<p>We consider an integral representation of the Lerch transcendent function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( \varvec{\Phi \left( z,s,a\right) } \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">Φ</mi> <mfenced close=")" open="("> <mi mathvariant="bold-italic">z</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">s</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">a</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( \varvec{\Phi \left( z,s,a\right) }\varvec{=}\varvec{\int }_{\varvec{0}}^{\varvec{1}} \varvec{h(t,z)g(t,s,a)dt} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold">Φ</mi> <mfenced close=")" open="("> <mi mathvariant="bold-italic">z</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">s</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">a</mi> </mfenced> </mrow> <mrow> <mo mathvariant="bold">=</mo> </mrow> <msubsup> <mrow> <mo mathvariant="bold">∫</mo> </mrow> <mrow> <mrow> <mn mathvariant="bold">0</mn> </mrow> </mrow> <mrow> <mn mathvariant="bold">1</mn> </mrow> </msubsup> <mrow> <mi mathvariant="bold-italic">h</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">t</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">z</mi> <mo mathvariant="bold" stretchy="false">)</mo> <mi mathvariant="bold-italic">g</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">t</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">s</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">a</mi> <mo mathvariant="bold" stretchy="false">)</mo> <mi mathvariant="bold-italic">d</mi> <mi mathvariant="bold-italic">t</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and two different analytical methods for the approximation of this integral transform to obtain new convergent expansions of the Lerch transcendent in the variable <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( \varvec{z} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">z</mi> </mrow> </math></EquationSource> </InlineEquation>. The first method uses multi-point Taylor expansions of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\( \varvec{h(t,z)} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">h</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">t</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">z</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> at certain appropriately selected base points that provides convergent expansions of the Lerch transcendent in terms of elementary functions of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\( \varvec{z} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">z</mi> </mrow> </math></EquationSource> </InlineEquation> uniformly valid in compact sets of the complex <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\( \varvec{z-} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">z</mi> <mo mathvariant="bold">-</mo> </mrow> </math></EquationSource> </InlineEquation>plane. The second method expands <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\( \varvec{g(t,s,a)} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">g</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">t</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">s</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">a</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in a Taylor series at a selected point in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\( \varvec{[0,1]} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo mathvariant="bold" stretchy="false">[</mo> <mn mathvariant="bold">0</mn> <mo mathvariant="bold">,</mo> <mn mathvariant="bold">1</mn> <mo mathvariant="bold" stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> giving a uniform convergent expansion of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\( \varvec{\Phi \left( z,s,a\right) } \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">Φ</mi> <mfenced close=")" open="("> <mi mathvariant="bold-italic">z</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">s</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">a</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> in terms of elementary functions of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\( \varvec{z} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">z</mi> </mrow> </math></EquationSource> </InlineEquation> valid in a large unbounded region of the complex plane. We provide explicit and/or recursive algorithms for the computation of the coefficients of the expansions. Numerical experiments illustrate the accuracy of the new approximations.</p>