<p>In this paper, we study the DEK-type orthogonal polynomials associated with the weight function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(w(x)=(1+x^2)^{-2}\textrm{e}^{-\frac{x^2}{2}}, \quad x\in (-\infty ,\infty ).\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <msup> <mtext>e</mtext> <mrow> <mo>-</mo> <mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> <mn>2</mn> </mfrac> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi>∞</mi> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> By using the ladder operator approach, we derive a series of difference and differential equations related to the recurrence coefficients <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\{\beta _{n}\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>β</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> of these polynomials. Additionally, we establish a class of confluent Heun equations with respect to the DEK-type orthogonal polynomials. Finally, we consider the asymptotic behavior of the smallest eigenvalue of large Hankel matrices generated by this weight function.</p>