<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> </mml

The <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>I</a:mi> <a:mo>=</a:mo> <a:mfrac> <a:mn>3</a:mn> <a:mn>2</a:mn> </a:mfrac> </a:math> <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>π</c:mi> <c:mi>K</c:mi> </c:math> <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>s</e:mi> </e:math> -wave scattering phase shift is computed by lattice quantum chromodynamics with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>N</g:mi> <g:mi>f</g:mi> </g:msub> <g:mo>=</g:mo> <g:mn>3</g:mn> </g:math> flavors of Asqtad-improved staggered fermions. The energy-eigenvalues of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>π</i:mi> <i:mi>K</i:mi> </i:math> systems at one center of mass frame and six moving frames using moving wall source technique are used to get phase shifts by Lüscher’s formula and its extensions. The calculations are good enough to acquire effective range expansion parameters: scattering length <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>a</k:mi> </k:math> , effective range <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>r</m:mi> </m:math> , and shape parameter <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>P</o:mi> </o:math> , which are in good agreement with our explicit analytical predictions in three-flavor chiral perturbation theory at next-to-leading order. All results are fairly consistent with experimental measurements, phenomenological studies, and lattice estimations. Numerical computations are implemented at a fine ( <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>a</q:mi> <q:mo>≈</q:mo> <q:mn>0.082</q:mn> <q:mtext> </q:mtext> <q:mtext> </q:mtext> <q:mi>fm</q:mi> </q:math> , <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msup> <s:mi>L</s:mi> <s:mn>3</s:mn> </s:msup> <s:mi>T</s:mi> <s:mo>=</s:mo> <s:mn>4</s:mn> <s:msup> <s:mn>0</s:mn> <s:mn>3</s:mn> </s:msup> <s:mn>96</s:mn> </s:math> ) lattice ensemble with physical quark masses.