Hypergeometric Properties

  1. Hello:

    I need to simplify the following if possible

    _2F_1(a,b;c;-x^2) - _2F_1(a+1,b+1;c+1;-x^2)

    In fact, a= 1/2 and c=3/2 and b>=1. In other words, the difference above that I am interested in is more specifically

    _2F_1(.5, b; 1.5; -x^2) - _2F_1(.5+1, b+1; 1.5+1; -x^2)

    I know that
    Arctan x = x* _2F_1(1/2, 1 ; 3/2; -x^2)
    which is a special case of the first term when b=1.

    But I am more interested in reducing the difference at the top for any b>=1. Can I express the difference above as one term (and hopefully not as hypergeometric fct)? And how?
  2. jcsd
  3. Using Maple, I get
    {{}_2F_1(1/2,b;\,3/2;\,-{x}^{2})}-{{}_2F_1(3/2,b+1;\,5/2;\,-{x}^{2})} =
    \sum _{k=0}^{\infty }{\frac { \left( -1 \right) ^{k}\Gamma \left( b+k \right) {x}^{2\,k}}{\Gamma \left( b \right) \Gamma \left( k+1 \right) \left( 2\,k+1 \right) }}-\sum _{k=0}^{\infty }3\,{\frac { \left( -1 \right) ^{k}\Gamma \left( 1+b+k \right) {x}^{2\,k}}{\Gamma \left( b+1 \right) \Gamma \left( k+1 \right) \left( 2\,k+3 \right) }}
    =\sum _{k=0}^{\infty }-{\frac { \left( -1 \right) ^{k}{x}^{2\,k}
    \left( 4\,b+6\,k+3 \right) \Gamma \left( b+k \right) }{ \left( 2\,k+
    3 \right) \left( 2\,k+1 \right) \Gamma \left( b+1 \right) \Gamma
    \left( k \right) }}
    =1/15\,{x}^{2} \left( 4\,b+9 \right)

  4. Thank you for checking in Maple. Hmmm... two things
    1. Mathematica software sometimes gives erroneous analytic solutions for integration. Do we fall into that kind of error w. Maple sometimes?
    2. If the above is analytically correct, can we represent the solution as exponential or trig functions?
  5. We have the property such as this

    Hypergeometric2F1[a,b,c,z] = (1-z)^(c-b-a)*Hypergeometric2F1[c-a,c-b,c,z]

    If we wanted to keep the 2nd term of the hypergeometric function constant, what would the r.h.s. be?

    Hypergeometric2F1[a,b,c,z] = something * Hypergeometric2F1[something,b,something,z]

    What would the somethings be?
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