Scholarlink

<p>In 2021, S. Hu and M.-S. Kim [On the Stieltjes constants and gamma functions with respect to alternating Hurwitz zeta functions, <i>J. Math. Anal. Appl.</i>, 509(1):125930, 2022] defined a new type of gamma function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\widetilde{\Gamma }\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation> from the alternating Hurwitz zeta function <i>ζ</i><sub><i>E</i></sub> and obtained some of its properties. In this paper, we further investigate the function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\widetilde{\Gamma }\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation> and obtain several properties in analogy to the classical gamma function Γ, including the integral representation, limit representation, recursive formula, special values, log-convexity, duplication and distribution formulas, and reflection equation. Furthermore, we also prove a Lerch-type formula, which shows that the derivative of <i>ζ</i><sub><i>E</i></sub>(<i>z, x</i>) can be represented by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\widetilde{\Gamma }\left(x\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">~</mo> </mover> <mfenced close=")" open="("> <mi>x</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

On gamma functions with respect to the alternating Hurwitz zeta functions

  • 期刊论文

  • 发表日期：

    2026年1月22日

查看全文
  • Wanyi Wang,
  • Su Hu,
  • Min-Soo Kim

摘要

In 2021, S. Hu and M.-S. Kim [On the Stieltjes constants and gamma functions with respect to alternating Hurwitz zeta functions, J. Math. Anal. Appl., 509(1):125930, 2022] defined a new type of gamma function \(\widetilde{\Gamma }\) Γ ~ from the alternating Hurwitz zeta function ζE and obtained some of its properties. In this paper, we further investigate the function \(\widetilde{\Gamma }\) Γ ~ and obtain several properties in analogy to the classical gamma function Γ, including the integral representation, limit representation, recursive formula, special values, log-convexity, duplication and distribution formulas, and reflection equation. Furthermore, we also prove a Lerch-type formula, which shows that the derivative of ζE(z, x) can be represented by \(\widetilde{\Gamma }\left(x\right)\) Γ ~ x .