adding the super delta continued fractions, after G.-N. Han

src/doc/en/reference/references/index.rst

@@ -3269,6 +3269,9 @@ REFERENCES:
 .. [Haj2000] \M. Hajiaghayi, *Consecutive Ones Property*, 2000.
              http://www-math.mit.edu/~hajiagha/pp11.ps
 
+.. [Han2016] \G.-N. Han, *Hankel continued fraction and its applications*,
+             Adv. in Math., 303, 2016, pp. 295-321.
+
 .. [Han1960] Haim Hanani,
              *On quadruple systems*,
              pages 145--157, vol. 12,

src/sage/rings/power_series_ring_element.pyx

@@ -1384,6 +1384,86 @@ cdef class PowerSeries(AlgebraElement):
             serie = (u - 1 - A * t) / (B * t ** 2)
         return tuple(resu)
 
+    def super_delta_fraction(self, delta):
+        r"""
+        Return the super delta continued fraction of ``self``.
+
+        This is a continued fraction of the following shape:
+
+        .. MATH::
+
+            \cfrac{v_0 x^{k_0}} {U_1(x) -
+            \cfrac{v_1 x^{k_0 + k_1 + \delta}} {U_2(x) -
+            \cfrac{v_2 x^{k_0 + k_1 + k_2 + \delta}} {U_3(x) - \cdots} } }
+
+        where each `U_j(x) = 1 + u_j(x) x`.
+
+        INPUT:
+
+        - ``delta`` -- positive integer, usually 2
+
+        OUTPUT: list of `(v_j, k_j, U_{j+1}(x))_{j \geq 0}`
+
+        REFERENCES:
+
+        - [Han2016]_
+
+        EXAMPLES::
+
+            sage: deg = 30
+            sage: PS = PowerSeriesRing(QQ, 'q', default_prec=deg+1)
+            sage: q = PS.gen()
+            sage: F = prod([(1+q**k).add_bigoh(deg+1) for k in range(1,deg)])
+            sage: F.super_delta_fraction(2)
+            [(1, 0, -q + 1),
+             (1, 1, q + 1),
+             (-1, 2, -q^3 + q^2 - q + 1),
+             (1, 1, q^2 + q + 1),
+             (-1, 0, -q + 1),
+             (-1, 1, q^2 + q + 1),
+             (-1, 0, -q + 1),
+             (1, 1, 3*q^2 + 2*q + 1),
+             (-4, 0, -q + 1)]
+
+        A Jacobi continued fraction::
+
+            sage: t = PowerSeriesRing(QQ, 't').gen()
+            sage: s = sum(factorial(k) * t**k for k in range(12)).O(12)
+            sage: s.super_delta_fraction(2)
+            [(1, 0, -t + 1),
+            (1, 0, -3*t + 1),
+            (4, 0, -5*t + 1),
+            (9, 0, -7*t + 1),
+            (16, 0, -9*t + 1),
+            (25, 0, -11*t + 1)]
+        """
+        q = self.parent().gen()
+        Gi, Gj = self.parent().one(), self
+        deg = self.prec()
+
+        list_vkU = []
+        di = Gi.valuation()
+        ci = Gi[di]
+
+        while deg >= 0:
+            dj = Gj.valuation()
+            cj = Gj[dj]
+            k, v = dj - di, cj / ci
+            c = v * q**k
+            gi = Gi.add_bigoh(dj + delta)
+            gj = Gj.add_bigoh(k + dj + delta)
+            U = (c * gi / gj).truncate()
+            Gk = (U * Gj - Gi * c) >> (k + delta)
+            deg -= 2 * k + delta
+            if deg < 0:
+                break
+            list_vkU.append((v, k, U))
+            if deg == 0 or Gk.degree() == -1:
+                break
+            di, ci, Gi, Gj = dj, cj, Gj, Gk
+
+        return list_vkU
+
     def stieltjes_continued_fraction(self):
         r"""
         Return the Stieltjes continued fraction of ``self``.