Ordinary differential operators and the integral representation of sums of certain power series
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K. A. Mirzoev and T. A. Safonova
Translated by: Alexey Alimov - Trans. Moscow Math. Soc. 2019, 133-151
- DOI: https://doi.org/10.1090/mosc/294
- Published electronically: April 1, 2020
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Abstract:
The explicit form of the eigenvalues and eigenfunctions is known for certain operators generated by symmetric differential expressions with constant coefficients and self-adjoint boundary conditions in the space of Lebesgue square-integrable functions on an interval, and their resolvents are known to be integral operators. According to the spectral theorem, the kernels of these resolvents satisfy a certain bilinear relation. Moreover, each such kernel is the Green’s function of some self-adjoint boundary value problem and the method of constructing it is well known. Consequently, the Green’s functions of these problems can be expanded in a series of eigenfunctions. In this paper, the identities obtained in this way are applied to construct an integral representation of sums of certain power series and special functions, and in particular, to evaluate sums of some converging number series.References
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press, New York-London, 1965. Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin; Translated from the Russian by Scripta Technica, Inc; Translation edited by Alan Jeffrey. MR 0197789
- Erich Kamke, Differentialgleichungen, B. G. Teubner, Stuttgart, 1977 (German). Lösungsmethoden und Lösungen. I: Gewöhnliche Differentialgleichungen; Neunte Auflage; Mit einem Vorwort von Detlef Kamke. MR 0466672
- K. A. Mirzoev and T. A. Safonova, Green’s function of ordinary differential operators and an integral representation of sums of certain power series, Doklady RAN 482 (2018), no. 5, 500–503 (Russian); English transl. Dokl. Math. 98 (2018), no. 2, 486-489.
- K. A. Mirzoev and T. A. Safonova, On the integral representation of the sums of some power series, Mat. Zametki 106 (2019), no. 3, 470–475 (Russian, with Russian summary); English transl., Math. Notes 106 (2019), no. 3-4, 468–472. MR 4017561, DOI 10.4213/mzm12373
- A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 2, Gordon & Breach Science Publishers, New York, 1986. Special functions; Translated from the Russian by N. M. Queen. MR 874987
- F. G. Tricomi, Integral equations, Pure and Applied Mathematics, Vol. V, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR 0094665
- Josef Stein, Table errata: Handbook of mathematical functions with formulas, graphs, and mathematical tables (Nat. Bur. Standards, Washington, D.C., 1964) edited by Milton Abramowitz and Irene A. Stegun, Math. Comp. 24 (1970), no. 110, 503. MR 415962, DOI 10.1090/S0025-5718-1970-0415962-5
- Djurdje Cvijović, New integral representations of the polylogarithm function, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2080, 897–905. MR 2310128, DOI 10.1098/rspa.2006.1794
- Djurdje Cvijović and Jacek Klinowski, Integral representations of the Riemann zeta function for odd-integer arguments, J. Comput. Appl. Math. 142 (2002), no. 2, 435–439. MR 1906742, DOI 10.1016/S0377-0427(02)00358-8
- Dean G. Duffy, Green’s functions with applications, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001. MR 1888091, DOI 10.1201/9781420034790
- Leonard Lewin, Polylogarithms and associated functions, North-Holland Publishing Co., New York-Amsterdam, 1981. With a foreword by A. J. Van der Poorten. MR 618278
Bibliographic Information
- K. A. Mirzoev
- Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia
- Email: mirzoev.karahan@mail.ru
- T. A. Safonova
- Affiliation: Northern (Arctic) Federal University named after M. V. Lomonosov, Arkhangelsk, Russia
- Email: t.safonova@narfu.ru
- Published electronically: April 1, 2020
- Additional Notes: The results of §§1–3 were obtained with the support of the Russian Science Foundation (grant no. 17-11-01215) and the results of §§4–5 were obtained with the support of the Russian Foundation for Basic Research (grant no. 18-01-00250).
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Moscow Math. Soc. 2019, 133-151
- MSC (2010): Primary 34B27, 34L10, 33E20
- DOI: https://doi.org/10.1090/mosc/294
- MathSciNet review: 4082865
Dedicated: Dedicated to A. A. Shkalikov on the occasion of his seventieth birthday