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• W4309326937 abstract "We introduce two convergent series expansions (direct and recursive) in terms of Bessel functions and the number of representations of an integer as a sum of squares for <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>N</mml:mi> </mml:math> -dimensional Madelung constants, <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:math> , where <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>s</mml:mi> </mml:math> is the exponent of the Madelung series (usually chosen as <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:math> ). The convergence of the Bessel function expansion is discussed in detail. Values for <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:math> for <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mo>,</mml:mo> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:math> and 6 for dimension up to <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>20</mml:mn> </mml:math> are presented. This work extends Zucker’s original analysis on <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>N</mml:mi> </mml:math> -dimensional Madelung constants for even dimensions up to <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>8</mml:mn> </mml:math> ." @default.
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• W4309326937 date "2022-11-01" @default.
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• W4309326937 title "The Madelung constant in <i>N</i> dimensions" @default.
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