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<p>Here, we establish a polynomial identity in three variables <i>a</i>,&#xa0;<i>b</i>,&#xa0;<i>c</i>, and with the degree of the polynomial given in terms of two integers <i>L</i>,&#xa0;<i>M</i>. By letting <i>L</i> and <i>M</i> tend to infinity, we get the 1993 Alladi–Gordon <i>q</i>-hypergeometric key-identity for the generalized Schur Theorem as well as the fundamental Lebesgue identity by two different choices of the variables. This polynomial identity provides a generalization and a unified approach to the Schur and Lebesgue theorems. We discuss other analytic identities for the Lebesgue and Schur theorems and also provide a key identity (<i>q</i>-hypergeometric) for Andrews’ deep refinement of the Alladi-Schur theorem. Finally, we discuss a new infinite hierarchy of identities, the first three of which relate to the partition theorems of Euler, Lebesgue, and Capparelli, and provide their polynomial versions as well.</p>

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Some q-hypergeometric identities associated with partition theorems of Lebesgue, Schur and Capparelli

  • 期刊论文

  • 发表日期：

    2025年11月15日

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  • Yazan Alamoudi,
  • Krishnaswami Alladi

摘要

Here, we establish a polynomial identity in three variables a, b, c, and with the degree of the polynomial given in terms of two integers L, M. By letting L and M tend to infinity, we get the 1993 Alladi–Gordon q-hypergeometric key-identity for the generalized Schur Theorem as well as the fundamental Lebesgue identity by two different choices of the variables. This polynomial identity provides a generalization and a unified approach to the Schur and Lebesgue theorems. We discuss other analytic identities for the Lebesgue and Schur theorems and also provide a key identity (q-hypergeometric) for Andrews’ deep refinement of the Alladi-Schur theorem. Finally, we discuss a new infinite hierarchy of identities, the first three of which relate to the partition theorems of Euler, Lebesgue, and Capparelli, and provide their polynomial versions as well.