# Holomorphic extension of the Collatz subsequence

UNDER CONSTRUCTION !!!

## Abstract

The Terras Sequence [1][2][3] is subsequence of the Collatz sequense [4]; this subsequence can be defined through the transfer function $$g$$ of the Collatz sequence. For integer values of $$n$$, the Collatx transfer function

$$g(n)=\left\{ \begin{array}{c} n/2 , \mathrm{ ~~if~~ } n/2 \in \mathbb N \\ (3n\!+\!1)/2 ~ \mathrm{~~over-vice} \end{array} \right.$$

The subsequence is defined with the transfer function $$h=g^2$$ , that means second iteration of $$g$$; for integer $$n$$,

$$h(n)=g(g(n))$$

The holomorphic extension of $$h$$ is suggested. The holomorphic extension $$F$$ of the sequence, generated with such a transfer function, is constructed,

$$h(F(z))=F(z\!+\!1)$$

## Extension of the transfer function

$$y=h(x)$$ and $$y=x$$ in the $$x,y$$ plane

The simple expression exactly fit values of $$g(g(n))$$ at integer values of $$n$$:
$$h(n)=\frac{1}{4} \left(4 n-(3 n\!+\!2) \sin \left(\frac{\pi n}{2}\right)-(n\!+\!1) \cos \left(\frac{\pi n}{2}\right)-(2 n\!+\!1) \cos (\pi n)+2\right)$$

This representation allows the straightforward extension for the complex $$n$$.

Function $$h$$ has several real fixed points; in particular, 0,1,2. Fixed point 2 is used below for the construction of the superfunction

## Regular iteration

Complex map of function $$F$$

Use of the regular iteration for the construction of a superfunction for a holomorphic transfer function at its fixed point is pretty straightforward [5][6][7]. Choose the fixed point 2, $$h(2)=2$$. In order to construct the superfunction $$F$$ of the transfer function $$h$$ at this fixed point, seethe the solution of equation

$$h(F(z))-F(z\!+\!1)=0$$

in th following form:

$$F(z)=2+\varepsilon + a_2 \varepsilon^2 + a_3 \varepsilon^3 + a_4 \varepsilon^4 + . . .$$

where $$\varepsilon=\exp(kz)$$ , and $$k\in \mathbb R$$ is constant. Then,

$$F(z\!+\!1)=2+q\varepsilon + a_2 q^2\varepsilon^2 + a_3 q^3\varepsilon^3 + a_4 q^4\varepsilon^4 + . . .$$

where $$q=\exp(k)$$.

Sibstitution the expressions for $$F(k)$$ and $$F(k\!+\!1)$$ into the equation above, and the expansion to the Taylor series with respect to $$\varepsilon$$ gives the equation for $$k$$ and the chain of equations for the coefficients $$a$$, determining

$$q=3/4+\pi$$~,~ id est, $$k=\log(3/4+\pi)\approx 1.3588184963564824177$$
$$a_2=\frac{12 \pi +17 \pi ^2}{2 \left(-3+8 \pi +16 \pi ^2\right)} \approx 0.57063727286450673698$$
$$a_3=-\frac{2 \pi ^2 \left(-369-1404 \pi -1171 \pi ^2+64 \pi ^3\right)}{3 (7+4 \pi ) \left(-3+8 \pi +16 \pi ^2\right)^2} \approx 0.14888910250614386537$$

and so on.

The approximation for $$F$$ above is food while $$-\Re(z)$$ is large; for other values of $$z$$, the representation

$$F(z)=h^n(F(z-n))$$

should be used at the integer $$n$$ large enough to get the required precision.

## Properties of $$F$$

Function $$F$$ has no singularities, it is entire function.

Function $$F$$ is periodic, its period $$T=\frac{2 \pi \rm i}{k}= \frac{2 \pi \rm i}{\ln(3/4+\pi)}\approx 4.6240063143291284184~ \mathrm i$$.

In the direction of negative real part of the argument, $$F$$ exponentially approached the fixed point 2 of the transfer function $$h$$.

At the real axis, the function grows monotonically, from 2, approaching the next fixed point of function $$h$$ (which is approximately 3.7).

As any other entire function, $$F$$ gets all the values (except, perhaps, one; for function $$F$$ it is 0), including all the integer values.

## Conclusion

In the similar way, the inverse of the Collatz sequence can be generalized. The holomorphic extensions can be use in the proof of the Collatz conjectures.

## References

1. Terras, R. "A Stopping Time Problem on the Positive Integers." Acta Arith. 30, 241-252, 1976.
2. Terras, R. "On the Existence of a Density." Acta Arith. 35, 101-102, 1979.
3. http://www.cs.uwaterloo.ca/journals/JIS/VOL11/Snapp/snapp.pdf Bart Snapp and Matt Tracy. The Collatz Problem and Analogues. Journal of Integer Sequences, Vol. 11 (2008), Article 08.4.7.
4. http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
5. http://www.springerlink.com/content/u712vtp4122544x4 D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98.
6. http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.

http://en.wikipedia.org/wiki/Collatz_conjecture Collatz Conjecture Calculation Center