Talk:Ordinary Differential Equations/Frobenius Solution to the Hypergeometric Equation

Comparison with Abramowitz and Stegun

The solutions where $\gamma$ is an integer not equal to one are both incorrect in this article. (See Abramowitz and Stegun "Handbook of Mathematical Functions" Dover Books on Advanced Mathematics ISBN 0-486-61272-4.)

For instance, in the case where $\gamma$ is a negative integer or zero so that the $c=0$ root of the indicial equation gives an infinite coefficient. However, we must evaluate $\frac{\partial y_b}{\partial c}$ at the $c=0$ root as well. We do not evaluate this derivative at $c=1-\gamma$ as the solution under discussion suggests. Doing so gives an entirely wrong result.

Evaluating $\frac{\partial y_b}{\partial c}$ at $c=0$ yields same result as described in Abramowitz and Stegun 15.5.20 and 15.5.21.

Similarly, if $c>2,3,4...=1+m$ then the $c=1=\gamma$ root throws up an infinity. In this case we must evaluate $\frac{\partial y_b}{\partial c}$ at $c=1-\gamma$ and not $c=0$. This gives the results  Abramowitz and Stegun 15.5.18 and 15.5.19.

The finite sums involved arise from coefficients which are zero at the root of the indicial equation in the first solution, but which have partial derivatives (w.r.t c) which are non zero at the the root x=zero of the indicial equation.Tethys sea (discuss • contribs) 15:46, 14 February 2016 (UTC) --Tethys sea (discuss • contribs) 14:25, 30 May 2016 (UTC)

Re-write of section 3.3
As I mentioned previously, the results in section 3.3 (with sub-sections gamma an integer <=0 and gamma >1) were wrong since y_2 was derived by evaluating the partial derivative at the wrong root of the indicial equation.

I have now replaced section 3.3 completely, starting with gamma 0 or negative I used the usual "replace a_0 with b_0 c" way to derive the solution.

However, this is not the standard solution as supplied (for instance) by Abramowitz and Stegun. I have therefore included a section describing (one of the ways) in which the standard solution can be arrived at. I have finished things off with writing down the standard solution for the case gamma >1.

I would be thankful for corrections regarding typos and html style (I am not used to html at all) and so on. --Tethys sea (discuss • contribs) 14:25, 30 May 2016 (UTC)