Some Appell–Dunkl Sequences
Tóm tắt
Some examples of Appell–Dunkl sequences are shown using determined operators. Specifically, Appell–Dunkl sequences whose generating functions are of the form
$$E_{\alpha }(xt)/(1\pm t^m)$$
, where the function
$$E_{\alpha }(xt)$$
is given in terms of Bessel functions. Particular cases of these examples are also generated by means of the inverse of the Dunkl operator.
Tài liệu tham khảo
Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series. Vol. 55. (1964)
Ben Cheikh, Y., Gaied, M.: Dunkl-Appell \(d\)-orthogonal polynomials. Integral Transf Spec. Funct. 18, 581–597 (2007)
Bouanani, A., Khériji, L., Ihsen Tounsi, M.: Characterization of \(q\)-Dunkl Appell symmetric orthogonal \(q\)-polynomials. Expo. Math. 28, 325–336 (2010)
Cholewinski, F.M.: The finite calculus associated with Bessel functions, Contemporary Mathematics, 75. American Mathematical Society, Providence, RI (1988)
Ciaurri, Ó., Durán, A., Pérez, M., Varona, J.L.: Bernoulli-Dunkl and Apostol-Euler-Dunkl polynomials with applications to series involving zeros of Bessel functions. J. Approx. Theory 235, 20–45 (2018)
Ciaurri, Ó., Mínguez Ceniceros, J., Varona, J.L.: Bernoulli-Dunkl and Euler-Dunkl polynomials and their generalizations. Rev. R. Acad. Cienc. Exactas Fís Nat. Ser. A Mat. RACSAM 113, 2853–2876 (2019)
Dilcher, K.: Bernoulli and Euler Polynomials, NIST handbook of mathematical functions (edited by F. W. F. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark), 587–599, National Institute of Standards and Technology, Washington, DC, and Cambridge University Press, Cambridge, 2010. Available online in http://dlmf.nist.gov/24
Dimovski, I.H., Hristov, V.Z.: Nonlocal operational calculi for Dunkl operators, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 030, 16 pp
Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311, 167–183 (1989)
Durán, A., Pérez, M., Varona, J.L.: Fourier-Dunkl system of the second kind and Euler-Dunkl polynomials. J. Approx. Theory 245, 23–39 (2019)
Graham, R., Knuth, D., Patashnik, O.: Concrete mathematics: a foundation for computer science, Second edition, Addison-Wesley, (1994)
Khan, S.A., Jagannathan, R.: On certain Appell polynomials and their generalizations based on the Tsallis \(q\)-exponential. Bull. Malays. Math. Sci. Soc. 45, 1453–1472 (2022)
Lebedev, N.N.: Special Functions and their Applications. Dover, New York (1972)
Loureiro, A.F., Maroni, P.: Quadratic decomposition of Appell sequences. Expo. Math. 26(2), 177–186 (2008)
Mínguez Ceniceros, J., Varona, J.L.: Asymptotic behavior of Bernoulli-Dunkl and Euler-Dunkl polynomials and their zeros. Funct. Approx. Comment. Math. 65, 211–226 (2021)
Olver, F.W.J., Maximon, L.C.: Bessel Functions, NIST handbook of mathematical functions (edited by F. W. F. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark), 215–286, National Institute of Standards and Technology, Washington, DC, and Cambridge University Press, Cambridge, 2010. Available online in http://dlmf.nist.gov/10
Rosenblum, M.: Generalized Hermite polynomials and the Bose-like oscillator calculus. Oper. Theory Adv. Appl. 73, 369–396 (1994)
Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge Univ. Press, Cambridge (1944)