We show that MacMahon’s sum-of-divisors q-series <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A_k(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C_k(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> arise naturally as the coefficients in the expansions of Gosper’s q-trigonometric functions. In particular, we express <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sin _q(\pi z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>sin</mo> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\cos _q(\pi z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>cos</mo> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in terms of MacMahon’s functions and use Gosper’s q-trigonometric identities to derive new convolution formulas for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A_k(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$C_k(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. As an application, we obtain well-known modular identities relating the Eisenstein series <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$E_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and certain <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-products.

Identities for MacMahon’s q-series through Gosper’s q-trigonometry

期刊论文
发表日期：

2026年7月3日

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Mohamed El Bachraoui

摘要

We show that MacMahon’s sum-of-divisors q-series $A_k(q)$ $A_{k} (q)$ and $C_k(q)$ $C_{k} (q)$ arise naturally as the coefficients in the expansions of Gosper’s q-trigonometric functions. In particular, we express $\sin _q(\pi z)$ ${sin}_{q} (π z)$ and $\cos _q(\pi z)$ ${cos}_{q} (π z)$ in terms of MacMahon’s functions and use Gosper’s q-trigonometric identities to derive new convolution formulas for $A_k(q)$ $A_{k} (q)$ and $C_k(q)$ $C_{k} (q)$ . As an application, we obtain well-known modular identities relating the Eisenstein series $E_2$ $E_{2}$ and certain $\eta $ $η$ -products.