A closed term for a sum that includes a coefficient which looks similar to Catalan coefficient.

Question

I have the following sum:

$$\sum_{k=0}^n \frac{1}{2k-1} {2k \choose k} (pq)^k$$

where $0<p=1-q<1$.

Is there a closed expression for this sum? if there is then how to prove it, if you have a reference even better.

Thanks.

Answer 1

Note that $p$ and $q$ don't matter, so we'll just call it $x$. According to WolframAlpha, we have

$$\sum_{k=0}^n\frac1{2k-1}\binom{2k}kx^k=-\frac{\binom{2n+2}{n+1}x^{n+1}_2F_1(1,n+\frac12;n+2;4x)+(2n+1)\sqrt{1-4x}}{2n+1}$$

One may note, however, that it looks much nicer if we let $n\to\infty$:

$$-\sqrt{1-4x}=\sum _{k=0}^{\infty }\frac1{2n-1}\binom{2k}kx^k$$

As seen on Wikipedia.