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 Fundam. Prikl. Mat., 1998, Volume 4, Issue 2, Pages 669–689 (Mi fpm326)

Representations for Appell's series $F_2(x,y)$ to the vicinity of the singular point $(1,1)$ and near the boundary of its domain of convergence

V. F. Tarasov

Bryansk State Technical University

Abstract: Exact analytical representations for Appell's series $F_2(x,y)$ to the vicinity of the singular point $(1,1)$ and the boundary of its domain of convergence are given. It is shown, that Appell's functions $F_2(1,1)$ and $F_3(1,1)$ have the property of mirror-like symmetry with respect to the center $j_0=-1/2$ under the change $j\mapsto-j-1$, $j\in\mathbb{Z}$, and they correlate between each other.

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Bibliographic databases:
UDC: 517.588

Citation: V. F. Tarasov, “Representations for Appell's series $F_2(x,y)$ to the vicinity of the singular point $(1,1)$ and near the boundary of its domain of convergence”, Fundam. Prikl. Mat., 4:2 (1998), 669–689

Citation in format AMSBIB
\Bibitem{Tar98} \by V.~F.~Tarasov \paper Representations for Appell's series $F_2(x,y)$ to the vicinity of the singular point $(1,1)$ and near the boundary of its domain of convergence \jour Fundam. Prikl. Mat. \yr 1998 \vol 4 \issue 2 \pages 669--689 \mathnet{http://mi.mathnet.ru/fpm326} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1801181} \zmath{https://zbmath.org/?q=an:0976.33010} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Tarasov V.F., “W. Gordon's integral (1929) and its representations by means of Appellis functions F-2, F-1 and F-3”, Modern Physics Letters B, 16:23–24 (2002), 895–899
2. Tarasov V.F., “W. Gordon's integral (1929) and its representations by means of Appell's functions F-2, F-1, and F-3”, Journal of Mathematical Physics, 44:3 (2003), 1449–1452
3. Tarasov V.F., “The Thomas-Fermi-gombas atom models in which the electrons are grouped into n- and nl-shells and a calculation of the atomic form factor”, International Journal of Modern Physics B, 18:3 (2004), 409–419
4. Tarasov V.F., “The Heun-Schrodinger radial equation for DH-atoms”, Modern Physics Letters B, 19:19–20 (2005), 981–989
5. Shpot M.A., “A massive Feynman integral and some reduction relations for Appell functions”, Journal of Mathematical Physics, 48:12 (2007), 123512
6. Tarasov V.F., “Exact analytical expressions and numerical values of diagonal matrix elements {$\langle r^k\rangle_{nlj}, \langle g\vert r^k\vert g\rangle, \langle g\vert r^k\vert f\rangle$} and {$\langle f\vert r^k\vert f\rangle$} with {D}irac's radial functions {$g(r)$} and {$f(r)$} of {H}-like atoms and the symmetry of {A}ppell's function {$F_2(1,1)$}”, International Journal of Modern Physics B, 22:29 (2008), 5175–5205
7. Tarasov V.F., “Multipole matrix elements $\langle nl|r^{\beta}| n' l' \rangle_{\nu}$ for H-like atoms, their asymptotics and applications $(AS \beta = 1, n \leq 4, n' \leq 10)$”, International Journal of Modern Physics B, 23:8 (2009), 2041–2067
8. Tarasov V.F., “Exact analytical expressions and numerical values of Slater's and Marvin's radial integrals of the type $R_k^{(\mu, \nu)} (11,21;12,22), F_k^{(\mu, \nu)} (1,2)$ and $G_k^{(\mu, \nu)} (1;2)$ with the kernel $r_<^{k+\mu}/r_>^{k+\mu}$ (as $k \geq 0, \mu \geq 0$ is an even and $\nu - \mu = 1,3,5, …$) for arbitrary $nl$-states of H-like atoms and $Z \leq 137$ by means of Appell's $F_2 (x,y)$ and Gauss's $_2F_1 (z)$ functions”, Internat J Modern Phys B, 24:27 (2010), 5387–5407
9. Tarasov V.F., “New properties of the {P}. {E}. {A}ppell hypergeometric series {$F_2(\alpha;\beta,\beta';\gamma,\gamma';x,y)$} to the vicinity of the singular point {$(1,1)$} and near the boundary of its domain of convergence {$D_2\colon\vert x\vert +\vert y\vert <1$}”, Internat J Modern Phys B, 24:22 (2010), 4181–4202
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