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How can I find the sum: $$\sum_{k=0}^{n} \binom{n-k}{k}x^{k}$$

Edit: The answer to this question is: $$\frac{{(1+\sqrt{1+4x})}^{n+1}-{(1-\sqrt{1+4x})}^{n+1}}{2^{n+1}\sqrt{1+4x}}$$ I don't know how to arrive at this answer.

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  • $\begingroup$ According to WolframAlpha, it seems that sum can only be expressed in terms of hypergeometric functions. Although for $x = 1$, the sum evaluates to the $n+1$-th Fibonacci number. $\endgroup$
    – JimmyK4542
    Jun 14, 2015 at 5:32
  • $\begingroup$ @JimmyK4542 I am fine with the result given in the link. What I do not understand is how to arrive at that result. $\endgroup$
    – Iguana
    Jun 14, 2015 at 5:37
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    $\begingroup$ the recurrence relation is $F(n)=F(n-1)+xF(n-2)$ $\endgroup$
    – VigneshM
    Jun 14, 2015 at 5:47
  • $\begingroup$ @VigneshManoharan How does one obtain this recurrence relation from the series? $\endgroup$
    – Iguana
    Jun 14, 2015 at 5:50

3 Answers 3

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This answer is similar to this answer and this answer. That is, compute the generating function of the given sequence:

$$ \begin{align} &\sum_{n=0}^\infty\sum_{k=0}^n\binom{n-k}{k}x^ky^n\\ &=\sum_{k=0}^\infty\sum_{n=k}^\infty\binom{n-k}{k}x^ky^n\tag{1}\\ &=\sum_{k=0}^\infty\sum_{n=0}^\infty\binom{n}{k}x^ky^{n+k}\tag{2}\\ &=\sum_{k=0}^\infty\frac{(xy^2)^k}{(1-y)^{k+1}}\tag{3}\\ &=\frac1{1-y}\frac1{1-\frac{xy^2}{1-y}}\tag{4}\\ &=\frac1{1-y-xy^2}\tag{5}\\ &=\frac1{\left(1-\frac{1-\sqrt{1+4x}}2y\right)\left(1-\frac{1+\sqrt{1+4x}}2y\right)}\tag{6}\\ &=\frac{\frac{-1+\sqrt{1+4x}}{2\sqrt{1+4x}}}{1-\frac{1-\sqrt{1+4x}}2y}+\frac{\frac{1+\sqrt{1+4x}}{2\sqrt{1+4x}}}{1-\frac{1+\sqrt{1+4x}}2y}\tag{7}\\ &=\frac1{\sqrt{1+4x}}\sum_{n=0}^\infty\left[\left(\frac{1+\sqrt{1+4x}}2\right)^{n+1}-\left(\frac{1-\sqrt{1+4x}}2\right)^{n+1}\right]y^n\tag{8} \end{align} $$ Explanation:
$(1)$: change order of summation
$(2)$: substitute $n\mapsto n+k$
$(3)$: $\sum\limits_{n\ge k}\binom{n}{k}y^n=\frac{y^k}{(1-y)^{k+1}}$
$(4)$: $\sum\limits_{k\ge 0}x^k=\frac1{1-x}$
$(5)$: simplify
$(6)$: quadratic formula
$(7)$: partial fractions
$(8)$: $\sum\limits_{n\ge 0}x^n=\frac1{1-x}$

Equating coefficients of $y^n$ yields $$ \sum_{k=0}^n\binom{n-k}{k}x^k =\frac1{\sqrt{1+4x}}\left[\left(\frac{1+\sqrt{1+4x}}2\right)^{n+1}-\left(\frac{1-\sqrt{1+4x}}2\right)^{n+1}\right] $$

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  • $\begingroup$ Having now looked at Marko Riedel's answer, I believe that we have used essentially the same methods, just presented a bit differently. $\endgroup$
    – robjohn
    Jun 15, 2015 at 10:24
  • $\begingroup$ I realized the same thing too. The reason why I accepted this answer is because it's presented better according to me. $\endgroup$
    – Iguana
    Jun 16, 2015 at 9:09
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Introduce the generating function $$f(z) = \sum_{n\ge 0} z^n \sum_{k=0}^n {n-k\choose k} x^k.$$

This becomes $$f(z) = \sum_{k\ge 0} x^k \sum_{n\ge k} z^n {n-k\choose k} \\ = \sum_{k\ge 0} x^k \sum_{n\ge 0} z^{n+k} {n\choose k} = \sum_{k\ge 0} x^k \sum_{n\ge k} z^{n+k} {n\choose k} \\ = \sum_{k\ge 0} x^k \sum_{n\ge 0} z^{n+2k} {n+k\choose k} = \sum_{k\ge 0} x^k z^{2k} \sum_{n\ge 0} z^{n} {n+k\choose k} \\ = \sum_{k\ge 0} x^k z^{2k} \frac{1}{(1-z)^{k+1}} = \frac{1}{1-z} \frac{1}{1-xz^2/(1-z)} \\ = \frac{1}{1-z-xz^2} = -\frac{1/x}{z^2+z/x-1/x}.$$

Solving $z^2+z/x-1/x$ we obtain $$\rho_{1,2} = \frac{-1 \pm\sqrt{1+4x}}{2x}$$

and we have that $$f(z) = -\frac{1}{x}\frac{1}{(z-\rho_1)(z-\rho_2)} \\ = -\frac{1}{x} \frac{1}{\rho_1-\rho_2} \left(\frac{1}{z-\rho_1}-\frac{1}{z-\rho_2}\right) \\ = -\frac{1}{x} \frac{1}{\rho_1-\rho_2} \left(\frac{1}{\rho_1}\frac{1}{z/\rho_1-1} -\frac{1}{\rho_2}\frac{1}{z/\rho_2-1}\right).$$

Extracting coefficients from this we obtain $$-\frac{1}{x} \frac{x}{\sqrt{1+4x}} \left(-\frac{1}{\rho_1^{n+1}} +\frac{1}{\rho_2^{n+1}}\right).$$

Now since $\rho_1\rho_2 = -1/x$ this becomes $$-\frac{1}{\sqrt{1+4x}} \left(-(-1)^{n+1} x^{n+1} \rho_2^{n+1} +(-1)^{n+1} x^{n+1} \rho_1^{n+1}\right).$$

Observe that $$(-1)^{n+1} x^{n+1} \rho_{1,2}^{n+1} = \frac{(1\mp\sqrt{1+4x})^{n+1}}{2^{n+1}}$$ so this finally becomes $$-\frac{1}{\sqrt{1+4x}} \frac{(1-\sqrt{1+4x})^{n+1}-(1+\sqrt{1+4x})^{n+1}} {2^{n+1}} \\ = \frac{1}{\sqrt{1+4x}} \frac{(1+\sqrt{1+4x})^{n+1}-(1-\sqrt{1+4x})^{n+1}} {2^{n+1}}.$$

Alternate derivation of the generating function.

Introduce $${n-k\choose k} = \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^{n-k}}{z^{k+1}} dz.$$

This yields for the sum $$\frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^{n}}{z} \sum_{k=0}^n \frac{x^k}{(1+z)^k z^k} \; dz \\ = \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^{n}}{z} \frac{x^{n+1}/(1+z)^{n+1}/z^{n+1}-1} {x/(1+z)/z-1} \; dz \\ = \frac{1}{2\pi i} \int_{|z|=\epsilon} (1+z)^{n+1} \frac{x^{n+1}/(1+z)^{n+1}/z^{n+1}-1} {x-z-z^2} \; dz.$$

We omit the part that does not contribute to get $$\frac{1}{2\pi i} \int_{|z|=\epsilon} (1+z)^{n+1} \frac{x^{n+1}/(1+z)^{n+1}/z^{n+1}} {x-z-z^2} \; dz \\ = \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{1}{z^{n+1}} \frac{x^{n+1}}{x-z-z^2} \; dz.$$

This is $$[z^n] \frac{x^{n+1}}{x-z-z^2} = x [z^n] x^{n} \frac{1}{x-z-z^2} \\ = x [z^n] \frac{1}{x-zx-z^2x^2} = [z^n] \frac{1}{1-z-z^2x}.$$

We have the same generating function as above, QED.

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  • $\begingroup$ (+1) It took me a while to see, but I think our approaches are similar. I don't quite understand the introduction of the terms with $\binom{n+k}{k}$, but perhaps you are using an identity with which I am unfamiliar. $\endgroup$
    – robjohn
    Jun 15, 2015 at 10:33
  • $\begingroup$ (+1). Thanks for commenting. I just realized my factorization of the generating function and the coefficient extraction is more complicated than it needs to be, I will leave it as is since the reader may consult your answer to see how it should be done. $\endgroup$ Jun 15, 2015 at 17:28
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$$\begin{align}F(n) = \sum_{k=0}^{n} \binom{n-k}{k} x^k &= \sum_{k=0}^{n} \left(\binom{n-k-1}{k} + \binom{n-k-1}{k-1}\right) x^k \\&=\sum_{k=0}^{n} \binom{n-k-1}{k} x^k +\sum_{k=0}^{n} \binom{n-k-1}{k-1} x^k\end{align}$$ $$\sum_{k=0}^{n} \binom{n-k-1}{k} x^k=\sum_{k=0}^{n-1} \binom{(n-1)-k}{k} x^k =F(n-1)$$ because when $k=n$, $\displaystyle \binom{(n-1)-k}{k}$ is $0$.

Similarly , $\displaystyle \sum_{k=0}^{n} \binom{n-k-1}{k-1} x^k=\sum_{k=-1}^{n-1} \binom{n-2-k}{k} x^{k+1} =xF(n-2)$ shifiting $k$ by $1$.

So , $F(n)=F(n-1)+xF(n-2)$

also you can get the closed form similar to fibonacci analysis.

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    $\begingroup$ thanks for the formatting @r9m $\endgroup$
    – VigneshM
    Jun 17, 2015 at 2:23

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