Difference between revisions of "Fibonacci zeta function"

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The Fibonacci zeta function is defined by
 
The Fibonacci zeta function is defined by
$$F(s)=\displaystyle\sum_{k=1}^{\infty} f_n^{-s},$$
+
$$F(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{F_n^z},$$
where $f_n$ denotes the $n$th term in the [[Fibonacci sequence]].
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where $F_n$ denotes the $n$th [[Fibonacci numbers|Fibonacci number]].
  
 
=Properties=
 
=Properties=
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[[Fibonacci zeta in terms of a sum of binomial coefficients]]<br />
<strong>Theorem:</strong> The number $F(1)$ is an [[irrational number]].
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[[Fibonacci zeta at 1 is irrational]]<br />
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[[Fibonacci zeta is transcendental at positive even integers]]<br />
<strong>Proof:</strong>  █
 
</div>
 
</div>
 
  
<div class="toccolours mw-collapsible mw-collapsed">
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=See Also=
<strong>Theorem:</strong> The number $F(2k)$ is a [[transcendental number]] for all $k=1,2,3,\ldots$.
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[[Fibonacci numbers]] <br />
<div class="mw-collapsible-content">
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[[Reciprocal Fibonacci constant]]<br />
<strong>Proof:</strong>  █
 
</div>
 
</div>
 
  
 
=References=
 
=References=
[http://www.mast.queensu.ca/~murty/fibon-tifr.pdf The Fibonacci zeta function by M. Ram Murty]<br />
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* {{PaperReference|The Fibonacci Zeta Function|1976|Maruti Ram Murty|prev=Fibonacci numbers|next=Binet's formula}}
 
[http://cc.oulu.fi/~tma/TAPANI20.pdf]<br />
 
[http://cc.oulu.fi/~tma/TAPANI20.pdf]<br />
 +
[http://www.fq.math.ca/Scanned/39-5/navas.pdf]<br />
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[[Category:SpecialFunction]]

Latest revision as of 00:25, 24 May 2017

The Fibonacci zeta function is defined by $$F(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{F_n^z},$$ where $F_n$ denotes the $n$th Fibonacci number.

Properties

Fibonacci zeta in terms of a sum of binomial coefficients
Fibonacci zeta at 1 is irrational
Fibonacci zeta is transcendental at positive even integers

See Also

Fibonacci numbers
Reciprocal Fibonacci constant

References

[1]
[2]