Let <Equation ID="Equ30"> <EquationSource Format="TEX">$ S(j)= \sum _{\nu =1}^\infty \frac{\nu }{16^\nu (2\nu -1)^2 (2\nu +1)(2\nu +j)}{2\nu \atopwithdelims ()\nu }^2, \quad j\in \{2,3,4,... \}. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>ν</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </munderover> <mfrac> <mi>ν</mi> <mrow> <msup> <mn>16</mn> <mi>ν</mi> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>ν</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>ν</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>ν</mi> <mo>+</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> <msup> <mfenced close=")" open="("> <mfrac linethickness="0pt"> <mrow> <mn>2</mn> <mi>ν</mi> </mrow> <mi>ν</mi> </mfrac> </mfenced> <mn>2</mn> </msup> <mo>,</mo> <mspace width="1em" /> <mi>j</mi> <mo>∈</mo> <mrow> <mo stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">}</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation>In 2022, N. Bhandari showed that for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$j\in \{3,4,5\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>j</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> there are rational numbers <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$b_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>b</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> such that <Equation ID="Equ31"> <EquationSource Format="TEX">$ 4 \pi S(j)=a_j-b_ jG, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mn>4</mn> <mi>π</mi> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mi>j</mi> </msub> <mo>-</mo> <msub> <mi>b</mi> <mi>j</mi> </msub> <mi>G</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where G denotes the Catalan constant. He conjectured that this representation holds for all <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$j\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>j</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. Here, we prove this conjecture. More precisely, we offer recursion formulas to determine the numbers <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$a_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> and bj <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(j\ge 3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>j</mi> <mo>≥</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> explicitly.

A class of series representations for Catalan’s constant

期刊论文
发表日期：

2026年4月7日

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HORST ALZER,
MAN KAM KWONG

摘要

Let $ S(j)= \sum _{\nu =1}^\infty \frac{\nu }{16^\nu (2\nu -1)^2 (2\nu +1)(2\nu +j)}{2\nu \atopwithdelims ()\nu }^2, \quad j\in \{2,3,4,... \}. $ $S (j) = \sum_{ν = 1}^{\infty} \frac{ν}{16^{ν} {(2 ν - 1)}^{2} (2 ν + 1) (2 ν + j)} {(\binom{2 ν}{ν})}^{2}, j \in {2, 3, 4, . . .} .$ In 2022, N. Bhandari showed that for $j\in \{3,4,5\}$ $j \in {3, 4, 5}$ there are rational numbers $a_j$ $a_{j}$ and $b_j$ $b_{j}$ such that $ 4 \pi S(j)=a_j-b_ jG, $ $4 π S (j) = a_{j} - b_{j} G,$ where G denotes the Catalan constant. He conjectured that this representation holds for all $j\ge 3$ $j \geq 3$ . Here, we prove this conjecture. More precisely, we offer recursion formulas to determine the numbers $a_j$ $a_{j}$ and bj $(j\ge 3)$ $(j \geq 3)$ explicitly.