The aforementioned convergent series in Ramanujan's “identity” is also similar to one that appears in a curious identity found in Chapter 15 in Ramanujan's second notebook, written in a more elegant, equivalent formulation on page 332 in the lost notebook. This formula may be regarded as a formula for class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815304059&_mathId=si1.gif&_user=111111111&_pii=S0001870815304059&_rdoc=1&_issn=00018708&md5=441bdc355ac020f293891bdf2862f795">class="imgLazyJSB inlineImage" height="20" width="31" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870815304059-si1.gif">class="mathContainer hidden">class="mathCode">, and in 1925, S. Wigert obtained a generalization giving a formula for class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815304059&_mathId=si2.gif&_user=111111111&_pii=S0001870815304059&_rdoc=1&_issn=00018708&md5=749ae913628c085295623434a63d02d7">class="imgLazyJSB inlineImage" height="20" width="32" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870815304059-si2.gif">class="mathContainer hidden">class="mathCode"> for any even integer class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815304059&_mathId=si3.gif&_user=111111111&_pii=S0001870815304059&_rdoc=1&_issn=00018708&md5=9038ee52a3dc255716c560621c9788a9" title="Click to view the MathML source">k≥2class="mathContainer hidden">class="mathCode">. We extend the work of Ramanujan and Wigert in this paper.
The Voronoï summation formula appears prominently in our study. In particular, we generalize work of J.R. Wilton and derive an analogue involving the sum of divisors function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815304059&_mathId=si1169.gif&_user=111111111&_pii=S0001870815304059&_rdoc=1&_issn=00018708&md5=9e188a7fcffda0d780d8d4f132cfc5c8" title="Click to view the MathML source">σs(n)class="mathContainer hidden">class="mathCode">.
The modified Bessel functions class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815304059&_mathId=si5.gif&_user=111111111&_pii=S0001870815304059&_rdoc=1&_issn=00018708&md5=1192aa7ef16fd9b178afeb7775e065fd" title="Click to view the MathML source">Ks(x)class="mathContainer hidden">class="mathCode"> arise in several contexts, as do Lommel functions. We establish here new series and integral identities involving modified Bessel functions and modified Lommel functions. Among other results, we establish a modular transformation for an infinite series involving class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815304059&_mathId=si1169.gif&_user=111111111&_pii=S0001870815304059&_rdoc=1&_issn=00018708&md5=9e188a7fcffda0d780d8d4f132cfc5c8" title="Click to view the MathML source">σs(n)class="mathContainer hidden">class="mathCode"> and modified Lommel functions. We also discuss certain obscure related work of N.S. Koshliakov. We define and discuss two new related classes of integral transforms, which we call Koshliakov transforms, because he first found elegant special cases of each.