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Let  G  be a group, and let σ be an involutive automorphism on  G . We determine the complex-valued solutions(   F  1   ,   F  2   ,  f  )    $(F_{1},F_{2},f)$    of the functional equationF  1    (  x  y  )   +   F  2    (  σ   (  y  )   x  )   =  f   (  x  )   f   (  y  )$F_{1}(xy)+F_{2}(\sigma (y)x)=f(x)f(y)$    (x  ,  y  ∈  G    $x,y\in G$   ) in terms of characters and additive functions. From this we find the complex-valued solutions ( f , g ) of the equationf   (  σ   (  y  )   x   z  0   )   +  g   (  x  y   z  0   )   =  2  f   (  x  )   f   (  y  )$f(\sigma (y)xz_{0})+g(xyz_{0})=2f(x)f(y)$   , where  z  0  is a fixed element in  G . This equation provides a joint generalization of many functional equations such as d'Alembert's, Kannappan's or Van Vleck's equations.

A joint generalization of Van Vleck's and Kannappan's equations on groups