Antilogarithms of second order in algebras with logarithms and their applications to special functions

• Autorzy: Przeworska-Rolewicz D.
• Języki publikacji: EN

Abstrakty

EN

In the paper [PR5] it was shown that the so-called special functions of Mathematical Physics can be obtained by means of antilogarithms of the second order for the usual differential operator ^j. The same method applied to a right invertible operator D in a commutative Leibniz algebra with logarithms permits to determine eigenvectors of linear equations of order two in D with coefficients in the algebra X under consideration by a reduction to the generalized Sturm-Liouville operator. It seems that, in a sense, the proposed method is an answer for the question of Gian-Carlo Rota concerning a unified approach to special functions (cf. [Rl], problem 4). Section 6 of the present paper is devoted to some summations formulae expressing special functions by means of exponentials. Note that, in general, we do not need any assumption about the Hilbert structure of the algebra X.

Słowa kluczowe

EN

Leibniz algebra; exponential; sine mapping; cosine mapping; logarithmic mapping of nth order; antilogarithmic mapping of nth order; symmetric differential operator of order two; Sturm-Liouville operator; eigenvector; special function; associate and orthogonal special functions; special element; hypergeometric functions; Legendre; Gauss; Chebyshev; Jacobi; Laguerre; Hermite functions; Helmholtz; Neumann; Henkel; Bessel; Euler; Gegenbauer; Weber; Airy; Ince; Mathieu; Sturm-Liouville equations and/or functions

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
• Rocznik: 2004
• Tom: Vol. 44, nr spec.
• Strony: 167--191
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Przeworska-Rolewicz D.

• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa 10, P.O. Box 21, Poland,

Bibliografia

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• [A2] Askey, Richard A., Theory and Application of Special Functions, Proc. Advanced Seminar, Math. Research Center, Univ. of Wisconsin, Madison, Wisconsin, March 31-April 2, 1975, Ed. Richard A. Askey, Academic Press, New York/San Francisco/London, 1975.
• [Kl] Kohno, Mitsuhiko, Global Analysis in Linear Differential Equations, Kluwer Akademic Publishers, Dordrecht/Boston/London, 1999.
• [MM] Margenau, H. and Murphy, G. M., The Mathematics of Physics and Chemistry, New York, 1950 (Polish ed. PWN, Warszawa, 1956).
• [PR1] Przeworska-Rolewicz, Danuta, Algebraic Analysis, D. Reidel and PWN-Polish Scientific Publishers, Dordrecht-Warszawa, 1988.
• [PR2] Przeworska-Rolewicz, Danuta, Logarithms and Antilogarithms. An Algebraic Analysis Approach. With Appendix by Z. Binderman, Kluwer Academic Publishers, Dordrecht, 1998.
• [PR3] Przeworska-Rolewicz, Danuta, Non-Leibniz algebras with logarithms do not have the trigonometric identity, In: Algebraic Analysis and Related Topics. Proc. Intern. Conf. Warszawa, 21-25 September, 1999, Banach Center Publ. 53, Warszawa, 2000, 177-189.
• [PR4] Przeworska-Rolewicz, Danuta, Isomorphisms preserving Leibniz condition, Fractional Calculus & Applied Analysis 2 2 (1999), 149-160.
• [PR5] Przeworska-Rolewicz, Danuta, Special functions as antilogarithms of second order, Fractional Calculus & Applied Analysis 3 2 (2000), 277-296.
• [PR6] Przeworska-Rolewicz, Danuta, Binomials formulae in commutative Leibniz algebras, Fractional Calculus & Applied Analysis 2 5 (2002), 195-207.
• [PR7] Przeworska-Rolewicz, Danuta, Generalized Sturm separation theorem, Demonstratio Math. 36 (2003), 735-746.
• [Rl] Rota, Gian-Carlo, Ten Mathematics Problems I will never solve. Invited address at the joint meeting of the Amer. Math. Soc. and the Mexican Math. Soc., Oaxaca, Mexico, December 6, 1997. DMV-Mitteilungen 2 (1998), 45-52.
• [R2] Rota, Gian-Carlo, Indiscrete Thoughts, Ed. F. Palombi. Birkhhäuser, Boston, 1997, 230-231.
• [Wl] Wawrzyńczyk, Antoni, Group Representation and Special Functions, Examples and Problems prepared by A. Strasburger, D. Reidel and PWN-Polish Scientific Publishers, Dordrecht/Boston/Lancaster, 1984 (Polish ed. PWN, Warszawa, 1978).

Uwagi

Dedicated to Professor Julian Musielak.