The main aim of this paper is to prove a general theorem, which is related with the theta functions of Jacobi and Ramanujan. Our proof of this theorem is accomplished by means of suitable decomposition and rearrangement of the infinite series involved therein. Furthermore, the theorem is applied in order to derive several presumably new values of Ramanujan’s theta functions <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1785_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi (\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1785_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi (\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and Jacobi’s theta functions <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1785_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\vartheta _2(0,\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ϑ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1785_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\vartheta _3(0,\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ϑ</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1785_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\vartheta _4(0,\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ϑ</mi> <mn>4</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, as well as to evaluate the sums of a large number of combinations of infinite series involving such exponential functions as <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1785_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$e^{-p\pi \left( qn+r\right) ^2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>p</mi> <mi>π</mi> <msup> <mfenced close=")" open="("> <mi>q</mi> <mi>n</mi> <mo>+</mo> <mi>r</mi> </mfenced> <mn>2</mn> </msup> </mrow> </msup> </math></EquationSource> </InlineEquation> for various positive integers p, q and r, where <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1785_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\geqq 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≧</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is the summation index. In particular, the evaluation of infinite sum with <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1785_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$p=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1785_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$q=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1785_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$r=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> was asked in an examination of Trinity College of the University of Cambridge in the year 1881.

A general theorem on the theta functions of Jacobi and Ramanujan with their applications

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发表日期：

2025年9月9日

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H. M. Srivastava,
Bhawna Gupta,
M. I. Qureshi,
M. S. Baboo

摘要

The main aim of this paper is to prove a general theorem, which is related with the theta functions of Jacobi and Ramanujan. Our proof of this theorem is accomplished by means of suitable decomposition and rearrangement of the infinite series involved therein. Furthermore, the theorem is applied in order to derive several presumably new values of Ramanujan’s theta functions $\varphi (\cdot )$ $φ (\cdot)$ and $\psi (\cdot )$ $ψ (\cdot)$ , and Jacobi’s theta functions $\vartheta _2(0,\cdot )$ $ϑ_{2} (0, \cdot)$ , $\vartheta _3(0,\cdot )$ $ϑ_{3} (0, \cdot)$ and $\vartheta _4(0,\cdot )$ $ϑ_{4} (0, \cdot)$ , as well as to evaluate the sums of a large number of combinations of infinite series involving such exponential functions as $e^{-p\pi \left( qn+r\right) ^2}$ $e^{- p π {(q n + r)}^{2}}$ for various positive integers p, q and r, where $n\geqq 0$ $n ≧ 0$ is the summation index. In particular, the evaluation of infinite sum with $p=1$ $p = 1$ , $q=2$ $q = 2$ and $r=1$ $r = 1$ was asked in an examination of Trinity College of the University of Cambridge in the year 1881.