Tag Archives: Beta function

and the generating function?

Well, in the previous post we were talking of the infinite sum of the Catalan numbers’  inverses, now let me tell you that the same method, but with $x^n$ included,  gonna give you the generating funcion

$\frac{2\,\left( {\sqrt{4-x}}\,\left(8+x\right)+12\,{\sqrt{x}}\,\arctan (\frac{{\sqrt{x}}}{{\sqrt{4 - x}}}) \right) }{\sqrt{\left( 4 - x \right)^5}}$

Yes, she (the formula) expands about the origin as

$1+x+\frac{x^2}{2}+\frac{x^3}{5}+\frac{x^4}{14}+\frac{x^5}{42}+\frac{x^6}{132}+\frac{x^7}{429}+\cdots$

cool! isn’t it?  By the way do not try to reproduce the expansion without software because just at first derivative you gonna get angry enough to want to pull your hair away! This function is not completely new at all -but strangely- it is hard to find it explicitly in the modern math literature.

A next Frage is: what the heck is it good for? $: )$

Exercise: insert $(-1)^nx^n$ in the beta fuction method explained earlier and tell me what you get… bye

Filed under Catalan numbers, math, numbers, sum of reciprocals

whole sum of the reciprocal Catalan numbers

In this little post I complete the details of the calculation

$2+\frac{4\sqrt{3}\pi}{27}=\sum_{n=0}^{\infty}\frac{n+1}{{2n\choose n}}=1+1+\frac{1}{2}+\frac{1}{5}+\frac{1}{14}+\frac{1}{42}+\cdots$

for the reciprocals $C_n=\frac{1}{n+1}{2n\choose n}$, the famous so called Catalan numbers. It seems this is well known but it is scarcely quoted anywhere: Cf1, Cf2

We begin by recalling the relation ${C_n}^{-1}=(2n+1)(n+1)\int_0^1t^n(1-t)^n{\rm{d}}t$, that is,  in terms of the Beta function. So

$\sum_{n=0}^{\infty}{C_n}^{-1}=\sum_{n=0}^{\infty}(2n+1)(n+1)\int_0^1t^n(1-t)^n{\rm{d}}t$

$=\int_0^1\left[\sum_{n=0}^{\infty}(2n+1)(n+1)t^n(1-t)^n\right]{\rm{d}}t$

$=\int_0^1\frac{1+3t - 3t^2}{{(1-t+t^2)}^3}{\rm{d}}t$

$=2+\frac{4\sqrt{3}\pi}{27}$

This was hinted to me, thanks to, by the professor Qiaochu Yuan and I am not ashamed to confess that the calculations were executed in mathematica-v5.  Why? well, the risk of introducing errors is a little-big.

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Filed under math