\(\(\(
\)\( (*\ 
      We\ specify\ the\ relation\ between\ the\ notation\ of\ the\ article\ and\ the\ notation\ used\ in\ this\ file, \ 
      using\ latex\ notation\ *) \)\(
\)\(
\)\( (*\ 
      This\ routine\ verifies\ that\ G_n\ achieves\ its\ absolute\ minimum\ at\ {t_n, \(-t_n\)}, \ 
      when\ n = \(7\ and\ \alpha_  
            7 = \(-3.215\)\)*) \)\(
\)\(
\)\( (*\ LegendreP[a, c, t] = 
        P_ {a}^{c} \((t)\)\ *) \)\(
\)\( (*\ 
      LegendreQ[a, c, t] = 
        Q_ {a}^{c} \((t)\)\ *) \)\(
\)\( (*\ 
      b\ denotes\ the\ parameter\ \alpha\ *) \)\(
\)\(P[
        b_, n_, t_] := 
      LegendreP[b + \((n - 3)\)/2, \(-\((n - 3)\)\)/2, t]

    p[b_, n_, t_] := 
      LegendreP[b + \((n - 3)\)/2, \ \(-\((n - 1)\)\)/2, 
        t]

    Q[n_, t_] := 
      LegendreQ[\((n - 1)\)/2, \((n - 3)\)/2, t]

    q[n_, t_] := 
      LegendreQ[\((n - 1)\)/2, \((n - 1)\)/2, 
        t]

 (*\ 
      En[b, n, t]\ denotes\ E_n \((t)\)\ *) 

    En[b_, n_, 
        t_] := \((P[b, n, t] + 
          P[b, n, \(-t\)])\)

 (*\ 
      Z[b, n, t]\ denotes\ Z_n \((t)\)\ *) 

    Z[b_, n_, 
        t_] := \((p[b, n, t] - 
          p[b, n, \(-t\)])\)

 (*\ 
      G[b, n, t]\ denotes\ G_n \((t)\)\ *) 

    G[b_, n_, 
        t_] := \((1 - t^2)\)^\((\(-\((n - 3)\)\)/
              2)\)\ {\((q[n, t] + 
                  b \((b + n - 2)\) 
                    Q[n, t]\ Z[b, n, t]/En[b, n, t])\)^2\  + \((1 - 
                    b)\)^2\ \((Q[n, 
                    t])\)^2}

 (*\ 
      The\ following\ command\ plots\ the\ graph\ of\ G_n \((t)\)\ on\ the\ interval\ \((\(-t_n\), \ t_n)\), 
      for\ n = \(7\ and\ alpha_  
            7 = \(-3.215\)\)\ \ *) 


    Plot[G[\(-3.215\), 7, t], {t, \(-0.5173305417372707\), 
        0.5173305417372707}]

 (*\ 
      The\ following\ command\ evaluates\ G_  7 \((0)\) - 
          G_  7 \((t_  7)\)\ for\ \alpha_  
            7 = \(-3.215\)\ \ *) 


    G[\(-3.215\), 7, 0] - G[\(-3.215\), 7, 0.5173305417372707]\)\)\)