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<p>We investigate the pattern and the range of parameters for which hypergeometric <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( _1F_2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>1</mn> <mrow /> </mmultiscripts> <msub> <mi>F</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> functions belong to the Laguerre-Pólya class with infinitely many real zeros. Our approach is based on the sampling theorem which provides sufficient conditions in terms of the sign of samples. In the case where the Lommel function comes into play, we use the differential equation method to determine the sign change, thereby specifying those parameters. As for the reality of all zeros of the Lommel function, our results give an extensive range of parameters, which turns out to be the best possible for the Struve function.</p>

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A sampling theorem of hypergeometric \( _1F_2\) functions and the Laguerre-Pólya class

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月5日

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  • Yong-Kum Cho,
  • Seok-Young Chung,
  • Young Woong Park

摘要

We investigate the pattern and the range of parameters for which hypergeometric \( _1F_2\) 1 F 2 functions belong to the Laguerre-Pólya class with infinitely many real zeros. Our approach is based on the sampling theorem which provides sufficient conditions in terms of the sign of samples. In the case where the Lommel function comes into play, we use the differential equation method to determine the sign change, thereby specifying those parameters. As for the reality of all zeros of the Lommel function, our results give an extensive range of parameters, which turns out to be the best possible for the Struve function.