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Tr. Mosk. Mat. Obs., 2019, Volume 80, Issue 2, Pages 157–177 (Mi mmo624)  

Ordinary differential operators and the integral representation of sums of certain power series

K. A. Mirzoeva, T. A. Safonovab

a Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia
b Northern (Arctic) Federal University named after M. V. Lomonosov, Arkhangelsk, Russia

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.

Key words and phrases: Green's function, polylogarithms and associated functions, integral representation of power series, Riemann $\zeta$-function, Euler digamma function.

Funding Agency Grant Number
Russian Science Foundation 17-11-01215
Russian Foundation for Basic Research 18-01-00250_a
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).


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English version:
Transactions of the Moscow Mathematical Society, 2019, 80, 133–151

UDC: 517.927.25, 517.521.15, 517.589
MSC: 34B27, 34L10, 33E20
Received: 15.04.2019

Citation: K. A. Mirzoev, T. A. Safonova, “Ordinary differential operators and the integral representation of sums of certain power series”, Tr. Mosk. Mat. Obs., 80, no. 2, MCCME, M., 2019, 157–177; Trans. Moscow Math. Soc., 80 (2019), 133–151

Citation in format AMSBIB
\Bibitem{MirSaf19}
\by K.~A.~Mirzoev, T.~A.~Safonova
\paper Ordinary differential operators and the integral representation of sums of certain power series
\serial Tr. Mosk. Mat. Obs.
\yr 2019
\vol 80
\issue 2
\pages 157--177
\publ MCCME
\publaddr M.
\mathnet{http://mi.mathnet.ru/mmo624}
\transl
\jour Trans. Moscow Math. Soc.
\yr 2019
\vol 80
\pages 133--151
\crossref{https://doi.org/10.1090/mosc/294}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85083790977}


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