A series representation for the Catalan constant

Abstract We prove the following series representation for the Catalan constant G : <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mfrac> <m:mn>3</m:mn> <m:mn>10</m:mn> </m:mfrac> <m:mo>-</m:mo> <m:mrow> <m:mn>48</m:mn> <m:mo>⁢</m:mo> <m:mrow> <m:munderover> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mi mathvariant="normal">∞</m:mi> </m:munderover> <m:mrow> <m:mfrac> <m:mrow> <m:mrow> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:msup> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mn>8</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mn>9</m:mn> </m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:msup> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mn>5</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:msup> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mn>20</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>29</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfrac> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>S</m:mi> <m:mi>n</m:mi> <m:mn>3</m:mn> </m:msubsup> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> G=\frac{3}{10}-48\sum_{n=0}^{\infty}\frac{4n^{2}+8n-9}{(4n^{2}-4n+5)(4n^{2}+20% n+29)}S_{n}^{3}, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:munderover> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mi>n</m:mi> </m:munderover> <m:mfrac> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>k</m:mi> </m:msup> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:mrow> </m:math> S_{n}=\sum_{k=0}^{n}\frac{(-1)^{k}}{2k+1} is the n th partial sum of the classical Leibniz series for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mi>π</m:mi> <m:mn>4</m:mn> </m:mfrac> </m:math> <jats:inline-graphic