The MGO construction¶

This page explains what [x]_q is in plain language, and why the coefficients qreals returns are trustworthy. The mathematics is from one paper: Morier-Genoud and Ovsienko, "q-deformed rationals and q-continued fractions", Forum Math. Sigma 8 (2020), e13, Definition 1.1 and Proposition 1.1.

From integers to rationals to reals¶

Start with the q-integer everyone knows:

[n]_q = 1 + q + q^2 + ... + q^{n-1}

Set q = 1 and it collapses back to n. The MGO idea is to extend the map x -> [x]_q so that it makes sense first for rationals and then for all reals, while keeping the q = 1 specialisation honest.

For a rational the route is the continued fraction. Write x = p/s as a regular continued fraction [a_1, a_2, ..., a_m], normalise it to even length, and fold the MGO formula through it: odd positions (counting from one) carry [a]_q with q^a above, even positions carry [a]_{q^{-1}} with q^{-a} above. The result [p/s]_q is an honest rational function of q. For example,

[3/2]_q = (1 + q + q^2) / (1 + q)
[1/2]_q = q / (1 + q)

Both specialise to the ordinary value at q = 1.

The series for an irrational¶

For an irrational x the continued fraction does not terminate, so [x]_q is not a rational function but a power series in q with integer coefficients. That series is the object people study: its coefficients carry arithmetic information about x, and which patterns appear in them is an open research question. The golden ratio, sqrt(2), pi, and e all give different and structured coefficient sequences.

q_real_truncated(x, N) returns the first N coefficients of that series.

The stable-coefficient guarantee¶

The depth of continued fraction needed is not a tuning knob. MGO Proposition 1.1 says: stop the continued fraction at the first point where the partial quotients sum to at least N + 1, and exactly that-sum-minus-one of the resulting power series coefficients agree with the true [x]_q. They will not change if you ask for more.

q_real_truncated(x, N) accumulates partial quotients until the sum clears N, so the N coefficients it returns are stable by that proposition. Two consequences:

For a rational p/s the continued fraction terminates, so the truncated series is just the Taylor expansion of the exact q_rational(p, s). The test suite checks that the two paths agree coefficient for coefficient.
The worst case for depth is the golden ratio, whose continued fraction is all ones, so reaching N stable coefficients takes depth N + 1. Everything else is faster.

Exactness¶

Coefficients are exact throughout. The series kernel holds each coefficient as a Python int and inverts series with Newton's iteration over the integers, so nothing is lost to floating point. This matters when the coefficients are used as data: there is no measurement noise to model around, and the same engine that generates an example can check a prediction about it.

Arithmetic, and two cautions¶

qreals also computes the series sum and product of two q-reals, the Jouteur q-negation, and a radius-of-convergence estimate. Two cautions, both spelled out in the correctness notes and the docstrings:

q_add and q_mul are the sum and product of the two series [x]_q and [y]_q, not [x + y]_q or [x * y]_q. The MGO map x -> [x]_q is not a ring homomorphism, so these differ already at the constant term.
q_neg is the PGL_2(Z)-action negation of Jouteur, an involution, not the coefficient-wise negation of [x]_q and not the MGO series of the real number -x. It is the tool behind Ovsienko's Example 6.4: for which real x is [x]_q + [-x]_q a finite Laurent polynomial?

Why a second engine¶

Every arithmetic result is cross-checked against an independent algorithm, the bihomographic q-Gosper engine (q_gosper, gosper_coeffs). It reaches the same quantity by a state machine over rational functions in q, never forming the two series separately, so agreement between the two paths is a real check rather than a re-run of one code path. The correctness notes map every public function to the theorem it computes and the check that confirms it.