Solve Recurrence Relation with Lambert W-function

[spiceman]

Dear forum people,

for a nonlinear software I am writing I am having a hard time to transform a
recurrence relation to an explicit function. Maybe someone can help me along the right lines...

The recurrence relation is of the form (an exponential type function)

y[n+1] = y[n] + k * log(1 + (y[n]/k))

Now I know that simple linear recurrence functions like y[n+1] = y[n] + y[n] transform to explicit functions like y = 2^(n-1), but the tricky part is the log(1+...).
I have a feeling that the Lambert W-function could be a solution to the series
log(1+log(1+log(...))), but I am stuck at the moment. Even Mathematica fails on this problem with its function 'Rsolve'. Does anybody have an idea?
BTW, n is positive integer > 0 and y[n] > 0.

Greets and thanks in advance!

 

[atomthick]

What is the range for the constant k?

 

[atomthick]

There are some special cases:

- if k = 0 you will have the limit of k * log (1+y(n)/k) = 0 therefore y(n+1) = y(n)
- if k -> infinity you will have k * log(1+y(n)/k)) = log((1+y(n)/k)^k) = log(e^y(n)) = y(n) and therefore y(n+1) = 2*y(n)

 

[Stephen Tashi]

Let
1 + y/k = s{i]

so y = k * s - 1

Can we solve

k*s[i+1] - 1 = k*s - 1 + k log(s)

s[i+1] - s = log( s )

 

[Stephen Tashi]

$$s[n] =\sum_{i=2}^{i=n} \log(s[i-1])$$

One thought is $$s[n] = \log((n-1)!)$$

 

[spiceman]

I forgot to mention that k is a constant > 0.
Usually something like 0.5.

Greets.

 

[spiceman]

Using y[0] = 1 and k = 1 the recurrence function will give
the following first 50 values:

1.000000 1.693147 2.683858 3.987818 5.594817 7.481101 9.618941
11.981580 14.545112 17.288858 20.195150 23.248922 26.437294 29.749198
33.175062 36.706558 40.336392 44.058135 47.866089 51.755172 55.720834
59.758976 63.865890 68.038212 72.272873 76.567063 80.918206 85.323927
89.782034 94.290495 98.847425 103.451069 108.099787 112.792050 117.526423 122.301559 127.116192 131.969130 136.859247 141.785480 146.746823 151.742323 156.771076 161.832221 166.924941 172.048458 177.202030 182.384948 187.596535 192.836145

Does anyone see a trend?