HP17bII+ Programming t-distribution

[Thomas Klemm]

(07-25-2015 11:29 AM)Don Shepherd Wrote:  I think I am getting outside of my area of comfort!

You might still remember Valentin's HP-15C Mini-challenge that was solved by Gerson using Euler's reflection formula:

\(\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin{(\pi z)}}\)

Now if you set \(z=\frac{1}{2}\) you get:

\(\Gamma(\tfrac{1}{2})\Gamma(\tfrac{1}{2}) = \frac{\pi}{\sin{(\tfrac{\pi}{2})}} = \frac{\pi}{1} = \pi \)

From this we can conclude that in fact: \(\Gamma(\tfrac{1}{2})=\sqrt{\pi}\)

Often I have to write down concrete examples to understand formulas.
Thus for \(\nu=6\) we can use \(\Gamma(x+1)=x \Gamma(x)\) to get:

\(\begin{align}
\Gamma(\frac{\nu+1}{2})&=\Gamma(\frac{7}{2})=\Gamma(\frac{5}{2}+1) \\
&=\frac{5}{2}\cdot\Gamma(\frac{5}{2}) \\
&=\frac{5}{2}\cdot\frac{3}{2}\cdot\Gamma(\frac{3}{2}) \\
&=\frac{5}{2}\cdot\frac{3}{2}\cdot\frac{1}{2}\cdot\Gamma(\frac{1}{2}) \\
&=\frac{5}{2}\cdot\frac{3}{2}\cdot\frac{1}{2}\cdot\sqrt{\pi} \\
\end{align}\)

We don't end up with a factorial but with the double factorial 5!! in the numerator.
Thus we just insert the missing values which happen to be even:

\(\begin{align}
\Gamma(\frac{7}{2})&=\frac{5}{2}\cdot\frac{3}{2}\cdot\frac{1}{2}\cdot\sqrt{\pi} \\
&=\frac{5}{2}\cdot\frac{4}{4}\cdot\frac{3}{2}\cdot\frac{2}{2}\cdot\frac{1}{2}​\cdot\sqrt{\pi} \\
&=\frac{5}{2}\cdot\frac{4}{2\cdot2}\cdot\frac{3}{2}\cdot\frac{2}{2\cdot 1}\cdot\frac{1}{2}\cdot\sqrt{\pi} \\\end{align}\)

Now we end up with 5! in the nominator. In the denominator we end up with the product of 2⁵ and 2!:

\(\Gamma(\frac{7}{2})=\frac{5!}{2^5\,2!}\,\sqrt{\pi}\)

Compare this to the general formula if you set \(\nu=6\):

\(\Gamma(\frac{6+1}{2})=\frac{(6-1)!}{2^{6-1}\,(\frac{6}{2}-1)!}\,\sqrt{\pi}\)

Quote:Something in my implementation of that equation causes the 17b to not be able to evaluate it at all.

I didn't enter your formula but as it's written right now the T=(…) is missing.
I assume that in this case the expression is just set to 0.
And this might explain that no solution was found.

HTH
Thomas