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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?

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# Hypergeometric Properties

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