Analytical solution for QCD $$\otimes $$ QED evolution

Abstract We present an analytical solution for the evolution of parton distributions incorporating mixed-order QCD $$\otimes $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>⊗</mml:mo> </mml:math> QED corrections, addressing both polarized and unpolarized cases. Using the Altarelli–Parisi kernels extended to mixed order, we solve the DGLAP equations exactly in Mellin N -space and derive the associated Wilson coefficients for the polarized structure function $$g_1.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Our analytical approach improves computational efficiency and enhances the precision of theoretical predictions in observables sensitive to QED corrections, with relevance for current and future phenomenological applications.