I am going to guess and I hope someone has a clearer idea.

The question is interesting from a (my) novice math perspective, the wording suggests it was moved from mathoverflow.se?

From a basic linguistic perspective, there is little to no difference. All you need is a slowly decreasing count. Both forms describe standard distributions, a concept that's naturally observed in nature. The specific formula of any such distribution depends on an accurate model. It doesn't hold much explanatory power, if the model isn't empiricly grounded, but it's a heuristic--we might speak of so called fudge factors. For the specifics you should check out datascience.se, or whatever it's called where statistics are treated (compression of text is rather important in signal processing).

V = k * n ^ b

but inverted, i.e. taking the square root (b=1/2) instead of the square; also, it has a random factor k instead of pi (=3.14...). This can be pictured various ways, for example as light cone projected onto a surface, or a stream of words onto a lexicon: Where the radious of a light cone increases linearly with distance, it's area increases squarely; if this area illuminated a text, the number of new words would increase linearly with distance from the lamp.

V ~ n^b

The second formular seems more elaborate, but in principle the same. I too have no idea what the extra variables are. Removing the logarithm we have f(w) = C * (r(w)-b)^(-alpha). And transposed 1/C * (r(w)-b)^a = 1 / f(w). This is in principle the same polynomial form as V=K*n^b with several new parameters, iff it were that V = 1 / f(w), k = 1/C, n = (r(w) - b), beta = -alpha.

• b is likely a threshold under which the distribution is useless, because if r(w)<b, then the logarithm of the difference (r(w) - b) is undefined. Perhaps that's the Basic vocabulary.

• If C is a constant as usual notation practice has it, then writing log(C) would be constant as well. This might just be a courtesy to ease solving for (w). It's inversely proportional to k, but that shouldn't trouble us now. I'm keen to assume that it means Corpus, but that gives me troubles. [todo]

• That leaves alpha to be explained, which seems to be a variable nudge factor determined per corpus by a specific statistical procedure for error correction.

The question is interesting from a (my) novice math perspective.

From a basic linguistic perspective, there is little to no difference between either form. All you need is a slowly decreasing count. Both forms describe standard distributions, a concept that's naturally observed in nature. The specific formula of any such distribution depends on an accurate model. It doesn't hold much explanatory power, if the model isn't empiricly grounded, but it's a heuristic--we might speak of so called fudge factors. For the specifics you should check out datascience.se, or whatever it's called where statistics are treated (compression of text is also rather important in signal processing).

V = k * n ^ beta

but inverted, i.e. taking the square root (beta=1/2) instead of the square; also, it has a random factor k instead of pi (=3.14...). This can be pictured various ways, for example as light cone projected onto a surface, or a stream of words onto a lexicon: Where the radious of a light cone increases linearly with distance, it's area increases squarely; if this area illuminated a text, the number of new words would increase linearly with distance from the lamp.

V ~ n ^ beta

The second formular is essentially the same. I too have no idea what the extra variables are. Removing the logarithm and transposing, we see

1. f(w) = C * (r(w)-b)^(-alpha).

2. 1/C * (r(w)-b)^a = 1 / f(w).

This is in principle the same polynomial form as V=K*n^b in either case, with several new parameters. It's not apparent why to choose the transposed form, which works as well, iff it were that V = 1 / f(w), k = 1/C, n = (r(w) - b), beta = alpha.

• b is likely a threshold under which the distribution is useless, because if r(w)<b, then the logarithm of the difference (r(w) - b) is undefined. Perhaps that's the Basic vocabulary.

that's the major difference in any case.

Another difference would be to focus on the transposed form.

• If C is a constant as usual notation practice has it, then writing log(C) would be constant as well. This might just be a courtesy to ease solving for (w). It's inversely proportional to k, but that shouldn't trouble us now. I'm keen to assume that it means Corpus, but that gives me troubles. [todo]

• That leaves alpha to be explained, which seems to be a variable nudge factor determined per corpus by a specific statistical procedure for error correction.

From a basic linguistic perspective, there is little to no difference, all you need is a slowly decreasing slope. Both describe standard distributions, a concepts that's naturally observed in nature, from the distribution of raindrops to the dispersion of a laser beam. The specific choice depends on an accurate model. If it's instead just chosen to fit the data somehow, it doesn't hold much explanatory power, but it's a heuristic. For the specifics you should check out datascience.se, or whatever it's called where statistics are treated (compression of text is also rather important in signal processing).

The first one, V = k * n ^ b, is akin to the area of a circle, A = pi * r ^ 2, but inverted (taking the square root) and with a random factor, instead of pi, which can be pictured various ways, e.g. as a circle projected onto a wavy area or through a lense (doesn't really matter unless there's a specific need). If b is not exactly 0.5 the picture is a little different, but not really. The point is, this appears as the inverse square law, e.g. if a light cone hits a wall further away, the radius will increase linearly, but the power per square are will diminish proportionally with the inverse square of the distance. A^1/2 ~ r. The length of a text, n, increases likewise proportionally with the number of new words, n^0.5 ~ V. In other words, the text grows squarely with each new new word. That's also proportional to the circumference.

The second one seems more elaborate. I too have no idea what the extra variables are. Removing the logarithm we have f(w) = C * (r(w)-b)^(-alpha). And transposed 1/C * (r(w)-b)^a = 1 / f(w). This is in principle the same polynomial form as V=K*n^b with several new parameters, if it were that V = 1 / f(w), k = 1/C, n = (r(w) - b), beta = -alpha.

• b is likely a threshold under which the distribution is useless, because if r(w)<b, then the logarithm of the difference (r(w) - b) is undefined. Perhaps that's the Basic vocabulary.

• If C is a constant as usual, then writing log(C), which would be constant as well, might just be a courtesy to ease solving for (w). It's inversely proportional to k, but that shouldn't trouble us now. One way or another it will be normalizing the results. I'm keen to assume that it means Corpus, but that gives me troubles. [todo]

• That leaves alpha to be explained, which seems to be a variable nudge factor determined per corpus by a specific statistical procedure for error correction.

The last one is crucial. Raising to a negative power of alpha (=reciproke of the power of alpha) is not quite the same as taking the square root (power of 0.5). But it is similar in effect because the ranges of the exponents are also different, b < 1 < alpha. The very important difference is that the number of new words will tend to zero as the number of typed words tends to infinity--which, if I may say so, is a rather real possibility with twitter ;-) While the old formula would require ever new words to grow the text.

From a basic linguistic perspective, there is little to no difference. All you need is a slowly decreasing count. Both forms describe standard distributions, a concept that's naturally observed in nature. The specific formula of any such distribution depends on an accurate model. It doesn't hold much explanatory power, if the model isn't empiricly grounded, but it's a heuristic--we might speak of so called fudge factors. For the specifics you should check out datascience.se, or whatever it's called where statistics are treated (compression of text is rather important in signal processing).

The first formular is akin to the area of a circle

V = k * n ^ b

A = pi * r ^ 2

but inverted, i.e. taking the square root (b=1/2) instead of the square; also, it has a random factor k instead of pi (=3.14...). This can be pictured various ways, for example as light cone projected onto a surface, or a stream of words onto a lexicon: Where the radious of a light cone increases linearly with distance, it's area increases squarely; if this area illuminated a text, the number of new words would increase linearly with distance from the lamp.

This only explains the inversion of the exponantial function, but the fudge factor is another matter, depending on the model. While the factor pi relates the circumference of a circle to its radius, a different factor implies first of all a different shape, either of the light cone, or alternatively of the surface (left as exercise to the reader to the reader), but it still grows linearly with distance. So it doesn't even make a difference in my simplistical model. 

In other words, if counting text length in number of words n, so the text grows linearly with each word, it should grow squarely if counting each newly introduced word.

V ~ n^b

Or vice-versa as the formular has it: The number of new words grows proportional to the square root of the number of total words.

The second formular seems more elaborate, but in principle the same. I too have no idea what the extra variables are. Removing the logarithm we have f(w) = C * (r(w)-b)^(-alpha). And transposed 1/C * (r(w)-b)^a = 1 / f(w). This is in principle the same polynomial form as V=K*n^b with several new parameters, iff it were that V = 1 / f(w), k = 1/C, n = (r(w) - b), beta = -alpha.

• b is likely a threshold under which the distribution is useless, because if r(w)<b, then the logarithm of the difference (r(w) - b) is undefined. Perhaps that's the Basic vocabulary.

• If C is a constant as usual notation practice has it, then writing log(C) would be constant as well. This might just be a courtesy to ease solving for (w). It's inversely proportional to k, but that shouldn't trouble us now. I'm keen to assume that it means Corpus, but that gives me troubles. [todo]

• That leaves alpha to be explained, which seems to be a variable nudge factor determined per corpus by a specific statistical procedure for error correction.

The last one is crucial. Raising to a negative power of alpha (=reciproke of the power of alpha) is not quite the same as taking the square root (power of 0.5). But it is similar in effect because the ranges of the exponents are also different than in the first formular; here we have b < 1 < alpha. The very important difference is that the number of new words will tend to zero as the number of typed words tends to infinity. In contrast, the old formula would require ever new words to grow the text.