Difference between revisions of "Fibonacci zeta in terms of a sum of binomial coefficients"
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
− | $$ | + | $$(\sqrt{5})^{-z} F(z)=-\displaystyle\sum_{j=0}^{\infty} (-1)^j {{-z} \choose j} \dfrac{1}{\phi^{z+2j}-(-1)^j} ,$$ |
− | where | + | where $F(z)$ denotes the [[Fibonacci zeta function]], $\displaystyle{{-z} \choose j}$ denotes a [[binomial coefficient]], and $\phi$ denotes the [[golden ratio]]. |
==Proof== | ==Proof== | ||
+ | By [[Binet's formula]], | ||
+ | $$F_n=\dfrac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}.$$ | ||
+ | Using the [[binomial series]], we see that | ||
+ | $$\dfrac{1}{F_k^z} = \left( \sqrt{5} \right)^z (\phi^k - (-\phi)^{-k})^{-z}=(\sqrt{5})^z \phi^{-zk} \left( 1 - \dfrac{(-\phi)^{-k}}{\phi^k} \right)^{-z}=(\sqrt{5})^z \phi^{-zk} \displaystyle\sum_{j=0}^{\infty} {{-z} \choose j} (-1)^j \left( \dfrac{(-\phi)^{-kj}}{\phi^{kj}} \right).$$ | ||
+ | Therefore compute | ||
+ | $$\begin{array}{ll} | ||
+ | (\sqrt{5})^{-z} F(z) &= (\sqrt{5})^{-z} \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{F_k^z} \\ | ||
+ | &= (\sqrt{5})^{-z} \displaystyle\sum_{k=1}^{\infty} \displaystyle\sum_{j=0}^{\infty} (\sqrt{5})^z \phi^{-zk} {{-z} \choose j} (-1)^j \left( \dfrac{(-\phi)^{-kj}}{\phi^{kj}} \right) \\ | ||
+ | &= \displaystyle\sum_{k=1}^{\infty} \displaystyle\sum_{j=0}^{\infty} \phi^{-zk} {{-z} \choose j} (-1)^j \left( \dfrac{(-\phi)^{-kj}}{\phi^{kj}} \right). | ||
+ | \end{array}$$ | ||
+ | Now we interchange the summations (justify this) and apply the [[geometric series]] (justify why it applies) to get | ||
+ | $$\begin{array}{ll} | ||
+ | (\sqrt{5})^{-z} F(z) &= \displaystyle\sum_{j=0}^{\infty} {{-z} \choose j} (-1)^j \displaystyle\sum_{k=1}^{\infty} \phi^{-zk} \left( \dfrac{(-\phi)^{-kj}}{\phi^{kj}} \right) \\ | ||
+ | &= \displaystyle\sum_{j=0}^{\infty} {{-z} \choose j} (-1)^j \displaystyle\sum_{k=1}^{\infty} \phi^{-zk} \left( \dfrac{(-\phi)^{-kj}}{\phi^{kj}} \right) \\ | ||
+ | &= \displaystyle\sum_{j=0}^{\infty} {{-z} \choose j} (-1)^j \displaystyle\sum_{k=1}^{\infty} \left((-1)^j \phi^{-z-2j} \right)^k \\ | ||
+ | &= \displaystyle\sum_{j=0}^{\infty} {{-z} \choose j} (-1)^j \dfrac{(-1)^j \phi^{-z-2j}}{1-(-1)^j \phi^{-z-2j}} \\ | ||
+ | &= \displaystyle\sum_{j=0}^{\infty} {{-z} \choose j} \dfrac{1}{\phi^{z+2j}-(-1)^j}, | ||
+ | \end{array}$$ | ||
+ | as was to be shown. | ||
==References== | ==References== | ||
+ | * {{PaperReference|The Fibonacci Zeta Function|1976|Maruti Ram Murty|prev=Binet's formula|next=findme}} | ||
[[Category:Theorem]] | [[Category:Theorem]] | ||
− | [[Category: | + | [[Category:Proven]] |
+ | [[Category:justify]] |
Latest revision as of 12:59, 11 August 2016
Theorem
The following formula holds: $$(\sqrt{5})^{-z} F(z)=-\displaystyle\sum_{j=0}^{\infty} (-1)^j {{-z} \choose j} \dfrac{1}{\phi^{z+2j}-(-1)^j} ,$$ where $F(z)$ denotes the Fibonacci zeta function, $\displaystyle{{-z} \choose j}$ denotes a binomial coefficient, and $\phi$ denotes the golden ratio.
Proof
By Binet's formula, $$F_n=\dfrac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}.$$ Using the binomial series, we see that $$\dfrac{1}{F_k^z} = \left( \sqrt{5} \right)^z (\phi^k - (-\phi)^{-k})^{-z}=(\sqrt{5})^z \phi^{-zk} \left( 1 - \dfrac{(-\phi)^{-k}}{\phi^k} \right)^{-z}=(\sqrt{5})^z \phi^{-zk} \displaystyle\sum_{j=0}^{\infty} {{-z} \choose j} (-1)^j \left( \dfrac{(-\phi)^{-kj}}{\phi^{kj}} \right).$$ Therefore compute $$\begin{array}{ll} (\sqrt{5})^{-z} F(z) &= (\sqrt{5})^{-z} \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{F_k^z} \\ &= (\sqrt{5})^{-z} \displaystyle\sum_{k=1}^{\infty} \displaystyle\sum_{j=0}^{\infty} (\sqrt{5})^z \phi^{-zk} {{-z} \choose j} (-1)^j \left( \dfrac{(-\phi)^{-kj}}{\phi^{kj}} \right) \\ &= \displaystyle\sum_{k=1}^{\infty} \displaystyle\sum_{j=0}^{\infty} \phi^{-zk} {{-z} \choose j} (-1)^j \left( \dfrac{(-\phi)^{-kj}}{\phi^{kj}} \right). \end{array}$$ Now we interchange the summations (justify this) and apply the geometric series (justify why it applies) to get $$\begin{array}{ll} (\sqrt{5})^{-z} F(z) &= \displaystyle\sum_{j=0}^{\infty} {{-z} \choose j} (-1)^j \displaystyle\sum_{k=1}^{\infty} \phi^{-zk} \left( \dfrac{(-\phi)^{-kj}}{\phi^{kj}} \right) \\ &= \displaystyle\sum_{j=0}^{\infty} {{-z} \choose j} (-1)^j \displaystyle\sum_{k=1}^{\infty} \phi^{-zk} \left( \dfrac{(-\phi)^{-kj}}{\phi^{kj}} \right) \\ &= \displaystyle\sum_{j=0}^{\infty} {{-z} \choose j} (-1)^j \displaystyle\sum_{k=1}^{\infty} \left((-1)^j \phi^{-z-2j} \right)^k \\ &= \displaystyle\sum_{j=0}^{\infty} {{-z} \choose j} (-1)^j \dfrac{(-1)^j \phi^{-z-2j}}{1-(-1)^j \phi^{-z-2j}} \\ &= \displaystyle\sum_{j=0}^{\infty} {{-z} \choose j} \dfrac{1}{\phi^{z+2j}-(-1)^j}, \end{array}$$ as was to be shown.