Identities involving the Gaussian hypergeometric function












0












$begingroup$


By applying the algorithm from Solving linear ordinary, 2nd order differential equations via global integral bases. to the five parameter family of ODEs defined in my first answer to Gauge transformation of differential equations I I have stumbled on two identities. Firstly we have:
begin{eqnarray}
&&!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!left[left(-311850 x^2+frac{1334025 x}{4}+frac{480249}{4}right)+left(-frac{1091475 x^3}{2}+frac{4002075 x^2}{8}+frac{1440747 x}{8}right) frac{d}{d x}right], _2F_1left(frac{3}{11},frac{2}{7};frac{4}{3};9 x^2right)=\
&&
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!left[left(-1715175 x^2+frac{1334025 x}{4}+frac{480249}{4}right)+left(frac{1334025 x}{4}-frac{12006225 x^3}{4}right) frac{d}{d x}right], _2F_1left(frac{3}{11},frac{2}{7};frac{1}{3};9 x^2right)
end{eqnarray}



In[334]:= x =.;

eX1 =  ( (480249/4 + (1334025 x)/4 - 311850 x^2) # + ((1440747 x)/

         8 + (4002075 x^2)/8 - (1091475 x^3)/2) D[#, 

        x]) & /@ {Hypergeometric2F1[3/11, 2/7, 4/3, 9 x^2]};

eX2 =  ((480249/4 + (1334025 x)/4 - 1715175 x^2) # + ((1334025 x)/

         4 - (12006225 x^3)/4) D[#, x]) & /@ { 

    Hypergeometric2F1[3/11, 2/7, 1/3, 9 x^2]};

FullSimplify[eX1 - eX2]



Out[337]= {0}


Secondly we have:



begin{eqnarray}
frac{12168 , _2F_1left(-frac{2}{33},-frac{1}{21};frac{2}{3};frac{1}{9}right)-11893 , _2F_1left(-frac{1}{21},frac{31}{33};frac{2}{3};frac{1}{9}right)}{1490148 ,
_2F_1left(frac{31}{33},frac{20}{21};frac{5}{3};frac{1}{9}right)+99200 , _2F_1left(frac{64}{33},frac{41}{21};frac{8}{3};frac{1}{9}right)} = frac{1}{4536}
end{eqnarray}



In[352]:= x =.;

eX1 =   (12168 Hypergeometric2F1[-(2/33), -(1/21), 2/3, 1/9] - 

    11893 Hypergeometric2F1[-(1/21), 31/33, 2/3, 1/9])/(

  1490148 Hypergeometric2F1[31/33, 20/21, 5/3, 1/9] + 

   99200 Hypergeometric2F1[64/33, 41/21, 8/3, 1/9]);

eX2 = 1/4536;

N[eX1 - eX2, 100]



During evaluation of In[352]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -(1/4536)+(12168 Hypergeometric2F1[-(2/33),-(1/21),2/3,1/9]-11893 Hypergeometric2F1[-(1/21),31/33,2/3,1/9])/(1490148 Hypergeometric2F1[31/33,20/21,5/3,1/9]+99200 Hypergeometric2F1[64/33,41/21,8/3,1/9]). >>



Out[355]= 0.*10^-152


The question is how would one go about proving either of those identities?






special-functions hypergeometric-function







share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    By applying the algorithm from Solving linear ordinary, 2nd order differential equations via global integral bases. to the five parameter family of ODEs defined in my first answer to Gauge transformation of differential equations I I have stumbled on two identities. Firstly we have:
    begin{eqnarray}
    &&!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!left[left(-311850 x^2+frac{1334025 x}{4}+frac{480249}{4}right)+left(-frac{1091475 x^3}{2}+frac{4002075 x^2}{8}+frac{1440747 x}{8}right) frac{d}{d x}right], _2F_1left(frac{3}{11},frac{2}{7};frac{4}{3};9 x^2right)=\
    &&
    !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!left[left(-1715175 x^2+frac{1334025 x}{4}+frac{480249}{4}right)+left(frac{1334025 x}{4}-frac{12006225 x^3}{4}right) frac{d}{d x}right], _2F_1left(frac{3}{11},frac{2}{7};frac{1}{3};9 x^2right)
    end{eqnarray}



    In[334]:= x =.;
    
eX1 =  ( (480249/4 + (1334025 x)/4 - 311850 x^2) # + ((1440747 x)/
    
         8 + (4002075 x^2)/8 - (1091475 x^3)/2) D[#, 
    
        x]) & /@ {Hypergeometric2F1[3/11, 2/7, 4/3, 9 x^2]};
    
eX2 =  ((480249/4 + (1334025 x)/4 - 1715175 x^2) # + ((1334025 x)/
    
         4 - (12006225 x^3)/4) D[#, x]) & /@ { 
    
    Hypergeometric2F1[3/11, 2/7, 1/3, 9 x^2]};
    
FullSimplify[eX1 - eX2]
    

    
Out[337]= {0}


    Secondly we have:



    begin{eqnarray}
    frac{12168 , _2F_1left(-frac{2}{33},-frac{1}{21};frac{2}{3};frac{1}{9}right)-11893 , _2F_1left(-frac{1}{21},frac{31}{33};frac{2}{3};frac{1}{9}right)}{1490148 ,
    _2F_1left(frac{31}{33},frac{20}{21};frac{5}{3};frac{1}{9}right)+99200 , _2F_1left(frac{64}{33},frac{41}{21};frac{8}{3};frac{1}{9}right)} = frac{1}{4536}
    end{eqnarray}



    In[352]:= x =.;
    
eX1 =   (12168 Hypergeometric2F1[-(2/33), -(1/21), 2/3, 1/9] - 
    
    11893 Hypergeometric2F1[-(1/21), 31/33, 2/3, 1/9])/(
    
  1490148 Hypergeometric2F1[31/33, 20/21, 5/3, 1/9] + 
    
   99200 Hypergeometric2F1[64/33, 41/21, 8/3, 1/9]);
    
eX2 = 1/4536;
    
N[eX1 - eX2, 100]
    

    
During evaluation of In[352]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -(1/4536)+(12168 Hypergeometric2F1[-(2/33),-(1/21),2/3,1/9]-11893 Hypergeometric2F1[-(1/21),31/33,2/3,1/9])/(1490148 Hypergeometric2F1[31/33,20/21,5/3,1/9]+99200 Hypergeometric2F1[64/33,41/21,8/3,1/9]). >>
    

    
Out[355]= 0.*10^-152


    The question is how would one go about proving either of those identities?






    special-functions hypergeometric-function







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      By applying the algorithm from Solving linear ordinary, 2nd order differential equations via global integral bases. to the five parameter family of ODEs defined in my first answer to Gauge transformation of differential equations I I have stumbled on two identities. Firstly we have:
      begin{eqnarray}
      &&!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!left[left(-311850 x^2+frac{1334025 x}{4}+frac{480249}{4}right)+left(-frac{1091475 x^3}{2}+frac{4002075 x^2}{8}+frac{1440747 x}{8}right) frac{d}{d x}right], _2F_1left(frac{3}{11},frac{2}{7};frac{4}{3};9 x^2right)=\
      &&
      !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!left[left(-1715175 x^2+frac{1334025 x}{4}+frac{480249}{4}right)+left(frac{1334025 x}{4}-frac{12006225 x^3}{4}right) frac{d}{d x}right], _2F_1left(frac{3}{11},frac{2}{7};frac{1}{3};9 x^2right)
      end{eqnarray}



      In[334]:= x =.;
      
eX1 =  ( (480249/4 + (1334025 x)/4 - 311850 x^2) # + ((1440747 x)/
      
         8 + (4002075 x^2)/8 - (1091475 x^3)/2) D[#, 
      
        x]) & /@ {Hypergeometric2F1[3/11, 2/7, 4/3, 9 x^2]};
      
eX2 =  ((480249/4 + (1334025 x)/4 - 1715175 x^2) # + ((1334025 x)/
      
         4 - (12006225 x^3)/4) D[#, x]) & /@ { 
      
    Hypergeometric2F1[3/11, 2/7, 1/3, 9 x^2]};
      
FullSimplify[eX1 - eX2]
      

      
Out[337]= {0}


      Secondly we have:



      begin{eqnarray}
      frac{12168 , _2F_1left(-frac{2}{33},-frac{1}{21};frac{2}{3};frac{1}{9}right)-11893 , _2F_1left(-frac{1}{21},frac{31}{33};frac{2}{3};frac{1}{9}right)}{1490148 ,
      _2F_1left(frac{31}{33},frac{20}{21};frac{5}{3};frac{1}{9}right)+99200 , _2F_1left(frac{64}{33},frac{41}{21};frac{8}{3};frac{1}{9}right)} = frac{1}{4536}
      end{eqnarray}



      In[352]:= x =.;
      
eX1 =   (12168 Hypergeometric2F1[-(2/33), -(1/21), 2/3, 1/9] - 
      
    11893 Hypergeometric2F1[-(1/21), 31/33, 2/3, 1/9])/(
      
  1490148 Hypergeometric2F1[31/33, 20/21, 5/3, 1/9] + 
      
   99200 Hypergeometric2F1[64/33, 41/21, 8/3, 1/9]);
      
eX2 = 1/4536;
      
N[eX1 - eX2, 100]
      

      
During evaluation of In[352]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -(1/4536)+(12168 Hypergeometric2F1[-(2/33),-(1/21),2/3,1/9]-11893 Hypergeometric2F1[-(1/21),31/33,2/3,1/9])/(1490148 Hypergeometric2F1[31/33,20/21,5/3,1/9]+99200 Hypergeometric2F1[64/33,41/21,8/3,1/9]). >>
      

      
Out[355]= 0.*10^-152


      The question is how would one go about proving either of those identities?






      special-functions hypergeometric-function







      share|cite|improve this question









      $endgroup$




      By applying the algorithm from Solving linear ordinary, 2nd order differential equations via global integral bases. to the five parameter family of ODEs defined in my first answer to Gauge transformation of differential equations I I have stumbled on two identities. Firstly we have:
      begin{eqnarray}
      &&!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!left[left(-311850 x^2+frac{1334025 x}{4}+frac{480249}{4}right)+left(-frac{1091475 x^3}{2}+frac{4002075 x^2}{8}+frac{1440747 x}{8}right) frac{d}{d x}right], _2F_1left(frac{3}{11},frac{2}{7};frac{4}{3};9 x^2right)=\
      &&
      !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!left[left(-1715175 x^2+frac{1334025 x}{4}+frac{480249}{4}right)+left(frac{1334025 x}{4}-frac{12006225 x^3}{4}right) frac{d}{d x}right], _2F_1left(frac{3}{11},frac{2}{7};frac{1}{3};9 x^2right)
      end{eqnarray}



      In[334]:= x =.;
      
eX1 =  ( (480249/4 + (1334025 x)/4 - 311850 x^2) # + ((1440747 x)/
      
         8 + (4002075 x^2)/8 - (1091475 x^3)/2) D[#, 
      
        x]) & /@ {Hypergeometric2F1[3/11, 2/7, 4/3, 9 x^2]};
      
eX2 =  ((480249/4 + (1334025 x)/4 - 1715175 x^2) # + ((1334025 x)/
      
         4 - (12006225 x^3)/4) D[#, x]) & /@ { 
      
    Hypergeometric2F1[3/11, 2/7, 1/3, 9 x^2]};
      
FullSimplify[eX1 - eX2]
      

      
Out[337]= {0}


      Secondly we have:



      begin{eqnarray}
      frac{12168 , _2F_1left(-frac{2}{33},-frac{1}{21};frac{2}{3};frac{1}{9}right)-11893 , _2F_1left(-frac{1}{21},frac{31}{33};frac{2}{3};frac{1}{9}right)}{1490148 ,
      _2F_1left(frac{31}{33},frac{20}{21};frac{5}{3};frac{1}{9}right)+99200 , _2F_1left(frac{64}{33},frac{41}{21};frac{8}{3};frac{1}{9}right)} = frac{1}{4536}
      end{eqnarray}



      In[352]:= x =.;
      
eX1 =   (12168 Hypergeometric2F1[-(2/33), -(1/21), 2/3, 1/9] - 
      
    11893 Hypergeometric2F1[-(1/21), 31/33, 2/3, 1/9])/(
      
  1490148 Hypergeometric2F1[31/33, 20/21, 5/3, 1/9] + 
      
   99200 Hypergeometric2F1[64/33, 41/21, 8/3, 1/9]);
      
eX2 = 1/4536;
      
N[eX1 - eX2, 100]
      

      
During evaluation of In[352]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -(1/4536)+(12168 Hypergeometric2F1[-(2/33),-(1/21),2/3,1/9]-11893 Hypergeometric2F1[-(1/21),31/33,2/3,1/9])/(1490148 Hypergeometric2F1[31/33,20/21,5/3,1/9]+99200 Hypergeometric2F1[64/33,41/21,8/3,1/9]). >>
      

      
Out[355]= 0.*10^-152


      The question is how would one go about proving either of those identities?






      special-functions hypergeometric-function




      special-functions hypergeometric-function






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 10 at 13:03









      PrzemoPrzemo

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