<p>In this paper, we develop a unified analytic-combinatorial framework for studying product-sum generating functions of the form <Equation ID="Equ83"> <EquationSource Format="TEX">$ \prod _{n=1}^\infty (1 - q^n)^{-a_n} = 1 + \sum _{n=1}^\infty b_n q^n, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <munderover> <mo>∏</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </munderover> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>q</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </munderover> <msub> <mi>b</mi> <mi>n</mi> </msub> <msup> <mi>q</mi> <mi>n</mi> </msup> <mo>,</mo> </mrow> </math></EquationSource> </Equation>focusing on the structural relations between the coefficient sequences <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(a_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(b_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We show that these sequences are intrinsically connected through the classical Euler and Möbius transforms, which provide an effective mechanism for translating between product and sum representations. Furthermore, we investigate the behavior of such generating functions under exponentiation and derive explicit identities relating the coefficients arising from powers of the generating functions. This approach unifies and extends several known results in partition theory and generating function identities, and offers a systematic method for deriving coefficient relations associated with infinite products.</p>

On Product-Sum Generating Functions

期刊论文
发表日期：

2026年5月27日

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Kwang-Wu Chen

摘要

In this paper, we develop a unified analytic-combinatorial framework for studying product-sum generating functions of the form $ \prod _{n=1}^\infty (1 - q^n)^{-a_n} = 1 + \sum _{n=1}^\infty b_n q^n, $ $\prod_{n = 1}^{\infty} {(1 - q^{n})}^{- a_{n}} = 1 + \sum_{n = 1}^{\infty} b_{n} q^{n},$ focusing on the structural relations between the coefficient sequences $(a_n)$ $(a_{n})$ and $(b_n)$ $(b_{n})$ . We show that these sequences are intrinsically connected through the classical Euler and Möbius transforms, which provide an effective mechanism for translating between product and sum representations. Furthermore, we investigate the behavior of such generating functions under exponentiation and derive explicit identities relating the coefficients arising from powers of the generating functions. This approach unifies and extends several known results in partition theory and generating function identities, and offers a systematic method for deriving coefficient relations associated with infinite products.