<p>We study the behavior of the signs of the coefficients of certain infinite products involving the Rogers–Ramanujan continued fraction. For example, if <Equation ID="Equ95"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1050_Article_Equ95.gif" Format="GIF" Height="48" Rendition="HTML" Resolution="72" Type="Linedraw" Width="235" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \sum _{n=0}^{\infty }A(n)q^{n}:= \dfrac{(q^2;q^5)_\infty ^5(q^3;q^5)_\infty ^5}{(q;q^5)_\infty ^5(q^4;q^5)_\infty ^5}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </munderover> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>q</mi> <mi>n</mi> </msup> <mo>:</mo> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>;</mo> <msup> <mi>q</mi> <mn>5</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mi>∞</mi> <mn>5</mn> </msubsup> <msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mn>3</mn> </msup> <mo>;</mo> <msup> <mi>q</mi> <mn>5</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mi>∞</mi> <mn>5</mn> </msubsup> </mrow> <mrow> <msubsup> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>;</mo> <msup> <mi>q</mi> <mn>5</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mi>∞</mi> <mn>5</mn> </msubsup> <msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mn>4</mn> </msup> <mo>;</mo> <msup> <mi>q</mi> <mn>5</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mi>∞</mi> <mn>5</mn> </msubsup> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>then <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1050_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$A(5n+1)>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1050_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$A(5n+2)>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1050_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$A(5n+3)>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1050_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$A(5n+4)<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We also find a few congruences satisfied by some coefficients. For example, for all nonnegative integers <i>n</i>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1050_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="167" /> </InlineMediaObject> <EquationSource Format="TEX">$A(9n+4)\equiv 0 \pmod 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mn>9</mn> <mi>n</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>≡</mo> <mn>0</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1050_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="183" /> </InlineMediaObject> <EquationSource Format="TEX">$ A(16n+13)\equiv 0 \pmod 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mn>16</mn> <mi>n</mi> <mo>+</mo> <mn>13</mn> <mo stretchy="false">)</mo> <mo>≡</mo> <mn>0</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1050_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="183" /> </InlineMediaObject> <EquationSource Format="TEX">$A(15n+r)\equiv 0\pmod {15}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mn>15</mn> <mi>n</mi> <mo>+</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mn>0</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>15</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1050_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="121" /> </InlineMediaObject> <EquationSource Format="TEX">$r\in \{4, 8, 13, 14\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>4</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>13</mn> <mo>,</mo> <mn>14</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Sign patterns and congruences of certain infinite products involving the Rogers–Ramanujan continued fraction

期刊论文
发表日期：

2025年3月14日

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Nayandeep Deka Baruah,
Abhishek Sarma

摘要

We study the behavior of the signs of the coefficients of certain infinite products involving the Rogers–Ramanujan continued fraction. For example, if $\begin{aligned} \sum _{n=0}^{\infty }A(n)q^{n}:= \dfrac{(q^2;q^5)_\infty ^5(q^3;q^5)_\infty ^5}{(q;q^5)_\infty ^5(q^4;q^5)_\infty ^5}, \end{aligned}$ $\begin{matrix} \sum_{n = 0}^{\infty} A (n) q^{n} : = \frac{{(q^{2}; q^{5})}_{\infty}^{5} {(q^{3}; q^{5})}_{\infty}^{5}}{{(q; q^{5})}_{\infty}^{5} {(q^{4}; q^{5})}_{\infty}^{5}}, \end{matrix}$ then $A(5n+1)>0$ $A (5 n + 1) > 0$ , $A(5n+2)>0$ $A (5 n + 2) > 0$ , $A(5n+3)>0$ $A (5 n + 3) > 0$ , and $A(5n+4)<0$ $A (5 n + 4) < 0$ . We also find a few congruences satisfied by some coefficients. For example, for all nonnegative integers n, $A(9n+4)\equiv 0 \pmod 3$ $A (9 n + 4) \equiv 0 (mod 3)$ , $ A(16n+13)\equiv 0 \pmod 4$ $A (16 n + 13) \equiv 0 (mod 4)$ , and $A(15n+r)\equiv 0\pmod {15}$ $A (15 n + r) \equiv 0 (mod 15)$ , where $r\in \{4, 8, 13, 14\}$ $r \in {4, 8, 13, 14}$ .