la demi-droite, C.R.A.S. Paris, t.286, Série A(1978), 399-402.

  11. Trimèche, K.: Generalized Wavelets and Hypergroups. Gordon and Breach Science Publishers, 1997.

  12. Watson, G.N.: A Treatise of Bessel Functions. Cambridge University Press, 1966.

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Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 1060, Tunis-Tunisie

    H. Kortas

  2. Ecole Supérieure des Sciences et Techniques de Tunis, 5 Rue Taha Hussein, 1008, Tunis-Tunisie

    M. Sifi

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  1. H. Kortas
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  2. M. Sifi
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Kortas, H., Sifi, M. Lévy–Khintchine Formula and Dual Convolution Semigroups Associated with Laguerre and Bessel Functions. Potential Analysis 15, 43–58 (2001). https://doi.org/10.1023/A:1011267200317

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  • DOI: https://doi.org/10.1023/A:1011267200317

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