Lazy Series#

Coefficients of lazy series are computed on demand. They have infinite precision, although equality can only be decided in special cases.

AUTHORS:

Kwankyu Lee (2019-02-24): initial version
Tejasvi Chebrolu, Martin Rubey, Travis Scrimshaw (2021-08): refactored and expanded functionality

EXAMPLES:

Laurent series over the integer ring are particularly useful as generating functions for sequences arising in combinatorics.

sage: L.<z> = LazyLaurentSeriesRing(ZZ)

The generating function of the Fibonacci sequence is:

sage: f = 1 / (1 - z - z^2)
sage: f
1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)

In principle, we can now compute any coefficient of \(f\):

sage: f.coefficient(100)
573147844013817084101

Which coefficients are actually computed depends on the type of implementation. For the sparse implementation, only the coefficients which are needed are computed.

sage: s = L(lambda n: n, valuation=0); s
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: s.coefficient(10)
10
sage: s._coeff_stream._cache
{0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5, 6: 6, 10: 10}

Using the dense implementation, all coefficients up to the required coefficient are computed.

sage: L.<x> = LazyLaurentSeriesRing(ZZ, sparse=False)
sage: s = L(lambda n: n, valuation=0); s
x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + O(x^7)
sage: s.coefficient(10)
10
sage: s._coeff_stream._cache
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

We can do arithmetic with lazy power series:

sage: f
1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)
sage: f^-1
1 - z - z^2 + O(z^7)
sage: f + f^-1
2 + z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)
sage: g = (f + f^-1)*(f - f^-1); g
4*z + 6*z^2 + 8*z^3 + 19*z^4 + 38*z^5 + 71*z^6 + O(z^7)

We call lazy power series whose coefficients are known to be eventually constant ‘exact’. In some cases, computations with such series are much faster. Moreover, these are the series where equality can be decided. For example:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: f = 1 + 2*z^2 / (1 - z)
sage: f - 2 / (1 - z) + 1 + 2*z
0

However, multivariate Taylor series are actually represented as streams of multivariate polynomials. Therefore, the only exact series in this case are polynomials:

sage: L.<x,y> = LazyPowerSeriesRing(ZZ)
sage: 1 / (1-x)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + O(x,y)^7

A similar statement is true for lazy symmetric functions:

sage: h = SymmetricFunctions(QQ).h()
sage: L = LazySymmetricFunctions(h)
sage: 1 / (1-L(h[1]))
h[] + h[1] + (h[1,1]) + (h[1,1,1]) + (h[1,1,1,1]) + (h[1,1,1,1,1]) + (h[1,1,1,1,1,1]) + O^7

We can change the base ring:

sage: h = g.change_ring(QQ)
sage: h.parent()
Lazy Laurent Series Ring in z over Rational Field
sage: h
4*z + 6*z^2 + 8*z^3 + 19*z^4 + 38*z^5 + 71*z^6 + O(z^7)
sage: hinv = h^-1; hinv
1/4*z^-1 - 3/8 + 1/16*z - 17/32*z^2 + 5/64*z^3 - 29/128*z^4 + 165/256*z^5 + O(z^6)
sage: hinv.valuation()
-1

class sage.rings.lazy_series.LazyCauchyProductSeries(parent, coeff_stream)#

Bases: LazyModuleElement

A class for series where multiplication is the Cauchy product.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = 1 / (1 - z)
sage: f
1 + z + z^2 + O(z^3)
sage: f * (1 - z)
1

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=True)
sage: f = 1 / (1 - z)
sage: f
1 + z + z^2 + O(z^3)

exp()#

Return the exponential series of self.

We use the identity

\[\exp(s) = 1 + \int s' \exp(s).\]

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: exp(z)
1 + z + 1/2*z^2 + 1/6*z^3 + 1/24*z^4 + 1/120*z^5 + 1/720*z^6 + O(z^7)
sage: exp(z + z^2)
1 + z + 3/2*z^2 + 7/6*z^3 + 25/24*z^4 + 27/40*z^5 + 331/720*z^6 + O(z^7)
sage: exp(0)
1
sage: exp(1 + z)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: exp(x+y)[4].factor()
(1/24) * (x + y)^4
sage: exp(x/(1-y)).polynomial(3)
1/6*x^3 + x^2*y + x*y^2 + 1/2*x^2 + x*y + x + 1

log()#

Return the series for the natural logarithm of self.

We use the identity

\[\log(s) = \int s' / s.\]

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: log(1/(1-z))
z + 1/2*z^2 + 1/3*z^3 + 1/4*z^4 + 1/5*z^5 + 1/6*z^6 + 1/7*z^7 + O(z^8)

sage: L.<x, y> = LazyPowerSeriesRing(QQ)
sage: log((1 + x/(1-y))).polynomial(3)
1/3*x^3 - x^2*y + x*y^2 - 1/2*x^2 + x*y + x

valuation()#

Return the valuation of self.

This method determines the valuation of the series by looking for a nonzero coefficient. Hence if the series happens to be zero, then it may run forever.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: s = 1/(1 - z) - 1/(1 - 2*z)
sage: s.valuation()
1
sage: t = z - z
sage: t.valuation()
+Infinity
sage: M = L(lambda n: n^2, 0)
sage: M.valuation()
1
sage: (M - M).valuation()
+Infinity

class sage.rings.lazy_series.LazyCompletionGradedAlgebraElement(parent, coeff_stream)#

Bases: LazyCauchyProductSeries

An element of a completion of a graded algebra that is computed lazily.

class sage.rings.lazy_series.LazyDirichletSeries(parent, coeff_stream)#

Bases: LazyModuleElement

A Dirichlet series where the coefficients are computed lazily.

EXAMPLES:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: f = L(constant=1)^2; f
1 + 2/2^z + 2/3^z + 3/4^z + 2/5^z + 4/6^z + 2/7^z + O(1/(8^z))
sage: f.coefficient(100) == number_of_divisors(100)
True

Lazy Dirichlet series is picklable:

sage: g = loads(dumps(f))
sage: g
1 + 2/2^z + 2/3^z + 3/4^z + 2/5^z + 4/6^z + 2/7^z + O(1/(8^z))
sage: g == f
True

is_unit()#

Return whether this element is a unit in the ring.

EXAMPLES:

sage: D = LazyDirichletSeriesRing(ZZ, "s")
sage: D([0, 2]).is_unit()
False

sage: D([-1, 2]).is_unit()
True

sage: D([3, 2]).is_unit()
False

sage: D = LazyDirichletSeriesRing(QQ, "s")
sage: D([3, 2]).is_unit()
True

valuation()#

Return the valuation of self.

This method determines the valuation of the series by looking for a nonzero coefficient. Hence if the series happens to be zero, then it may run forever.

EXAMPLES:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: mu = L(moebius); mu.valuation()
0
sage: (mu - mu).valuation()
+Infinity
sage: g = L(constant=1, valuation=2)
sage: g.valuation()
log(2)
sage: (g*g).valuation()
2*log(2)

class sage.rings.lazy_series.LazyLaurentSeries(parent, coeff_stream)#

Bases: LazyCauchyProductSeries

A Laurent series where the coefficients are computed lazily.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)

We can build a series from a function and specify if the series eventually takes a constant value:

sage: f = L(lambda i: i, valuation=-3, constant=-1, degree=3)
sage: f
-3*z^-3 - 2*z^-2 - z^-1 + z + 2*z^2 - z^3 - z^4 - z^5 + O(z^6)
sage: f[-2]
-2
sage: f[10]
-1
sage: f[-5]
0

sage: f = L(lambda i: i, valuation=-3)
sage: f
-3*z^-3 - 2*z^-2 - z^-1 + z + 2*z^2 + 3*z^3 + O(z^4)
sage: f[20]
20

Anything that converts into a polynomial can be input, where we can also specify the valuation or if the series eventually takes a constant value:

sage: L([-5,2,0,5])
-5 + 2*z + 5*z^3
sage: L([-5,2,0,5], constant=6)
-5 + 2*z + 5*z^3 + 6*z^4 + 6*z^5 + 6*z^6 + O(z^7)
sage: L([-5,2,0,5], degree=6, constant=6)
-5 + 2*z + 5*z^3 + 6*z^6 + 6*z^7 + 6*z^8 + O(z^9)
sage: L([-5,2,0,5], valuation=-2, degree=3, constant=6)
-5*z^-2 + 2*z^-1 + 5*z + 6*z^3 + 6*z^4 + 6*z^5 + O(z^6)
sage: L([-5,2,0,5], valuation=5)
-5*z^5 + 2*z^6 + 5*z^8
sage: L({-2:9, 3:4}, constant=2, degree=5)
9*z^-2 + 4*z^3 + 2*z^5 + 2*z^6 + 2*z^7 + O(z^8)

We can also perform arithmetic:

sage: f = 1 / (1 - z - z^2)
sage: f
1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)
sage: f.coefficient(100)
573147844013817084101
sage: f = (z^-2 - 1 + 2*z) / (z^-1 - z + 3*z^2)
sage: f
z^-1 - z^2 - z^4 + 3*z^5 + O(z^6)

However, we may not always be able to know when a result is exactly a polynomial:

sage: f * (z^-1 - z + 3*z^2)
z^-2 - 1 + 2*z + O(z^5)

approximate_series(prec, name=None)#

Return the Laurent series with absolute precision prec approximated from this series.

INPUT:

prec – an integer
name – name of the variable; if it is None, the name of the variable of the series is used

OUTPUT: a Laurent series with absolute precision prec

EXAMPLES:

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: z = L.gen()
sage: f = (z - 2*z^3)^5/(1 - 2*z)
sage: f
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + 32*z^10 - 16*z^11 + O(z^12)
sage: g = f.approximate_series(10)
sage: g
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + O(z^10)
sage: g.parent()
Power Series Ring in z over Integer Ring
sage: h = (f^-1).approximate_series(3)
sage: h
z^-5 - 2*z^-4 + 10*z^-3 - 20*z^-2 + 60*z^-1 - 120 + 280*z - 560*z^2 + O(z^3)
sage: h.parent()
Laurent Series Ring in z over Integer Ring

compose(g, check)#

Return the composition of self with g.

Given two Laurent series \(f\) and \(g\) over the same base ring, the composition \((f \circ g)(z) = f(g(z))\) is defined if and only if:

\(g = 0\) and \(val(f) >= 0\),
\(g\) is non-zero and \(f\) has only finitely many non-zero coefficients,
\(g\) is non-zero and \(val(g) > 0\).

INPUT:

g – other series

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: f = z^2 + 1 + z
sage: f(0)
1
sage: f(L(0))
1
sage: f(f)
3 + 3*z + 4*z^2 + 2*z^3 + z^4
sage: g = z^-3/(1-2*z); g
z^-3 + 2*z^-2 + 4*z^-1 + 8 + 16*z + 32*z^2 + 64*z^3 + O(z^4)
sage: f(g)
z^-6 + 4*z^-5 + 12*z^-4 + 33*z^-3 + 82*z^-2 + 196*z^-1 + 457 + O(z)
sage: g^2 + 1 + g
z^-6 + 4*z^-5 + 12*z^-4 + 33*z^-3 + 82*z^-2 + 196*z^-1 + 457 + O(z)
sage: f(int(2))
7

sage: f = z^-2 + z + 4*z^3
sage: f(f)
4*z^-6 + 12*z^-3 + z^-2 + 48*z^-1 + 12 + O(z)
sage: f^-2 + f + 4*f^3
4*z^-6 + 12*z^-3 + z^-2 + 48*z^-1 + 12 + O(z)
sage: f(g)
4*z^-9 + 24*z^-8 + 96*z^-7 + 320*z^-6 + 960*z^-5 + 2688*z^-4 + 7169*z^-3 + O(z^-2)
sage: g^-2 + g + 4*g^3
4*z^-9 + 24*z^-8 + 96*z^-7 + 320*z^-6 + 960*z^-5 + 2688*z^-4 + 7169*z^-3 + O(z^-2)

sage: f = z^-3 + z^-2 + 1 / (1 + z^2); f
z^-3 + z^-2 + 1 - z^2 + O(z^4)
sage: g = z^3 / (1 + z - z^3); g
z^3 - z^4 + z^5 - z^7 + 2*z^8 - 2*z^9 + O(z^10)
sage: f(g)
z^-9 + 3*z^-8 + 3*z^-7 - z^-6 - 4*z^-5 - 2*z^-4 + z^-3 + O(z^-2)
sage: g^-3 + g^-2 + 1 / (1 + g^2)
z^-9 + 3*z^-8 + 3*z^-7 - z^-6 - 4*z^-5 - 2*z^-4 + z^-3 + O(z^-2)

sage: f = z^-3
sage: g = z^-2 + z^-1
sage: g^(-3)
z^6 - 3*z^7 + 6*z^8 - 10*z^9 + 15*z^10 - 21*z^11 + 28*z^12 + O(z^13)
sage: f(g)
z^6 - 3*z^7 + 6*z^8 - 10*z^9 + 15*z^10 - 21*z^11 + 28*z^12 + O(z^13)

sage: f = z^2 + z^3
sage: g = z^-3 + z^-2
sage: f^-3 + f^-2
z^-6 - 3*z^-5 + 7*z^-4 - 12*z^-3 + 18*z^-2 - 25*z^-1 + 33 + O(z)
sage: g(f)
z^-6 - 3*z^-5 + 7*z^-4 - 12*z^-3 + 18*z^-2 - 25*z^-1 + 33 + O(z)
sage: g^2 + g^3
z^-9 + 3*z^-8 + 3*z^-7 + 2*z^-6 + 2*z^-5 + z^-4
sage: f(g)
z^-9 + 3*z^-8 + 3*z^-7 + 2*z^-6 + 2*z^-5 + z^-4

sage: f = L(lambda n: n, valuation=0); f
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: f(z^2)
z^2 + 2*z^4 + 3*z^6 + O(z^7)

sage: f = L(lambda n: n, valuation=-2); f
-2*z^-2 - z^-1 + z + 2*z^2 + 3*z^3 + 4*z^4 + O(z^5)
sage: f3 = f(z^3); f3
-2*z^-6 - z^-3 + O(z)
sage: [f3[i] for i in range(-6,13)]
[-2, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4]

We compose a Laurent polynomial with a generic element:

sage: R.<x> = QQ[]
sage: f = z^2 + 1 + z^-1
sage: g = x^2 + x + 3
sage: f(g)
(x^6 + 3*x^5 + 12*x^4 + 19*x^3 + 37*x^2 + 28*x + 31)/(x^2 + x + 3)
sage: f(g) == g^2 + 1 + g^-1
True

We compose with another lazy Laurent series:

sage: LS.<y> = LazyLaurentSeriesRing(QQ)
sage: f = z^2 + 1 + z^-1
sage: fy = f(y); fy
y^-1 + 1 + y^2
sage: fy.parent() is LS
True
sage: g = y - y
sage: f(g)
Traceback (most recent call last):
...
ZeroDivisionError: the valuation of the series must be nonnegative

sage: g = 1 - y
sage: f(g)
3 - y + 2*y^2 + y^3 + y^4 + y^5 + O(y^6)
sage: g^2 + 1 + g^-1
3 - y + 2*y^2 + y^3 + y^4 + y^5 + O(y^6)

sage: f = L(lambda n: n, valuation=0); f
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: f(0)
0
sage: f(y)
y + 2*y^2 + 3*y^3 + 4*y^4 + 5*y^5 + 6*y^6 + O(y^7)
sage: fp = f(y - y)
sage: fp == 0
True
sage: fp.parent() is LS
True

sage: f = z^2 + 3 + z
sage: f(y - y)
3

With both of them sparse:

sage: L.<z> = LazyLaurentSeriesRing(QQ, sparse=True)
sage: LS.<y> = LazyLaurentSeriesRing(QQ, sparse=True)
sage: f = L(lambda n: 1, valuation=0); f
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
sage: f(y^2)
1 + y^2 + y^4 + y^6 + O(y^7)

sage: fp = f - 1 + z^-2; fp
z^-2 + z + z^2 + z^3 + z^4 + O(z^5)
sage: fpy = fp(y^2); fpy
y^-4 + y^2 + O(y^3)
sage: fpy.parent() is LS
True
sage: [fpy[i] for i in range(-4,11)]
[1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]

sage: g = LS(valuation=2, constant=1); g
y^2 + y^3 + y^4 + O(y^5)
sage: fg = f(g); fg
1 + y^2 + y^3 + 2*y^4 + 3*y^5 + 5*y^6 + O(y^7)
sage: 1 + g + g^2 + g^3 + g^4 + g^5 + g^6
1 + y^2 + y^3 + 2*y^4 + 3*y^5 + 5*y^6 + O(y^7)

sage: h = LS(lambda n: 1 if n % 2 else 0, valuation=2); h
y^3 + y^5 + y^7 + O(y^9)
sage: fgh = fg(h); fgh
1 + y^6 + O(y^7)
sage: [fgh[i] for i in range(0, 15)]
[1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 6, 13]
sage: t = 1 + h^2 + h^3 + 2*h^4 + 3*h^5 + 5*h^6
sage: [t[i] for i in range(0, 15)]
[1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 6, 13]

We look at mixing the sparse and the dense:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: f = L(lambda n: 1, valuation=0); f
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
sage: g = LS(lambda n: 1, valuation=1); g
y + y^2 + y^3 + y^4 + y^5 + y^6 + y^7 + O(y^8)
sage: f(g)
1 + y + 2*y^2 + 4*y^3 + 8*y^4 + 16*y^5 + 32*y^6 + O(y^7)

sage: f = z^-2 + 1 + z
sage: g = 1/(y*(1-y)); g
y^-1 + 1 + y + O(y^2)
sage: f(g)
y^-1 + 2 + y + 2*y^2 - y^3 + 2*y^4 + y^5 + y^6 + y^7 + O(y^8)
sage: g^-2 + 1 + g == f(g)
True

sage: f = z^-3 + z^-2 + 1
sage: g = 1/(y^2*(1-y)); g
y^-2 + y^-1 + 1 + O(y)
sage: f(g)
1 + y^4 - 2*y^5 + 2*y^6 - 3*y^7 + 3*y^8 - y^9
sage: g^-3 + g^-2 + 1 == f(g)
True
sage: z(y)
y

We look at cases where the composition does not exist. \(g = 0\) and \(val(f) < 0\):

sage: g = L(0)
sage: f = z^-1 + z^-2
sage: f.valuation() < 0
True
sage: f(g)
Traceback (most recent call last):
...
ZeroDivisionError: the valuation of the series must be nonnegative

\(g \neq 0\) and \(val(g) \leq 0\) and \(f\) has infinitely many non-zero coefficients:

sage: g = z^-1 + z^-2
sage: g.valuation() <= 0
True
sage: f = L(lambda n: n, valuation=0)
sage: f(g)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

sage: f = L(lambda n: n, valuation=1)
sage: f(1 + z)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

We compose the exponential with a Dirichlet series:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: e = L(lambda n: 1/factorial(n), 0)
sage: D = LazyDirichletSeriesRing(QQ, "s")
sage: g = D(constant=1)-1; g
1/(2^s) + 1/(3^s) + 1/(4^s) + O(1/(5^s))

sage: e(g)[0:10]
[0, 1, 1, 1, 3/2, 1, 2, 1, 13/6, 3/2]

sage: sum(g^k/factorial(k) for k in range(10))[0:10]
[0, 1, 1, 1, 3/2, 1, 2, 1, 13/6, 3/2]

sage: g = D([0,1,0,1,1,2]); g
1/(2^s) + 1/(4^s) + 1/(5^s) + 2/6^s
sage: e(g)[0:10]
[0, 1, 1, 0, 3/2, 1, 2, 0, 7/6, 0]
sage: sum(g^k/factorial(k) for k in range(10))[0:10]
[0, 1, 1, 0, 3/2, 1, 2, 0, 7/6, 0]

sage: e(D([1,0,1]))
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

sage: e5 = L(e, degree=5); e5
1 + z + 1/2*z^2 + 1/6*z^3 + 1/24*z^4
sage: e5(g)
1 + 1/(2^s) + 3/2/4^s + 1/(5^s) + 2/6^s + O(1/(8^s))
sage: sum(e5[k] * g^k for k in range(5))
1 + 1/(2^s) + 3/2/4^s + 1/(5^s) + 2/6^s + O(1/(8^s))

The output parent is always the common parent between the base ring of \(f\) and the parent of \(g\) or extended to the corresponding lazy series:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: R.<x> = ZZ[]
sage: parent(z(x))
Univariate Polynomial Ring in x over Rational Field
sage: parent(z(R.zero()))
Univariate Polynomial Ring in x over Rational Field
sage: parent(z(0))
Rational Field
sage: f = 1 / (1 - z)
sage: f(x)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + O(x^7)
sage: three = L(3)(x^2); three
3
sage: parent(three)
Univariate Polynomial Ring in x over Rational Field

Consistency check when \(g\) is an uninitialized series between a polynomial \(f\) as both a polynomial and a lazy series:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: f = 1 + z
sage: g = L.undefined(valuation=0)
sage: f(g) == f.polynomial()(g)
True

compositional_inverse()#

Return the compositional inverse of self.

Given a Laurent series \(f\), the compositional inverse is a Laurent series \(g\) over the same base ring, such that \((f \circ g)(z) = f(g(z)) = z\).

The compositional inverse exists if and only if:

\(val(f) = 1\), or
\(f = a + b z\) with \(a, b \neq 0\), or
\(f = a/z\) with \(a \neq 0\)

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: (2*z).revert()
1/2*z
sage: (2/z).revert()
2*z^-1
sage: (z-z^2).revert()
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

sage: s = L(degree=1, constant=-1)
sage: s.revert()
-z - z^2 - z^3 + O(z^4)

sage: s = L(degree=1, constant=1)
sage: s.revert()
z - z^2 + z^3 - z^4 + z^5 - z^6 + z^7 + O(z^8)

derivative(*args)#

Return the derivative of the Laurent series.

Multiple variables and iteration counts may be supplied; see the documentation of sage.calculus.functional.derivative() function for details.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: z.derivative()
1
sage: (1+z+z^2).derivative(3)
0
sage: (1/z).derivative()
-z^-2
sage: (1/(1-z)).derivative(z)
1 + 2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + O(z^7)

is_unit()#

Return whether this element is a unit in the ring.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: (2*z).is_unit()
False

sage: (1 + 2*z).is_unit()
True

sage: (1 + 2*z^-1).is_unit()
False

sage: (z^3 + 4 - z^-2).is_unit()
True

polynomial(degree=None, name=None)#

Return self as a Laurent polynomial if self is actually so.

INPUT:

degree – None or an integer
name – name of the variable; if it is None, the name of the variable of the series is used

OUTPUT:

A Laurent polynomial if the valuation of the series is negative or a polynomial otherwise.

If degree is not None, the terms of the series of degree greater than degree are first truncated. If degree is None and the series is not a polynomial or a Laurent polynomial, a ValueError is raised.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = L([1,0,0,2,0,0,0,3], valuation=5); f
z^5 + 2*z^8 + 3*z^12
sage: f.polynomial()
3*z^12 + 2*z^8 + z^5

revert()#

Return the compositional inverse of self.

Given a Laurent series \(f\), the compositional inverse is a Laurent series \(g\) over the same base ring, such that \((f \circ g)(z) = f(g(z)) = z\).

The compositional inverse exists if and only if:

\(val(f) = 1\), or
\(f = a + b z\) with \(a, b \neq 0\), or
\(f = a/z\) with \(a \neq 0\)

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: (2*z).revert()
1/2*z
sage: (2/z).revert()
2*z^-1
sage: (z-z^2).revert()
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

sage: s = L(degree=1, constant=-1)
sage: s.revert()
-z - z^2 - z^3 + O(z^4)

sage: s = L(degree=1, constant=1)
sage: s.revert()
z - z^2 + z^3 - z^4 + z^5 - z^6 + z^7 + O(z^8)

class sage.rings.lazy_series.LazyModuleElement(parent, coeff_stream)#

Bases: Element

A lazy sequence with a module structure given by term-wise addition and scalar multiplication.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: M = L(lambda n: n, valuation=0)
sage: N = L(lambda n: 1, valuation=0)
sage: M[0:10]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: N[0:10]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Two sequences can be added:

sage: O = M + N
sage: O[0:10]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

Two sequences can be subtracted:

sage: P = M - N
sage: P[0:10]
[-1, 0, 1, 2, 3, 4, 5, 6, 7, 8]

A sequence can be multiplied by a scalar:

sage: Q = 2 * M
sage: Q[0:10]
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]

The negation of a sequence can also be found:

sage: R = -M
sage: R[0:10]
[0, -1, -2, -3, -4, -5, -6, -7, -8, -9]

arccos()#

Return the arccos of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(RR)
sage: arccos(z)
1.57079632679490 - 1.00000000000000*z + 0.000000000000000*z^2
 - 0.166666666666667*z^3 + 0.000000000000000*z^4
 - 0.0750000000000000*z^5 + O(1.00000000000000*z^7)

sage: L.<z> = LazyLaurentSeriesRing(SR)
sage: arccos(z/(1-z))
1/2*pi - z - z^2 - 7/6*z^3 - 3/2*z^4 - 83/40*z^5 - 73/24*z^6 + O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(SR)
sage: arccos(x/(1-y))
1/2*pi + (-x) + (-x*y) + ((-1/6)*x^3-x*y^2) + ((-1/2)*x^3*y-x*y^3)
 + ((-3/40)*x^5-x^3*y^2-x*y^4) + ((-3/8)*x^5*y+(-5/3)*x^3*y^3-x*y^5) + O(x,y)^7

arccot()#

Return the arctangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(RR)
sage: arccot(z)
1.57079632679490 - 1.00000000000000*z + 0.000000000000000*z^2
 + 0.333333333333333*z^3 + 0.000000000000000*z^4
 - 0.200000000000000*z^5 + O(1.00000000000000*z^7)

sage: L.<z> = LazyLaurentSeriesRing(SR)
sage: arccot(z/(1-z))
1/2*pi - z - z^2 - 2/3*z^3 + 4/5*z^5 + 4/3*z^6 + O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(SR)
sage: acot(x/(1-y))
1/2*pi + (-x) + (-x*y) + (1/3*x^3-x*y^2) + (x^3*y-x*y^3)
 + ((-1/5)*x^5+2*x^3*y^2-x*y^4) + (-x^5*y+10/3*x^3*y^3-x*y^5) + O(x,y)^7

arcsin()#

Return the arcsin of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: arcsin(z)
z + 1/6*z^3 + 3/40*z^5 + 5/112*z^7 + O(z^8)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: asin(x/(1-y))
x + x*y + (1/6*x^3+x*y^2) + (1/2*x^3*y+x*y^3)
 + (3/40*x^5+x^3*y^2+x*y^4) + (3/8*x^5*y+5/3*x^3*y^3+x*y^5)
 + (5/112*x^7+9/8*x^5*y^2+5/2*x^3*y^4+x*y^6) + O(x,y)^8

arcsinh()#

Return the inverse of the hyperbolic sine of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: asinh(z)
z - 1/6*z^3 + 3/40*z^5 - 5/112*z^7 + O(z^8)

arcsinh is an alias:

sage: arcsinh(z)
z - 1/6*z^3 + 3/40*z^5 - 5/112*z^7 + O(z^8)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: asinh(x/(1-y))
x + x*y + (-1/6*x^3+x*y^2) + (-1/2*x^3*y+x*y^3) + (3/40*x^5-x^3*y^2+x*y^4)
 + (3/8*x^5*y-5/3*x^3*y^3+x*y^5) + (-5/112*x^7+9/8*x^5*y^2-5/2*x^3*y^4+x*y^6) + O(x,y)^8

arctan()#

Return the arctangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: arctan(z)
z - 1/3*z^3 + 1/5*z^5 - 1/7*z^7 + O(z^8)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: atan(x/(1-y))
x + x*y + (-1/3*x^3+x*y^2) + (-x^3*y+x*y^3) + (1/5*x^5-2*x^3*y^2+x*y^4)
 + (x^5*y-10/3*x^3*y^3+x*y^5) + (-1/7*x^7+3*x^5*y^2-5*x^3*y^4+x*y^6) + O(x,y)^8

arctanh()#

Return the inverse of the hyperbolic tangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: atanh(z)
z + 1/3*z^3 + 1/5*z^5 + 1/7*z^7 + O(z^8)

arctanh is an alias:

sage: arctanh(z)
z + 1/3*z^3 + 1/5*z^5 + 1/7*z^7 + O(z^8)

sage: L.<x, y> = LazyPowerSeriesRing(QQ)
sage: atanh(x/(1-y))
x + x*y + (1/3*x^3+x*y^2) + (x^3*y+x*y^3) + (1/5*x^5+2*x^3*y^2+x*y^4)
 + (x^5*y+10/3*x^3*y^3+x*y^5) + (1/7*x^7+3*x^5*y^2+5*x^3*y^4+x*y^6) + O(x,y)^8

change_ring(ring)#

Return self with coefficients converted to elements of ring.

INPUT:

ring – a ring

EXAMPLES:

Dense Implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=False)
sage: s = 2 + z
sage: t = s.change_ring(QQ)
sage: t^-1
1/2 - 1/4*z + 1/8*z^2 - 1/16*z^3 + 1/32*z^4 - 1/64*z^5 + 1/128*z^6 + O(z^7)
sage: M = L(lambda n: n, valuation=0); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: N = M.change_ring(QQ)
sage: N.parent()
Lazy Laurent Series Ring in z over Rational Field
sage: M.parent()
Lazy Laurent Series Ring in z over Integer Ring

Sparse Implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=True)
sage: M = L(lambda n: n, valuation=0); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: M.parent()
Lazy Laurent Series Ring in z over Integer Ring
sage: N = M.change_ring(QQ)
sage: N.parent()
Lazy Laurent Series Ring in z over Rational Field
sage: M^-1
z^-1 - 2 + z + O(z^6)

A Dirichlet series example:

sage: L = LazyDirichletSeriesRing(ZZ, 'z')
sage: s = L(constant=2)
sage: t = s.change_ring(QQ)
sage: t.parent()
Lazy Dirichlet Series Ring in z over Rational Field
sage: t^-1
1/2 - 1/2/2^z - 1/2/3^z - 1/2/5^z + 1/2/6^z - 1/2/7^z + O(1/(8^z))

A Taylor series example:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: s = 2 + z
sage: t = s.change_ring(QQ)
sage: t^-1
1/2 - 1/4*z + 1/8*z^2 - 1/16*z^3 + 1/32*z^4 - 1/64*z^5 + 1/128*z^6 + O(z^7)
sage: t.parent()
Lazy Taylor Series Ring in z over Rational Field

coefficient(n)#

Return the homogeneous degree n part of the series.

INPUT:

n – integer; the degree

For a series f, the slice f[start:stop] produces the following:

if start and stop are integers, return the list of terms with given degrees
if start is None, return the list of terms beginning with the valuation
if stop is None, return a lazy_list_generic instead.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = z / (1 - 2*z^3)
sage: [f[n] for n in range(20)]
[0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32, 0, 0, 64]
sage: f[0:20]
[0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32, 0, 0, 64]
sage: f[:20]
[1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32, 0, 0, 64]
sage: f[::3]
lazy list [1, 2, 4, ...]

sage: M = L(lambda n: n, valuation=0)
sage: [M[n] for n in range(20)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=True)
sage: M = L(lambda n: n, valuation=0)
sage: [M[n] for n in range(20)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

Similarly for multivariate series:

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: sin(x*y)[:11]
[x*y, 0, 0, 0, -1/6*x^3*y^3, 0, 0, 0, 1/120*x^5*y^5]
sage: sin(x*y)[2::4]
lazy list [x*y, -1/6*x^3*y^3, 1/120*x^5*y^5, ...]

Similarly for Dirichlet series:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: L(lambda n: n)[1:11]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

coefficients(n=None)#

Return the first \(n\) non-zero coefficients of self.

INPUT:

n – (optional) the number of non-zero coefficients to return

If the series has fewer than \(n\) non-zero coefficients, only these are returned.

If n is None, a lazy_list_generic with all non-zero coefficients is returned instead.

Warning

If there are fewer than \(n\) non-zero coefficients, but this cannot be detected, this method will not return.

EXAMPLES:

sage: L.<x> = LazyPowerSeriesRing(QQ)
sage: f = L([1,2,3])
sage: f.coefficients(5)
doctest:...: DeprecationWarning: the method coefficients now only returns the non-zero coefficients. Use __getitem__ instead.
See https://trac.sagemath.org/32367 for details.
[1, 2, 3]

sage: f = sin(x)
sage: f.coefficients(5)
[1, -1/6, 1/120, -1/5040, 1/362880]

sage: L.<x, y> = LazyPowerSeriesRing(QQ)
sage: f = sin(x^2+y^2)
sage: f.coefficients(5)
[1, 1, -1/6, -1/2, -1/2]

sage: f.coefficients()
lazy list [1, 1, -1/6, ...]

sage: L.<x> = LazyPowerSeriesRing(GF(2))
sage: f = L(lambda n: n)
sage: f.coefficients(5)
[1, 1, 1, 1, 1]

cos()#

Return the cosine of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: cos(z)
1 - 1/2*z^2 + 1/24*z^4 - 1/720*z^6 + O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: cos(x/(1-y)).polynomial(4)
1/24*x^4 - 3/2*x^2*y^2 - x^2*y - 1/2*x^2 + 1

cosh()#

Return the cosh of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: cosh(z)
1 + 1/2*z^2 + 1/24*z^4 + 1/720*z^6 + O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: cosh(x/(1-y))
1 + 1/2*x^2 + x^2*y + (1/24*x^4+3/2*x^2*y^2) + (1/6*x^4*y+2*x^2*y^3)
 + (1/720*x^6+5/12*x^4*y^2+5/2*x^2*y^4) + O(x,y)^7

cot()#

Return the cotangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: cot(z)
z^-1 - 1/3*z - 1/45*z^3 - 2/945*z^5 + O(z^6)

sage: L.<x> = LazyLaurentSeriesRing(QQ)
sage: cot(x/(1-x)).polynomial(4)
x^-1 - 1 - 1/3*x - 1/3*x^2 - 16/45*x^3 - 2/5*x^4

coth()#

Return the hyperbolic cotangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: coth(z)
z^-1 + 1/3*z - 1/45*z^3 + 2/945*z^5 + O(z^6)

sage: coth(z + z^2)
z^-1 - 1 + 4/3*z - 2/3*z^2 + 44/45*z^3 - 16/15*z^4 + 884/945*z^5 + O(z^6)

csc()#

Return the cosecant of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: csc(z)
z^-1 + 1/6*z + 7/360*z^3 + 31/15120*z^5 + O(z^6)

sage: L.<x> = LazyLaurentSeriesRing(QQ)
sage: csc(x/(1-x)).polynomial(4)
x^-1 - 1 + 1/6*x + 1/6*x^2 + 67/360*x^3 + 9/40*x^4

csch()#

Return the hyperbolic cosecant of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: csch(z)
z^-1 - 1/6*z + 7/360*z^3 - 31/15120*z^5 + O(z^6)

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: csch(z/(1-z))
z^-1 - 1 - 1/6*z - 1/6*z^2 - 53/360*z^3 - 13/120*z^4 - 787/15120*z^5 + O(z^6)

define(s)#

Define an equation by self = s.

INPUT:

s – a lazy series

EXAMPLES:

We begin by constructing the Catalan numbers:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: C = L.undefined()
sage: C.define(1 + z*C^2)
sage: C
1 + z + 2*z^2 + 5*z^3 + 14*z^4 + 42*z^5 + 132*z^6 + O(z^7)
sage: binomial(2000, 1000) / C[1000]
1001

The Catalan numbers but with a valuation 1:

sage: B = L.undefined(valuation=1)
sage: B.define(z + B^2)
sage: B
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

We can define multiple series that are linked:

sage: s = L.undefined()
sage: t = L.undefined()
sage: s.define(1 + z*t^3)
sage: t.define(1 + z*s^2)
sage: s[0:9]
[1, 1, 3, 9, 34, 132, 546, 2327, 10191]
sage: t[0:9]
[1, 1, 2, 7, 24, 95, 386, 1641, 7150]

A bigger example:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: A = L.undefined(valuation=5)
sage: B = L.undefined()
sage: C = L.undefined(valuation=2)
sage: A.define(z^5 + B^2)
sage: B.define(z^5 + C^2)
sage: C.define(z^2 + C^2 + A^2)
sage: A[0:15]
[0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 5, 4, 14, 10, 48]
sage: B[0:15]
[0, 0, 0, 0, 1, 1, 2, 0, 5, 0, 14, 0, 44, 0, 138]
sage: C[0:15]
[0, 0, 1, 0, 1, 0, 2, 0, 5, 0, 15, 0, 44, 2, 142]

Counting binary trees:

sage: L.<z> = LazyPowerSeriesRing(QQ)
sage: s = L.undefined(valuation=1)
sage: s.define(z + (s^2+s(z^2))/2)
sage: s[0:9]
[0, 1, 1, 1, 2, 3, 6, 11, 23]

The \(q\)-Catalan numbers:

sage: R.<q> = ZZ[]
sage: L.<z> = LazyLaurentSeriesRing(R)
sage: s = L.undefined(valuation=0)
sage: s.define(1+z*s*s(q*z))
sage: s
1 + z + (q + 1)*z^2 + (q^3 + q^2 + 2*q + 1)*z^3
 + (q^6 + q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 3*q + 1)*z^4
 + (q^10 + q^9 + 2*q^8 + 3*q^7 + 5*q^6 + 5*q^5 + 7*q^4 + 7*q^3 + 6*q^2 + 4*q + 1)*z^5
 + (q^15 + q^14 + 2*q^13 + 3*q^12 + 5*q^11 + 7*q^10 + 9*q^9 + 11*q^8
    + 14*q^7 + 16*q^6 + 16*q^5 + 17*q^4 + 14*q^3 + 10*q^2 + 5*q + 1)*z^6 + O(z^7)

We count unlabeled ordered trees by total number of nodes and number of internal nodes:

sage: R.<q> = QQ[]
sage: Q.<z> = LazyPowerSeriesRing(R)
sage: leaf = z
sage: internal_node = q * z
sage: L = Q(constant=1, degree=1)
sage: T = Q.undefined(valuation=1)
sage: T.define(leaf + internal_node * L(T))
sage: T[0:6]
[0, 1, q, q^2 + q, q^3 + 3*q^2 + q, q^4 + 6*q^3 + 6*q^2 + q]

Similarly for Dirichlet series:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: g = L(constant=1, valuation=2)
sage: F = L.undefined()
sage: F.define(1 + g*F)
sage: F[:16]
[1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3]
sage: oeis(_)                                                       # optional, internet
0: A002033: Number of perfect partitions of n.
1: A074206: Kalmár's [Kalmar's] problem: number of ordered factorizations of n.
...

sage: F = L.undefined()
sage: F.define(1 + g*F*F)
sage: F[:16]
[1, 1, 1, 3, 1, 5, 1, 10, 3, 5, 1, 24, 1, 5, 5]

We can compute the Frobenius character of unlabeled trees:

sage: m = SymmetricFunctions(QQ).m()
sage: s = SymmetricFunctions(QQ).s()
sage: L = LazySymmetricFunctions(m)
sage: E = L(lambda n: s[n], valuation=0)
sage: X = L(s[1])
sage: A = L.undefined()
sage: A.define(X*E(A, check=False))
sage: A[:6]
[m[1],
 2*m[1, 1] + m[2],
 9*m[1, 1, 1] + 5*m[2, 1] + 2*m[3],
 64*m[1, 1, 1, 1] + 34*m[2, 1, 1] + 18*m[2, 2] + 13*m[3, 1] + 4*m[4],
 625*m[1, 1, 1, 1, 1] + 326*m[2, 1, 1, 1] + 171*m[2, 2, 1] + 119*m[3, 1, 1] + 63*m[3, 2] + 35*m[4, 1] + 9*m[5]]

euler()#

Return the Euler function evaluated at self.

The Euler function is defined as

\[\phi(z) = (z; z)_{\infty} = \sum_{n=0}^{\infty} (-1)^n q^{(3n^2-n)/2}.\]

EXAMPLES:

sage: L.<q> = LazyLaurentSeriesRing(ZZ)
sage: phi = L.euler()
sage: (q + q^2).euler() - phi(q + q^2)
O(q^7)

exp()#

Return the exponential series of self.

EXAMPLES:

sage: L = LazyDirichletSeriesRing(QQ, "s")
sage: Z = L(constant=1, valuation=2)
sage: exp(Z)
1 + 1/(2^s) + 1/(3^s) + 3/2/4^s + 1/(5^s) + 2/6^s + 1/(7^s) + O(1/(8^s))

hypergeometric(a, b)#

Return the \({}_{p}F_{q}\)-hypergeometric function \(\,_pF_{q}\) where \((p,q)\) is the parameterization of self.

INPUT:

a – the first parameter of the hypergeometric function
b – the second parameter of the hypergeometric function

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: z.hypergeometric([1, 1], [1])
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
sage: z.hypergeometric([], []) - exp(z)
O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: (x+y).hypergeometric([1, 1], [1]).polynomial(4)
x^4 + 4*x^3*y + 6*x^2*y^2 + 4*x*y^3 + y^4 + x^3 + 3*x^2*y
 + 3*x*y^2 + y^3 + x^2 + 2*x*y + y^2 + x + y + 1

lift_to_precision(absprec=None)#

Return another element of the same parent with absolute precision at least absprec, congruent to this element modulo the precision of this element.

Since the precision of a lazy series is infinity, this method returns the series itself, and the argument is ignored.

EXAMPLES:

sage: P.<t> = PowerSeriesRing(QQ, default_prec=2)
sage: R.<z> = LazyPowerSeriesRing(P)
sage: f = R(lambda n: 1/(1-t)^n)
sage: f
1 + ((1+t+O(t^2))*z) + ((1+2*t+O(t^2))*z^2)
  + ((1+3*t+O(t^2))*z^3)
  + ((1+4*t+O(t^2))*z^4)
  + ((1+5*t+O(t^2))*z^5)
  + ((1+6*t+O(t^2))*z^6) + O(z^7)
sage: f.lift_to_precision()
1 + ((1+t+O(t^2))*z) + ((1+2*t+O(t^2))*z^2)
  + ((1+3*t+O(t^2))*z^3)
  + ((1+4*t+O(t^2))*z^4)
  + ((1+5*t+O(t^2))*z^5)
  + ((1+6*t+O(t^2))*z^6) + O(z^7)

log()#

Return the series for the natural logarithm of self.

EXAMPLES:

sage: L = LazyDirichletSeriesRing(QQ, "s")
sage: Z = L(constant=1)
sage: log(Z)
1/(2^s) + 1/(3^s) + 1/2/4^s + 1/(5^s) + 1/(7^s) + O(1/(8^s))

map_coefficients(f)#

Return the series with f applied to each nonzero coefficient of self.

INPUT:

func – function that takes in a coefficient and returns a new coefficient

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: m = L(lambda n: n, valuation=0); m
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: m.map_coefficients(lambda c: c + 1)
2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + O(z^7)

Similarly for Dirichlet series:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: s = L(lambda n: n-1); s
1/(2^z) + 2/3^z + 3/4^z + 4/5^z + 5/6^z + 6/7^z + O(1/(8^z))
sage: s.map_coefficients(lambda c: c + 1)
2/2^z + 3/3^z + 4/4^z + 5/5^z + 6/6^z + 7/7^z + O(1/(8^z))

Similarly for multivariate power series:

sage: L.<x, y> = LazyPowerSeriesRing(QQ)
sage: f = 1/(1-(x+y)); f
1 + (x+y) + (x^2+2*x*y+y^2) + (x^3+3*x^2*y+3*x*y^2+y^3)
 + (x^4+4*x^3*y+6*x^2*y^2+4*x*y^3+y^4)
 + (x^5+5*x^4*y+10*x^3*y^2+10*x^2*y^3+5*x*y^4+y^5)
 + (x^6+6*x^5*y+15*x^4*y^2+20*x^3*y^3+15*x^2*y^4+6*x*y^5+y^6)
 + O(x,y)^7
sage: f.map_coefficients(lambda c: c^2)
1 + (x+y) + (x^2+4*x*y+y^2) + (x^3+9*x^2*y+9*x*y^2+y^3)
 + (x^4+16*x^3*y+36*x^2*y^2+16*x*y^3+y^4)
 + (x^5+25*x^4*y+100*x^3*y^2+100*x^2*y^3+25*x*y^4+y^5)
 + (x^6+36*x^5*y+225*x^4*y^2+400*x^3*y^3+225*x^2*y^4+36*x*y^5+y^6)
 + O(x,y)^7

Similarly for lazy symmetric functions:

sage: p = SymmetricFunctions(QQ).p()
sage: L = LazySymmetricFunctions(p)
sage: f = 1/(1-2*L(p[1])); f
p[] + 2*p[1] + (4*p[1,1]) + (8*p[1,1,1]) + (16*p[1,1,1,1])
 + (32*p[1,1,1,1,1]) + (64*p[1,1,1,1,1,1]) + O^7
sage: f.map_coefficients(lambda c: log(c, 2))
p[1] + (2*p[1,1]) + (3*p[1,1,1]) + (4*p[1,1,1,1])
 + (5*p[1,1,1,1,1]) + (6*p[1,1,1,1,1,1]) + O^7

prec()#

Return the precision of the series, which is infinity.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = 1/(1 - z)
sage: f.prec()
+Infinity

q_pochhammer(q=None)#

Return the infinite q-Pochhammer symbol \((a; q)_{\infty}\), where \(a\) is self.

This is also one version of the quantum dilogarithm or the \(q\)-Exponential function.

INPUT:

q – (default: \(q \in \QQ(q)\)) the parameter \(q\)

EXAMPLES:

sage: q = ZZ['q'].fraction_field().gen()
sage: L.<z> = LazyLaurentSeriesRing(q.parent())
sage: qp = L.q_pochhammer(q)
sage: (z + z^2).q_pochhammer(q) - qp(z + z^2)
O(z^7)

sec()#

Return the secant of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sec(z)
1 + 1/2*z^2 + 5/24*z^4 + 61/720*z^6 + O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: sec(x/(1-y)).polynomial(4)
5/24*x^4 + 3/2*x^2*y^2 + x^2*y + 1/2*x^2 + 1

sech()#

Return the hyperbolic secant of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sech(z)
1 - 1/2*z^2 + 5/24*z^4 - 61/720*z^6 + O(z^7)

sage: L.<x, y> = LazyPowerSeriesRing(QQ)
sage: sech(x/(1-y))
1 + (-1/2*x^2) + (-x^2*y) + (5/24*x^4-3/2*x^2*y^2) + (5/6*x^4*y-2*x^2*y^3)
 + (-61/720*x^6+25/12*x^4*y^2-5/2*x^2*y^4) + O(x,y)^7

set(s)#

Define an equation by self = s.

INPUT:

s – a lazy series

EXAMPLES:

We begin by constructing the Catalan numbers:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: C = L.undefined()
sage: C.define(1 + z*C^2)
sage: C
1 + z + 2*z^2 + 5*z^3 + 14*z^4 + 42*z^5 + 132*z^6 + O(z^7)
sage: binomial(2000, 1000) / C[1000]
1001

The Catalan numbers but with a valuation 1:

sage: B = L.undefined(valuation=1)
sage: B.define(z + B^2)
sage: B
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

We can define multiple series that are linked:

sage: s = L.undefined()
sage: t = L.undefined()
sage: s.define(1 + z*t^3)
sage: t.define(1 + z*s^2)
sage: s[0:9]
[1, 1, 3, 9, 34, 132, 546, 2327, 10191]
sage: t[0:9]
[1, 1, 2, 7, 24, 95, 386, 1641, 7150]

A bigger example:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: A = L.undefined(valuation=5)
sage: B = L.undefined()
sage: C = L.undefined(valuation=2)
sage: A.define(z^5 + B^2)
sage: B.define(z^5 + C^2)
sage: C.define(z^2 + C^2 + A^2)
sage: A[0:15]
[0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 5, 4, 14, 10, 48]
sage: B[0:15]
[0, 0, 0, 0, 1, 1, 2, 0, 5, 0, 14, 0, 44, 0, 138]
sage: C[0:15]
[0, 0, 1, 0, 1, 0, 2, 0, 5, 0, 15, 0, 44, 2, 142]

Counting binary trees:

sage: L.<z> = LazyPowerSeriesRing(QQ)
sage: s = L.undefined(valuation=1)
sage: s.define(z + (s^2+s(z^2))/2)
sage: s[0:9]
[0, 1, 1, 1, 2, 3, 6, 11, 23]

The \(q\)-Catalan numbers:

sage: R.<q> = ZZ[]
sage: L.<z> = LazyLaurentSeriesRing(R)
sage: s = L.undefined(valuation=0)
sage: s.define(1+z*s*s(q*z))
sage: s
1 + z + (q + 1)*z^2 + (q^3 + q^2 + 2*q + 1)*z^3
 + (q^6 + q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 3*q + 1)*z^4
 + (q^10 + q^9 + 2*q^8 + 3*q^7 + 5*q^6 + 5*q^5 + 7*q^4 + 7*q^3 + 6*q^2 + 4*q + 1)*z^5
 + (q^15 + q^14 + 2*q^13 + 3*q^12 + 5*q^11 + 7*q^10 + 9*q^9 + 11*q^8
    + 14*q^7 + 16*q^6 + 16*q^5 + 17*q^4 + 14*q^3 + 10*q^2 + 5*q + 1)*z^6 + O(z^7)

We count unlabeled ordered trees by total number of nodes and number of internal nodes:

sage: R.<q> = QQ[]
sage: Q.<z> = LazyPowerSeriesRing(R)
sage: leaf = z
sage: internal_node = q * z
sage: L = Q(constant=1, degree=1)
sage: T = Q.undefined(valuation=1)
sage: T.define(leaf + internal_node * L(T))
sage: T[0:6]
[0, 1, q, q^2 + q, q^3 + 3*q^2 + q, q^4 + 6*q^3 + 6*q^2 + q]

Similarly for Dirichlet series:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: g = L(constant=1, valuation=2)
sage: F = L.undefined()
sage: F.define(1 + g*F)
sage: F[:16]
[1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3]
sage: oeis(_)                                                       # optional, internet
0: A002033: Number of perfect partitions of n.
1: A074206: Kalmár's [Kalmar's] problem: number of ordered factorizations of n.
...

sage: F = L.undefined()
sage: F.define(1 + g*F*F)
sage: F[:16]
[1, 1, 1, 3, 1, 5, 1, 10, 3, 5, 1, 24, 1, 5, 5]

We can compute the Frobenius character of unlabeled trees:

sage: m = SymmetricFunctions(QQ).m()
sage: s = SymmetricFunctions(QQ).s()
sage: L = LazySymmetricFunctions(m)
sage: E = L(lambda n: s[n], valuation=0)
sage: X = L(s[1])
sage: A = L.undefined()
sage: A.define(X*E(A, check=False))
sage: A[:6]
[m[1],
 2*m[1, 1] + m[2],
 9*m[1, 1, 1] + 5*m[2, 1] + 2*m[3],
 64*m[1, 1, 1, 1] + 34*m[2, 1, 1] + 18*m[2, 2] + 13*m[3, 1] + 4*m[4],
 625*m[1, 1, 1, 1, 1] + 326*m[2, 1, 1, 1] + 171*m[2, 2, 1] + 119*m[3, 1, 1] + 63*m[3, 2] + 35*m[4, 1] + 9*m[5]]

shift(n)#

Return self with the indices shifted by n.

For example, a Laurent series is multiplied by the power \(z^n\), where \(z\) is the variable of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = 1 / (1 + 2*z)
sage: f
1 - 2*z + 4*z^2 - 8*z^3 + 16*z^4 - 32*z^5 + 64*z^6 + O(z^7)
sage: f.shift(3)
z^3 - 2*z^4 + 4*z^5 - 8*z^6 + 16*z^7 - 32*z^8 + 64*z^9 + O(z^10)
sage: f << -3  # shorthand
z^-3 - 2*z^-2 + 4*z^-1 - 8 + 16*z - 32*z^2 + 64*z^3 + O(z^4)
sage: g = z^-3 + 3 + z^2
sage: g.shift(5)
z^2 + 3*z^5 + z^7
sage: L([2,0,3], valuation=2, degree=7, constant=1) << -2
2 + 3*z^2 + z^5 + z^6 + z^7 + O(z^8)

sage: D = LazyDirichletSeriesRing(QQ, 't')
sage: f = D([0,1,2]); f
1/(2^t) + 2/3^t
sage: f.shift(3)
1/(5^t) + 2/6^t

sin()#

Return the sine of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sin(z)
z - 1/6*z^3 + 1/120*z^5 - 1/5040*z^7 + O(z^8)

sage: sin(1 + z)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: sin(x/(1-y)).polynomial(3)
-1/6*x^3 + x*y^2 + x*y + x

sinh()#

Return the sinh of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sinh(z)
z + 1/6*z^3 + 1/120*z^5 + 1/5040*z^7 + O(z^8)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: sinh(x/(1-y))
x + x*y + (1/6*x^3+x*y^2) + (1/2*x^3*y+x*y^3)
 + (1/120*x^5+x^3*y^2+x*y^4) + (1/24*x^5*y+5/3*x^3*y^3+x*y^5)
 + (1/5040*x^7+1/8*x^5*y^2+5/2*x^3*y^4+x*y^6) + O(x,y)^8

sqrt()#

Return self^(1/2).

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sqrt(1+z)
1 + 1/2*z - 1/8*z^2 + 1/16*z^3 - 5/128*z^4 + 7/256*z^5 - 21/1024*z^6 + O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: sqrt(1+x/(1-y))
1 + 1/2*x + (-1/8*x^2+1/2*x*y) + (1/16*x^3-1/4*x^2*y+1/2*x*y^2)
 + (-5/128*x^4+3/16*x^3*y-3/8*x^2*y^2+1/2*x*y^3)
 + (7/256*x^5-5/32*x^4*y+3/8*x^3*y^2-1/2*x^2*y^3+1/2*x*y^4)
 + (-21/1024*x^6+35/256*x^5*y-25/64*x^4*y^2+5/8*x^3*y^3-5/8*x^2*y^4+1/2*x*y^5)
 + O(x,y)^7

This also works for Dirichlet series:

sage: D = LazyDirichletSeriesRing(SR, "s")
sage: Z = D(constant=1)
sage: f = sqrt(Z)
sage: f
1 + 1/2/2^s + 1/2/3^s + 3/8/4^s + 1/2/5^s + 1/4/6^s + 1/2/7^s + O(1/(8^s))
sage: f*f - Z
O(1/(8^s))

tan()#

Return the tangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: tan(z)
z + 1/3*z^3 + 2/15*z^5 + 17/315*z^7 + O(z^8)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: tan(x/(1-y)).polynomial(5)
2/15*x^5 + 2*x^3*y^2 + x*y^4 + x^3*y + x*y^3 + 1/3*x^3 + x*y^2 + x*y + x

tanh()#

Return the tanh of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: tanh(z)
z - 1/3*z^3 + 2/15*z^5 - 17/315*z^7 + O(z^8)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: tanh(x/(1-y))
x + x*y + (-1/3*x^3+x*y^2) + (-x^3*y+x*y^3) + (2/15*x^5-2*x^3*y^2+x*y^4)
 + (2/3*x^5*y-10/3*x^3*y^3+x*y^5) + (-17/315*x^7+2*x^5*y^2-5*x^3*y^4+x*y^6) + O(x,y)^8

truncate(d)#

Return this series with its terms of degree >= d truncated.

INPUT:

d – integer; the degree from which the series is truncated

EXAMPLES:

Dense implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=False)
sage: alpha = 1/(1-z)
sage: alpha
1 + z + z^2 + O(z^3)
sage: beta = alpha.truncate(5)
sage: beta
1 + z + z^2 + z^3 + z^4
sage: alpha - beta
z^5 + z^6 + z^7 + O(z^8)
sage: M = L(lambda n: n, valuation=0); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: M.truncate(4)
z + 2*z^2 + 3*z^3

Sparse Implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=True)
sage: M = L(lambda n: n, valuation=0); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: M.truncate(4)
z + 2*z^2 + 3*z^3

Series which are known to be exact can also be truncated:

sage: M = z + z^2 + z^3 + z^4
sage: M.truncate(4)
z + z^2 + z^3

class sage.rings.lazy_series.LazyPowerSeries(parent, coeff_stream)#

Bases: LazyCauchyProductSeries

A Taylor series where the coefficients are computed lazily.

EXAMPLES:

sage: L.<x, y> = LazyPowerSeriesRing(ZZ)
sage: f = 1 / (1 - x^2 + y^3); f
1 + x^2 + (-y^3) + x^4 + (-2*x^2*y^3) + (x^6+y^6) + O(x,y)^7
sage: P.<x, y> = PowerSeriesRing(ZZ, default_prec=101)
sage: g = 1 / (1 - x^2 + y^3); f[100] - g[100]
0

Lazy Taylor series is picklable:

sage: g = loads(dumps(f))
sage: g
1 + x^2 + (-y^3) + x^4 + (-2*x^2*y^3) + (x^6+y^6) + O(x,y)^7
sage: g == f
True

compose(check, *g)#

Return the composition of self with g.

The arity of self must be equal to the number of arguments provided.

Given a Taylor series \(f\) of arity \(n\) and a tuple of Taylor series \(g = (g_1,\dots, g_n)\) over the same base ring, the composition \(f \circ g\) is defined if and only if for each \(1\leq k\leq n\):

\(g_i\) is zero, or
setting all variables except the \(i`th in `f\) to zero yields a polynomial, or
\(val(g_i) > 0\).

If \(f\) is a univariate ‘exact’ series, we can check whether \(f\) is a actually a polynomial. However, if \(f\) is a multivariate series, we have no way to test whether setting all but one variable of \(f\) to zero yields a polynomial, except if \(f\) itself is ‘exact’ and therefore a polynomial.

INPUT:

g – other series, all can be coerced into the same parent

EXAMPLES:

sage: L.<x, y, z> = LazyPowerSeriesRing(QQ)
sage: M.<a, b> = LazyPowerSeriesRing(ZZ)
sage: g1 = 1 / (1 - x)
sage: g2 = x + y^2
sage: p = a^2 + b + 1
sage: p(g1, g2) - g1^2 - g2 - 1
O(x,y,z)^7

The number of mappings from a set with \(m\) elements to a set with \(n\) elements:

sage: M.<a> = LazyPowerSeriesRing(QQ)
sage: Ea = M(lambda n: 1/factorial(n))
sage: Ex = L(lambda n: 1/factorial(n)*x^n)
sage: Ea(Ex*y)[5]
1/24*x^4*y + 2/3*x^3*y^2 + 3/4*x^2*y^3 + 1/6*x*y^4 + 1/120*y^5

So, there are \(3! 2! 2/3 = 8\) mappings from a three element set to a two element set.

We perform the composition with a lazy Laurent series:

sage: N.<w> = LazyLaurentSeriesRing(QQ)
sage: f1 = 1 / (1 - w)
sage: f2 = cot(w / (1 - w))
sage: p(f1, f2)
w^-1 + 1 + 5/3*w + 8/3*w^2 + 164/45*w^3 + 23/5*w^4 + 5227/945*w^5 + O(w^6)

We perform the composition with a lazy Dirichlet series:

sage: D = LazyDirichletSeriesRing(QQ, "s")
sage: g = D(constant=1)-1; g
1/(2^s) + 1/(3^s) + 1/(4^s) + O(1/(5^s))
sage: f = 1 / (1 - x - y*z); f
1 + x + (x^2+y*z) + (x^3+2*x*y*z) + (x^4+3*x^2*y*z+y^2*z^2)
 + (x^5+4*x^3*y*z+3*x*y^2*z^2)
 + (x^6+5*x^4*y*z+6*x^2*y^2*z^2+y^3*z^3)
 + O(x,y,z)^7
sage: fog = f(g, g, g); fog
1 + 1/(2^s) + 1/(3^s) + 3/4^s + 1/(5^s) + 5/6^s + O(1/(7^s))
sage: fg = 1 / (1 - g - g*g); fg
1 + 1/(2^s) + 1/(3^s) + 3/4^s + 1/(5^s) + 5/6^s + 1/(7^s) + O(1/(8^s))
sage: fog - fg
O(1/(7^s))

sage: f = 1 / (1 - 2*a)
sage: f(g)
1 + 2/2^s + 2/3^s + 6/4^s + 2/5^s + 10/6^s + 2/7^s + O(1/(8^s))
sage: 1 / (1 - 2*g)
1 + 2/2^s + 2/3^s + 6/4^s + 2/5^s + 10/6^s + 2/7^s + O(1/(8^s))

The output parent is always the common parent between the base ring of \(f\) and the parent of \(g\) or extended to the corresponding lazy series:

sage: T.<x,y> = LazyPowerSeriesRing(QQ)
sage: R.<a,b,c> = ZZ[]
sage: S.<v> = R[]
sage: L.<z> = LaurentPolynomialRing(ZZ)
sage: parent(x(a, b))
Multivariate Polynomial Ring in a, b, c over Rational Field
sage: parent(x(CC(2), a))
Multivariate Polynomial Ring in a, b, c over Complex Field with 53 bits of precision
sage: parent(x(0, 0))
Rational Field
sage: f = 1 / (1 - x - y); f
1 + (x+y) + (x^2+2*x*y+y^2) + (x^3+3*x^2*y+3*x*y^2+y^3)
 + (x^4+4*x^3*y+6*x^2*y^2+4*x*y^3+y^4)
 + (x^5+5*x^4*y+10*x^3*y^2+10*x^2*y^3+5*x*y^4+y^5)
 + (x^6+6*x^5*y+15*x^4*y^2+20*x^3*y^3+15*x^2*y^4+6*x*y^5+y^6)
 + O(x,y)^7
sage: f(a^2, b*c)
1 + (a^2+b*c) + (a^4+2*a^2*b*c+b^2*c^2) + (a^6+3*a^4*b*c+3*a^2*b^2*c^2+b^3*c^3) + O(a,b,c)^7
sage: f(v, v^2)
1 + v + 2*v^2 + 3*v^3 + 5*v^4 + 8*v^5 + 13*v^6 + O(v^7)
sage: f(z, z^2 + z)
1 + 2*z + 5*z^2 + 12*z^3 + 29*z^4 + 70*z^5 + 169*z^6 + O(z^7)
sage: three = T(3)(a^2, b); three
3
sage: parent(three)
Multivariate Polynomial Ring in a, b, c over Rational Field

compositional_inverse()#

Return the compositional inverse of self.

Given a Taylor series \(f\), the compositional inverse is a Laurent series \(g\) over the same base ring, such that \((f \circ g)(z) = f(g(z)) = z\).

The compositional inverse exists if and only if:

\(val(f) = 1\), or
\(f = a + b z\) with \(a, b \neq 0\)

EXAMPLES:

sage: L.<z> = LazyPowerSeriesRing(QQ)
sage: (2*z).revert()
1/2*z
sage: (z-z^2).revert()
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

sage: s = L(degree=1, constant=-1)
sage: s.revert()
-z - z^2 - z^3 + O(z^4)

sage: s = L(degree=1, constant=1)
sage: s.revert()
z - z^2 + z^3 - z^4 + z^5 - z^6 + z^7 + O(z^8)

compute_coefficients(i)#

Computes all the coefficients of self up to i.

This method is deprecated, it has no effect anymore.

derivative(*args)#

Return the derivative of the Taylor series.

Multiple variables and iteration counts may be supplied; see the documentation of sage.calculus.functional.derivative() function for details.

EXAMPLES:

sage: T.<z> = LazyPowerSeriesRing(ZZ)
sage: z.derivative()
1
sage: (1+z+z^2).derivative(3)
0
sage: (1/(1-z)).derivative()
1 + 2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + O(z^7)

sage: R.<q> = QQ[]
sage: L.<x, y> = LazyPowerSeriesRing(R)
sage: f = 1/(1-q*x+y); f
1 + (q*x-y) + (q^2*x^2+(-2*q)*x*y+y^2)
 + (q^3*x^3+(-3*q^2)*x^2*y+3*q*x*y^2-y^3)
 + (q^4*x^4+(-4*q^3)*x^3*y+6*q^2*x^2*y^2+(-4*q)*x*y^3+y^4)
 + (q^5*x^5+(-5*q^4)*x^4*y+10*q^3*x^3*y^2+(-10*q^2)*x^2*y^3+5*q*x*y^4-y^5)
 + (q^6*x^6+(-6*q^5)*x^5*y+15*q^4*x^4*y^2+(-20*q^3)*x^3*y^3+15*q^2*x^2*y^4+(-6*q)*x*y^5+y^6)
 + O(x,y)^7
sage: f.derivative(q)
x + (2*q*x^2+(-2)*x*y) + (3*q^2*x^3+(-6*q)*x^2*y+3*x*y^2)
 + (4*q^3*x^4+(-12*q^2)*x^3*y+12*q*x^2*y^2+(-4)*x*y^3)
 + (5*q^4*x^5+(-20*q^3)*x^4*y+30*q^2*x^3*y^2+(-20*q)*x^2*y^3+5*x*y^4)
 + (6*q^5*x^6+(-30*q^4)*x^5*y+60*q^3*x^4*y^2+(-60*q^2)*x^3*y^3+30*q*x^2*y^4+(-6)*x*y^5)
 + O(x,y)^7

exponential()#

Return the exponential series of self.

This method is deprecated, use exp() instead.

is_unit()#

Return whether this element is a unit in the ring.

EXAMPLES:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: (2*z).is_unit()
False

sage: (1 + 2*z).is_unit()
True

sage: (3 + 2*z).is_unit()
False

sage: L.<x,y> = LazyPowerSeriesRing(ZZ)
sage: (-1 + 2*x + 3*x*y).is_unit()
True

polynomial(degree=None, names=None)#

Return self as a polynomial if self is actually so.

INPUT:

degree – None or an integer
names – names of the variables; if it is None, the name of the variables of the series is used

OUTPUT:

If degree is not None, the terms of the series of degree greater than degree are first truncated. If degree is None and the series is not a polynomial polynomial, a ValueError is raised.

EXAMPLES:

sage: L.<x,y> = LazyPowerSeriesRing(ZZ)
sage: f = x^2 + y*x - x + 2; f
2 + (-x) + (x^2+x*y)
sage: f.polynomial()
x^2 + x*y - x + 2

revert()#

Return the compositional inverse of self.

Given a Taylor series \(f\), the compositional inverse is a Laurent series \(g\) over the same base ring, such that \((f \circ g)(z) = f(g(z)) = z\).

The compositional inverse exists if and only if:

\(val(f) = 1\), or
\(f = a + b z\) with \(a, b \neq 0\)

EXAMPLES:

sage: L.<z> = LazyPowerSeriesRing(QQ)
sage: (2*z).revert()
1/2*z
sage: (z-z^2).revert()
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

sage: s = L(degree=1, constant=-1)
sage: s.revert()
-z - z^2 - z^3 + O(z^4)

sage: s = L(degree=1, constant=1)
sage: s.revert()
z - z^2 + z^3 - z^4 + z^5 - z^6 + z^7 + O(z^8)

class sage.rings.lazy_series.LazyPowerSeries_gcd_mixin#

Bases: object

A lazy power series that also implements the GCD algorithm.

gcd(other)#

Return the greatest common divisor of self and other.

EXAMPLES:

sage: L.<x> = LazyPowerSeriesRing(QQ)
sage: a = 16*x^5 / (1 - 5*x)
sage: b = (22*x^2 + x^8) / (1 - 4*x^2)
sage: a.gcd(b)
x^2

xgcd(f)#

Return the extended gcd of self and f.

OUTPUT:

A triple (g, s, t) such that g is the gcd of self and f, and s and t are cofactors satisfying the Bezout identity

\[g = s \cdot \mathrm{self} + t \cdot f.\]

EXAMPLES:

sage: L.<x> = LazyPowerSeriesRing(QQ)
sage: a = 16*x^5 / (1 - 2*x)
sage: b = (22*x^3 + x^8) / (1 - 3*x^2)
sage: g, s, t = a.xgcd(b)
sage: g
x^3
sage: s
1/22 - 41/242*x^2 - 8/121*x^3 + 120/1331*x^4 + 1205/5324*x^5 + 316/14641*x^6 + O(x^7)
sage: t
1/22 - 41/242*x^2 - 8/121*x^3 + 120/1331*x^4 + 1205/5324*x^5 + 316/14641*x^6 + O(x^7)

sage: LazyPowerSeriesRing.options.halting_precision(20)  # verify up to degree 20

sage: g == s * a + t * b
True

sage: a = 16*x^5 / (1 - 2*x)
sage: b = (-16*x^5 + x^8) / (1 - 3*x^2)
sage: g, s, t = a.xgcd(b)
sage: g
x^5
sage: s
1/16 - 1/16*x - 3/16*x^2 + 1/8*x^3 - 17/256*x^4 + 9/128*x^5 + 1/128*x^6 + O(x^7)
sage: t
1/16*x - 1/16*x^2 - 3/16*x^3 + 1/8*x^4 - 17/256*x^5 + 9/128*x^6 + 1/128*x^7 + O(x^8)
sage: g == s * a + t * b
True

sage: L.<x> = LazyPowerSeriesRing(GF(2))
sage: a = L(lambda n: n % 2, valuation=3); a
x^3 + x^5 + x^7 + x^9 + O(x^10)
sage: b = L(lambda n: binomial(n,2) % 2, valuation=3); b
x^3 + x^6 + x^7 + O(x^10)
sage: g, s, t = a.xgcd(b)
sage: g
x^3
sage: s
1 + x + x^3 + x^4 + x^5 + O(x^7)
sage: t
x + x^2 + x^4 + x^5 + x^6 + O(x^8)
sage: g == s * a + t * b
True

sage: LazyPowerSeriesRing.options._reset()  # reset the options

class sage.rings.lazy_series.LazySymmetricFunction(parent, coeff_stream)#

Bases: LazyCompletionGradedAlgebraElement

A symmetric function where each degree is computed lazily.

EXAMPLES:

sage: s = SymmetricFunctions(ZZ).s()
sage: L = LazySymmetricFunctions(s)

arithmetic_product(check, *args)#

Return the arithmetic product of self with g.

The arithmetic product is a binary operation \(\boxdot\) on the ring of symmetric functions which is bilinear in its two arguments and satisfies

\[p_{\lambda} \boxdot p_{\mu} = \prod\limits_{i \geq 1, j \geq 1} p_{\mathrm{lcm}(\lambda_i, \mu_j)}^{\mathrm{gcd}(\lambda_i, \mu_j)}\]

for any two partitions \(\lambda = (\lambda_1, \lambda_2, \lambda_3, \dots )\) and \(\mu = (\mu_1, \mu_2, \mu_3, \dots )\) (where \(p_{\nu}\) denotes the power-sum symmetric function indexed by the partition \(\nu\), and \(p_i\) denotes the \(i\)-th power-sum symmetric function). This is enough to define the arithmetic product if the base ring is torsion-free as a \(\ZZ\)-module; for all other cases the arithmetic product is uniquely determined by requiring it to be functorial in the base ring. See http://mathoverflow.net/questions/138148/ for a discussion of this arithmetic product.

Warning

The operation \(f \boxdot g\) was originally defined only for symmetric functions \(f\) and \(g\) without constant term. We extend this definition using the convention that the least common multiple of any integer with \(0\) is \(0\).

If \(f\) and \(g\) are two symmetric functions which are homogeneous of degrees \(a\) and \(b\), respectively, then \(f \boxdot g\) is homogeneous of degree \(ab\).

The arithmetic product is commutative and associative and has unity \(e_1 = p_1 = h_1\).

For species \(M\) and \(N\) such that \(M[\varnothing] = N[\varnothing] = \varnothing\), their arithmetic product is the species \(M \boxdot N\) of “\(M\)-assemblies of cloned \(N\)-structures”. This operation is defined and several examples are given in [MM2008].

INPUT:

g – a cycle index series having the same parent as self
check – (default: True) a Boolean which, when set to False, will cause input checks to be skipped

OUTPUT:

The arithmetic product of self with g.

EXAMPLES:

For \(C\) the species of (oriented) cycles and \(L_{+}\) the species of nonempty linear orders, \(C \boxdot L_{+}\) corresponds to the species of “regular octopuses”; a \((C \boxdot L_{+})\)-structure is a cycle of some length, each of whose elements is an ordered list of a length which is consistent for all the lists in the structure.

sage: R.<q> = QQ[]
sage: p = SymmetricFunctions(R).p()
sage: m = SymmetricFunctions(R).m()
sage: L = LazySymmetricFunctions(m)

sage: C = species.CycleSpecies().cycle_index_series()
sage: c = L(lambda n: C[n])
sage: Lplus = L(lambda n: p([1]*n), valuation=1)
sage: r = c.arithmetic_product(Lplus); r
m[1] + (3*m[1,1]+2*m[2])
 + (8*m[1,1,1]+4*m[2,1]+2*m[3])
 + (42*m[1,1,1,1]+21*m[2,1,1]+12*m[2,2]+7*m[3,1]+3*m[4])
 + (144*m[1,1,1,1,1]+72*m[2,1,1,1]+36*m[2,2,1]+24*m[3,1,1]+12*m[3,2]+6*m[4,1]+2*m[5])
 + ...
 + O^7

In particular, the number of regular octopuses is:

sage: [r[n].coefficient([1]*n) for n in range(8)]
[0, 1, 3, 8, 42, 144, 1440, 5760]

It is shown in [MM2008] that the exponential generating function for regular octopuses satisfies \((C \boxdot L_{+}) (x) = \sum_{n \geq 1} \sigma (n) (n - 1)! \frac{x^{n}}{n!}\) (where \(\sigma (n)\) is the sum of the divisors of \(n\)).

sage: [sum(divisors(i))*factorial(i-1) for i in range(1,8)]
[1, 3, 8, 42, 144, 1440, 5760]

AUTHORS:

Andrew Gainer-Dewar (2013)

REFERENCES:

[MM2008]

compositional_inverse()#

Return the compositional inverse of self.

Given a symmetric function \(f\), the compositional inverse is a symmetric function \(g\) over the same base ring, such that \(f \circ g = p_1\). Thus, it is the inverse with respect to plethystic substitution.

The compositional inverse exists if and only if:

\(val(f) = 1\), or
\(f = a + b p_1\) with \(a, b \neq 0\).

EXAMPLES:

sage: h = SymmetricFunctions(QQ).h()
sage: L = LazySymmetricFunctions(h)
sage: f = L(lambda n: h[n]) - 1
sage: f(f.revert())
h[1] + O^7

ALGORITHM:

Let \(F\) be a symmetric function with valuation \(1\), i.e., whose constant term vanishes and whose degree one term equals \(b p_1\). Then

\[(F - b p_1) \circ G = F \circ G - b p_1 \circ G = p_1 - b G,\]

and therefore \(G = (p_1 - (F - b p_1) \circ G) / b\), which allows recursive computation of \(G\).

See also

The compositional inverse \(\Omega\) of the symmetric function \(h_1 + h_2 + \dots\) can be handled much more efficiently using specialized methods. See LogarithmCycleIndexSeries()

AUTHORS:

Andrew Gainer-Dewar
Martin Rubey

derivative_with_respect_to_p1(n=1)#

Return the symmetric function obtained by taking the derivative of self with respect to the power-sum symmetric function \(p_1\) when the expansion of self in the power-sum basis is considered as a polynomial in \(p_k\)’s (with \(k \geq 1\)).

This is the same as skewing self by the first power-sum symmetric function \(p_1\).

INPUT:

n – (default: 1) nonnegative integer which determines which power of the derivative is taken

EXAMPLES:

The species \(E\) of sets satisfies the relationship \(E' = E\):

sage: h = SymmetricFunctions(QQ).h()
sage: T = LazySymmetricFunctions(h)
sage: E = T(lambda n: h[n])
sage: E - E.derivative_with_respect_to_p1()
O^6

The species \(C\) of cyclic orderings and the species \(L\) of linear orderings satisfy the relationship \(C' = L\):

sage: p = SymmetricFunctions(QQ).p()
sage: C = T(lambda n: (sum(euler_phi(k)*p([k])**(n//k) for k in divisors(n))/n if n > 0 else 0))
sage: L = T(lambda n: p([1]*n))
sage: L - C.derivative_with_respect_to_p1()
O^6

functorial_composition(*args)#

Return the functorial composition of self and g.

Let \(X\) be a finite set of cardinality \(m\). For a group action of the symmetric group \(g: S_n \to S_X\) and a (possibly virtual) representation of the symmetric group on \(X\), \(f: S_X \to GL(V)\), the functorial composition is the (virtual) representation of the symmetric group \(f \Box g: S_n \to GL(V)\) given by \(\sigma \mapsto f(g(\sigma))\).

This is more naturally phrased in the language of combinatorial species. Let \(F\) and \(G\) be species, then their functorial composition is the species \(F \Box G\) with \((F \Box G) [A] = F[ G[A] ]\). In other words, an \((F \Box G)\)-structure on a set \(A\) of labels is an \(F\)-structure whose labels are the set of all \(G\)-structures on \(A\).

The Frobenius character (or cycle index series) of \(F \Box G\) can be computed as follows, see section 2.2 of [BLL]):

\[\sum_{n \geq 0} \frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_{n}} \operatorname{fix} F[ (G[\sigma])_{1}, (G[\sigma])_{2}, \ldots ] \, p_{1}^{\sigma_{1}} p_{2}^{\sigma_{2}} \cdots.\]

Warning

The operation \(f \Box g\) only makes sense when \(g\) corresponds to a permutation representation, i.e., a group action.

EXAMPLES:

The species \(G\) of simple graphs can be expressed in terms of a functorial composition: \(G = \mathfrak{p} \Box \mathfrak{p}_{2}\), where \(\mathfrak{p}\) is the SubsetSpecies.:

sage: R.<q> = QQ[]
sage: h = SymmetricFunctions(R).h()
sage: m = SymmetricFunctions(R).m()
sage: L = LazySymmetricFunctions(m)
sage: P = L(lambda n: sum(q^k*h[n-k]*h[k] for k in range(n+1)))
sage: P2 = L(lambda n: h[2]*h[n-2], valuation=2)
sage: P.functorial_composition(P2)[:4]
[m[],
 m[1],
 (q+1)*m[1, 1] + (q+1)*m[2],
 (q^3+3*q^2+3*q+1)*m[1, 1, 1] + (q^3+2*q^2+2*q+1)*m[2, 1] + (q^3+q^2+q+1)*m[3]]

For example, there are:

sage: P.functorial_composition(P2)[4].coefficient([4])[3]
3

unlabelled graphs on 4 vertices and 3 edges, and:

sage: P.functorial_composition(P2)[4].coefficient([2,2])[3]
8

labellings of their vertices with two 1’s and two 2’s.

The symmetric function \(h_1 \sum_n h_n\) is the neutral element with respect to functorial composition:

sage: p = SymmetricFunctions(QQ).p()
sage: h = SymmetricFunctions(QQ).h()
sage: e = SymmetricFunctions(QQ).e()
sage: L = LazySymmetricFunctions(h)
sage: E = L(lambda n: h[n])
sage: Ep = p[1]*E.derivative_with_respect_to_p1(); Ep
h[1] + (h[1,1]) + (h[2,1]) + (h[3,1]) + (h[4,1]) + (h[5,1]) + O^7
sage: f = L(lambda n: h[n-n//2, n//2])
sage: f - Ep.functorial_composition(f)
O^7

The functorial composition distributes over the sum:

sage: F1 = L(lambda n: h[n])
sage: F2 = L(lambda n: e[n])
sage: f1 = F1.functorial_composition(f)
sage: f2 = F2.functorial_composition(f)
sage: (F1 + F2).functorial_composition(f) - f1 - f2  # long time
O^7

is_unit()#

Return whether this element is a unit in the ring.

EXAMPLES:

sage: m = SymmetricFunctions(ZZ).m()
sage: L = LazySymmetricFunctions(m)

sage: L(2*m[1]).is_unit()
False

sage: L(-1 + 2*m[1]).is_unit()
True

sage: L(2 + m[1]).is_unit()
False

sage: m = SymmetricFunctions(QQ).m()
sage: L = LazySymmetricFunctions(m)

sage: L(2 + 3*m[1]).is_unit()
True

plethysm(check, *args)#

Return the composition of self with g.

The arity of self must be equal to the number of arguments provided.

Given a lazy symmetric function \(f\) of arity \(n\) and a tuple of lazy symmetric functions \(g = (g_1,\dots, g_n)\) over the same base ring, the composition (or plethysm) \((f \circ g)\) is defined if and only if for each \(1\leq k\leq n\):

\(g_i = 0\), or
setting all alphabets except the \(i`th in `f\) to zero yields a symmetric function with only finitely many non-zero coefficients, or
\(val(g) > 0\).

If \(f\) is a univariate ‘exact’ lazy symmetric function, we can check whether \(f\) has only finitely many non-zero coefficients. However, if \(f\) has larger arity, we have no way to test whether setting all but one alphabets of \(f\) to zero yields a polynomial, except if \(f\) itself is ‘exact’ and therefore a symmetric function with only finitely many non-zero coefficients.

INPUT:

g – other (lazy) symmetric functions

Todo

Allow specification of degree one elements.

EXAMPLES:

sage: P.<q> = QQ[]
sage: s = SymmetricFunctions(P).s()
sage: L = LazySymmetricFunctions(s)
sage: f = s[2]
sage: g = s[3]
sage: L(f)(L(g)) - L(f(g))
0

sage: f = s[2] + s[2,1]
sage: g = s[1] + s[2,2]
sage: L(f)(L(g)) - L(f(g))
0

sage: L(f)(g) - L(f(g))
0

sage: f = s[2] + s[2,1]
sage: g = s[1] + s[2,2]
sage: L(f)(L(q*g)) - L(f(q*g))
0

The Frobenius character of the permutation action on set partitions is a plethysm:

sage: s = SymmetricFunctions(QQ).s()
sage: S = LazySymmetricFunctions(s)
sage: E1 = S(lambda n: s[n], valuation=1)
sage: E = 1 + E1
sage: P = E(E1)
sage: P[:5]
[s[], s[1], 2*s[2], s[2, 1] + 3*s[3], 2*s[2, 2] + 2*s[3, 1] + 5*s[4]]

The plethysm with a tensor product is also implemented:

sage: s = SymmetricFunctions(QQ).s()
sage: X = tensor([s[1],s[[]]])
sage: Y = tensor([s[[]],s[1]])
sage: S = LazySymmetricFunctions(s)
sage: S2 = LazySymmetricFunctions(tensor([s, s]))
sage: A = S(s[1,1,1])
sage: B = S2(X+Y)
sage: A(B)
(s[]#s[1,1,1]+s[1]#s[1,1]+s[1,1]#s[1]+s[1,1,1]#s[])

sage: H = S(lambda n: s[n])
sage: H(S2(X*Y))
(s[]#s[]) + (s[1]#s[1]) + (s[1,1]#s[1,1]+s[2]#s[2])
 + (s[1,1,1]#s[1,1,1]+s[2,1]#s[2,1]+s[3]#s[3]) + O^7
sage: H(S2(X+Y))
(s[]#s[]) + (s[]#s[1]+s[1]#s[]) + (s[]#s[2]+s[1]#s[1]+s[2]#s[])
 + (s[]#s[3]+s[1]#s[2]+s[2]#s[1]+s[3]#s[])
 + (s[]#s[4]+s[1]#s[3]+s[2]#s[2]+s[3]#s[1]+s[4]#s[])
 + (s[]#s[5]+s[1]#s[4]+s[2]#s[3]+s[3]#s[2]+s[4]#s[1]+s[5]#s[])
 + (s[]#s[6]+s[1]#s[5]+s[2]#s[4]+s[3]#s[3]+s[4]#s[2]+s[5]#s[1]+s[6]#s[])
 + O^7

plethystic_inverse()#

Return the compositional inverse of self.

The compositional inverse exists if and only if:

\(val(f) = 1\), or
\(f = a + b p_1\) with \(a, b \neq 0\).

EXAMPLES:

sage: h = SymmetricFunctions(QQ).h()
sage: L = LazySymmetricFunctions(h)
sage: f = L(lambda n: h[n]) - 1
sage: f(f.revert())
h[1] + O^7

ALGORITHM:

Let \(F\) be a symmetric function with valuation \(1\), i.e., whose constant term vanishes and whose degree one term equals \(b p_1\). Then

\[(F - b p_1) \circ G = F \circ G - b p_1 \circ G = p_1 - b G,\]

and therefore \(G = (p_1 - (F - b p_1) \circ G) / b\), which allows recursive computation of \(G\).

See also

The compositional inverse \(\Omega\) of the symmetric function \(h_1 + h_2 + \dots\) can be handled much more efficiently using specialized methods. See LogarithmCycleIndexSeries()

AUTHORS:

Andrew Gainer-Dewar
Martin Rubey

revert()#

Return the compositional inverse of self.

The compositional inverse exists if and only if:

\(val(f) = 1\), or
\(f = a + b p_1\) with \(a, b \neq 0\).

EXAMPLES:

sage: h = SymmetricFunctions(QQ).h()
sage: L = LazySymmetricFunctions(h)
sage: f = L(lambda n: h[n]) - 1
sage: f(f.revert())
h[1] + O^7

ALGORITHM:

Let \(F\) be a symmetric function with valuation \(1\), i.e., whose constant term vanishes and whose degree one term equals \(b p_1\). Then

\[(F - b p_1) \circ G = F \circ G - b p_1 \circ G = p_1 - b G,\]

and therefore \(G = (p_1 - (F - b p_1) \circ G) / b\), which allows recursive computation of \(G\).

See also

The compositional inverse \(\Omega\) of the symmetric function \(h_1 + h_2 + \dots\) can be handled much more efficiently using specialized methods. See LogarithmCycleIndexSeries()

AUTHORS:

Andrew Gainer-Dewar
Martin Rubey

symmetric_function(degree=None)#

Return self as a symmetric function if self is actually so.

INPUT:

degree – None or an integer

OUTPUT:

If degree is not None, the terms of the series of degree greater than degree are first truncated. If degree is None and the series is not a polynomial polynomial, a ValueError is raised.

EXAMPLES:

sage: s = SymmetricFunctions(QQ).s()
sage: S = LazySymmetricFunctions(s)
sage: elt = S(s[2])
sage: elt.symmetric_function()
s[2]