For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k\in \mathbb {N}_{0}=\mathbb {N}\cup \{0\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">N</mi> <mn>0</mn> </msub> <mo>=</mo> <mi mathvariant="double-struck">N</mi> <mo>∪</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$x\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$F(a,b;a+b+k;x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>k</mi> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Li_{k+1}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <msub> <mi>i</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the k-balanced hypergeometric function and the polylogarithm function, respectively. In this paper, the authors show the monotonicity properties of certain combinations defined in terms of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$F(a,b;a+b+k;x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>k</mi> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$Li_{k+1}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <msub> <mi>i</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for a suitable region of (a, b). These results extends the corresponding results of the zero-balanced hypergeometric function for the cases that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$k=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, in which cases if we take <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$a=b=1/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, the results are obtained for the complete complete elliptic integral of the first kind.

Monotonicity Properties for k-Balanced Hypergeometric Functions

期刊论文
发表日期：

2026年3月17日

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Xiao-Yan Ma,
Chen-Tao Yu,
Gen-Hong Zhong

摘要

For $k\in \mathbb {N}_{0}=\mathbb {N}\cup \{0\}$ $k \in N_{0} = N \cup {0}$ and $x\in (0,1)$ $x \in (0, 1)$ , let $F(a,b;a+b+k;x)$ $F (a, b; a + b + k; x)$ and $Li_{k+1}(x)$ $L i_{k + 1} (x)$ be the k-balanced hypergeometric function and the polylogarithm function, respectively. In this paper, the authors show the monotonicity properties of certain combinations defined in terms of $F(a,b;a+b+k;x)$ $F (a, b; a + b + k; x)$ and $Li_{k+1}(x)$ $L i_{k + 1} (x)$ for a suitable region of (a, b). These results extends the corresponding results of the zero-balanced hypergeometric function for the cases that $k=0$ $k = 0$ , in which cases if we take $a=b=1/2$ $a = b = 1 / 2$ , the results are obtained for the complete complete elliptic integral of the first kind.