ASKSAGE: Sage Q&A Forum - Latest question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 10 Jan 2019 19:06:35 -0600Homomorphisms lifted from base ring in PowerSeriesRing do not preserve precisionhttp://ask.sagemath.org/question/45002/homomorphisms-lifted-from-base-ring-in-powerseriesring-do-not-preserve-precision/Hi all,
Homomorphisms which are lifted from the base ring seem to be unaware that precision exists in power/Laurent series rings. For example:
sage: R.<x> = PowerSeriesRing(ZZ)
sage: f = Hom(ZZ, ZZ)([1])
sage: Rf = Hom(R, R)(f); Rf
Ring endomorphism of Power Series Ring in x over Integer Ring
Defn: Induced from base ring by
Ring endomorphism of Integer Ring
Defn: 1 |--> 1
sage: Rf(1 + x + O(x^2))
1 + x
Can someone confirm that the expected output should be 1 + x + O(x^2), and that this is a bug?
Thanks,
Henryliu.henry.hlThu, 10 Jan 2019 19:06:35 -0600http://ask.sagemath.org/question/45002/formal series over InfinitePolynomialRinghttp://ask.sagemath.org/question/44056/formal-series-over-infinitepolynomialring/ I apologize if the question does not belong here. This is my first try to using sage and I find the documentation hard to read/search. I am trying to work with symbolic power series over a non-Noetherian ring. So for example I have:
sage: P.<x> = InfinitePolynomialRing(QQ)
sage: R.<t> = PowerSeriesRing(P)
And I'd like to consider the series $f(t) = \sum x_n t^n$ as an element of R. But my first try
<pre><code>
sage: sum(x[n]*t^n,(n,0,oo))
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-27-d7d20df9d062> in <module>()
----> 1 sum(x[n]*t**n,(n,Integer(0),oo))
/usr/lib64/python2.7/site-packages/sage/rings/polynomial/infinite_polynomial_ring.pyc in __getitem__(self, i)
1433 alpha_1
1434 """
-> 1435 if int(i) != i:
1436 raise ValueError("The index (= %s) must be an integer" % i)
1437 i = int(i)
TypeError: int() argument must be a string or a number, not 'function'
</code></pre>
I'll appreciate any help or if you can point me to the documentation where to read about this. heluaniWed, 24 Oct 2018 10:31:33 -0500http://ask.sagemath.org/question/44056/Valuation error when composing power serieshttp://ask.sagemath.org/question/43999/valuation-error-when-composing-power-series/ I'm trying to work with power series with coefficients in a ring consisting of elements $f/g$, where $f$ is a polynomial in several variables over the integers and g is a polynomial in one of those variables with unit constant coefficient. Ultimately, I'd like to be able to do all of the following:
- Compose power series over this ring
- Invert single-variable power series over this ring whose degree-1 coefficients are units
- Reduce the coefficients of a power series modulo some ideal of the ring
I thought I would do this by starting with the Laurent series over $\mathbf{Z}$ in the variable that occurs in the denominator, constructing a polynomial ring over this by adjoining the other variables, and letting my power series take coefficients in this ring. But I'm getting an error I don't understand when I try to compose series. The following example illustrates the problem:
M1.<b> = LaurentSeriesRing(ZZ)
M.<a,c> = PolynomialRing(M1)
R.<t1,t2> = PowerSeriesRing(M)
X = t1 + t2 + O(t1,t2)^3
R1.<t> = PowerSeriesRing(M)
f = X(t,t^2); f
This returns the type error `Substitution defined only for elements of positive valuation, unless self has infinite precision` despite the fact that replacing the last line by `t.valuation()` or `(t^2).valuation()` returns `1` or `2`, respectively.
The problem seems to result from a combination of factors. If I replace the second line with either
M.<a,c> = PolynomialRing(ZZ)
or
M.<a> = PolynomialRing(M1)
the error vanishes and the code returns the expected `t + t^2 + O(t)^3`. It also gives no error if I define `R` and `R1` to be polynomial rings, rather than power series rings, over `M` (and correspondingly delete the big-O notation).
Any ideas about what is triggering the error, or another way to construct these power series?Annie CarterFri, 19 Oct 2018 19:03:23 -0500http://ask.sagemath.org/question/43999/Solving a power series equationhttp://ask.sagemath.org/question/42778/solving-a-power-series-equation/I'd like to solve the equation `f(t) == 1` where `f` is a power series:
var('n,t')
a1 = float(4 + sqrt(8))
b1 = float(3 + sqrt(8))
c1 = b1
an = 4*n*(n-1)+a1*n
an1 = 4*n*(n+1) + a1*(n+1)
anr = b1 + 2*n*(4*n+a1)
bn = 1+2*n*(n-1)+n*(b1-1)
bn1 = 1+2*n*(n+1)+(n+1)*(b1-1)
bnr = c1 + 2*n*(2*n+b1-1)
def f(t):
return (1+b1)^-t + 3/2*a1^-t + 1/2*(2*b1)^-t + b1^-t + 2*sum(an^-t +an1^-t + anr^-t + 2*bn^-t +2*bn1^-t + bnr^-t ,n,1,1000).n()
I've tried simply plugging in values for t and I estimate that if f(t) = 1 then t is approximately 1.51. I'd like to possibly do something like
solve(f(t) == 1,t)
but this appears to lock sage up and I have to interrupt the process. The true f function is actually an infinite series, but I am truncating it to the first 1000 terms. I know there are methods of solving series equations such as the method of regula falsi. Does Sage have anything like this? Thanks!Daniel LFri, 29 Jun 2018 12:52:53 -0500http://ask.sagemath.org/question/42778/Help summing an infinite serieshttp://ask.sagemath.org/question/40512/help-summing-an-infinite-series/I have been trying to evaluate the following series using SageMath.
sum((x^x)/(factorial(x)*exp(x)) - 1/sqrt(2*pi*x), x, 1, +oo).n()
The answer should be -0.084069508727655... (-2/3 - zeta(1/2)/sqrt(Pi)), but I get the following error when the upper limit is above 96:
RuntimeError: maximum recursion depth exceeded while calling a Python object
Is there any way SageMath could give me even the first few digits of the infinite sum? This sum seems to evaluate just fine in Maple and Mathematica, but not in any open source program I have tried, including Axiom, Maxima, PARI, and SymPy.
Any help would be greatly appreciated.
lkklSat, 06 Jan 2018 18:16:21 -0600http://ask.sagemath.org/question/40512/working with coefficients of formal serieshttp://ask.sagemath.org/question/40256/working-with-coefficients-of-formal-series/ I want to define a truncated series or polynomial of arbitrary degree and then work algebraically with the polynomial to solve for various quantities in terms of the coefficients. When I write something like
i,n,z=var('i,n,z')
c=function('c')
p=z^(-4)+sum(c(i)/z^i,i,0,2)
p
this returns
(z^2*c(0) + z*c(1) + c(2))/z^2 + 1/z^4
But if I try to define an arbitrary polynomial of this type
p(n)=z^(-2n) + sum(c(i)/z^i,i,0,2n-2)
p(2)
returns
z^4 + sum(z^(-i)*c(i), i, 0, 2)
What is the crucial difference here?
I had a similar problem when working with truncated power series over the ring `PowerSeriesRing(SR)`. I want to manipulate these expressions algebraically as elements of a ring then ask for info about certain coefficients. But working with formal sums and substituting values of n returns power series coefficients, e.g. the following expression as the coefficient of z^-4 (after some computations...f and g are symbolic functions)
1/4*(sum(z^i*f(i), i, 1, 3)*sum(z^i*g(i), i, 1, 3) + 2)^2 + sum(z^i*f(i), i, 1, 3)*sum(z^i*g(i), i, 1, 3)
How do I get sage to work with the series and also give info about the coefficients, i.e. multiply series expressions but then expand them out in z? I've tried `expand()` in this setting with mixed success.charleslebarronWed, 20 Dec 2017 08:44:55 -0600http://ask.sagemath.org/question/40256/How to truncate a power series in two variables?http://ask.sagemath.org/question/39983/how-to-truncate-a-power-series-in-two-variables/I would like to truncate power-series by setting $y$ to 0, in order to express $y = x^3 - xy^2$ as a power series in $x$ by recursively plugging in the equation for $y$, then truncating.
R.<x, y> = PowerSeriesRing(QQ, default_prec = 20)
f = x^3 + x*y^2
j = f(x, f(x, f(x, f(x, f(x,y))))).expand()
j + O(y)
---------------------------------------------------------------------------
ArithmeticError Traceback (most recent call last)
<ipython-input-20-fe9380f2a091> in <module>()
----> 1 j + O(y)
/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/rings/big_oh.py in O(*x, **kwds)
154 elif hasattr(x, 'O'):
155 return x.O(**kwds)
--> 156 raise ArithmeticError("O(%s) not defined" % (x,))
ArithmeticError: O(y) not defined
I have found that other variants, such as` j + O(x, y)^{31}`, `j.truncate(31)`, and `j + R.O(31)` also do not work. masseygirlWed, 06 Dec 2017 17:09:09 -0600http://ask.sagemath.org/question/39983/How can I recurse a power series in two variables?http://ask.sagemath.org/question/39960/how-can-i-recurse-a-power-series-in-two-variables/ I would like very much to express, for example,
R.<x, y> = PowerSeriesRing(QQ, default_prec = 20)
g(x, g(x, g(x, x)))
Or,
f(x, f(x, f(x, f(x, f(x, f(x, f(x, f(x, f(x, f(x, f(x, f(x, (f(x, (f(x, f(x,y))))))))))))))))).expand()
In a more elegant way, for a specified number of self-compositions in one variable. I have only been able to find the sage function of composition for one variable polynomials, not for nesting two variable power series.masseygirlTue, 05 Dec 2017 18:59:43 -0600http://ask.sagemath.org/question/39960/How can I compose 2 power series in one variable with their compositional inverse get a power series in two variables?http://ask.sagemath.org/question/39739/how-can-i-compose-2-power-series-in-one-variable-with-their-compositional-inverse-get-a-power-series-in-two-variables/I would like to compose a power series $\ell$ defined in $x$ to get a power series $\ell^{-1}(\ell(x) + \ell(y))$ as a power series $f(x, y)$ in two variables, $x$ and $y$. In the code below I call l := $\ell$, and e := $\ell^{-1}$.
PREC = 20
R.<x, y> = PowerSeriesRing( QQ, default_prec=PREC )
f = exp( 1/3 * log( 1-x^3 ) )
print f
w = 1/f
l = w.integral(x)
e = l.reverse()
g = e(l(x) + l(y)) ??
I find immediately the following issue, let alone the issue of composing:
e = l.reverse()
AttributeError: 'MPowerSeriesRing_generic_with_category.element_class' object has no attribute 'reverse'
Once I have this two variable power series $f(x, y)$, I would like to output $f(x, (f(x, ..., f(x,x)))$, composed with itself $n$-times for a natural number $n$.masseygirlWed, 22 Nov 2017 23:38:05 -0600http://ask.sagemath.org/question/39739/Power series with coeffs in SymmetricFunctions()http://ask.sagemath.org/question/37460/power-series-with-coeffs-in-symmetricfunctions/The following gives an error:
sym = SymmetricFunctions(QQ)
R.<t> = PowerSeriesRing(sym)
saying that base_ring is not a commutative ring. I can check sym.categories(), and indeed it is not, but I think it should be. Is there an easy way to make this work?
Thanks!paragonFri, 28 Apr 2017 15:42:00 -0500http://ask.sagemath.org/question/37460/Derivative of O(x^0)http://ask.sagemath.org/question/36301/derivative-of-ox0/Consider a power series
R.<x> = PowerSeries(SR)
f = 1 + O(x^2)
f.derivative()
This gives us `O(x^1)` as we would expect. However, if we do
f = O(x^0)
f.derivative()
we get `O(x^-1)` instead of `O(x^0)` again. Is this a bug or am I missing something?NullInfinitySun, 15 Jan 2017 08:16:30 -0600http://ask.sagemath.org/question/36301/How to compute the multiplicative inverse of a series?http://ask.sagemath.org/question/24904/how-to-compute-the-multiplicative-inverse-of-a-series/After a formula of Sergei N. Gladkovskii the logarithm of
the generating function for the Pell numbers is the
multiplicative inverse of sqrt(2) coth(sqrt(2) x)-1
seen as an exponential generating function.
In the words of Maple:
seq(k!*coeff(series(1/(sqrt(2)*coth(sqrt(2)*x)-1),x=0,k+2),x,k),k=0..9);
0, 1, 2, 2, -8, -56, -112, 848, 9088, 25216
How can I do this with Sage?
**Added:**
If I want a *compositional inverse* this is what I do:
SR -> taylor -> PowerSeriesRing(SR) -> reversion() -> egf_to_ogf() -> padded_list()
If I want a *multiplicative inverse* this is what I want to do:
SR -> taylor -> PowerSeriesRing(SR) -> inversion() -> egf_to_ogf() -> padded_list()
Isn't this what one expects naturally? But I cannot find an 'inversion'.
**Solution d'après kcrisman**
x = SR.var('x')
gf = (sqrt(2)*coth(sqrt(2)*x)-1)^-1
taylor(gf,x,0,9).power_series(SR).egf_to_ogf().padded_list()
Peter LuschnyTue, 18 Nov 2014 08:48:08 -0600http://ask.sagemath.org/question/24904/How to convert a Taylor polynomial to a power series?http://ask.sagemath.org/question/24777/how-to-convert-a-taylor-polynomial-to-a-power-series/With Maple I can write
g := 2/(1+x+sqrt((1+x)*(1-3*x)));
t := taylor(g,x=0,6);
coeffs(convert(t,polynom));
end get
1, 1, 1, 3, 6
Trying to do the same with Sage I tried
var('x')
g = 2/(1+x+sqrt((1+x)*(1-3*x)))
taylor(g, x, 0, n)
and get
NotImplementedError
Wrong arguments passed to taylor. See taylor? for more details.
I could not find the details I am missing by typing 'taylor?'. Then I tried
g = 2/(1+x+sqrt((1+x)*(1-3*x)))
def T(g, n): return taylor(g, x, 0, n)
T(g, 5)
and got
6*x^5 + 3*x^4 + x^3 + x^2 + O(0) + 1
which is almost what I want (although I fail to understand this 'workaround').
But when I tried next to convert this Taylor polynomial to a power series
g = 2/(1+x+sqrt((1+x)*(1-3*x)))
def T(g, n): return taylor(g, x, 0, n)
w = T(g, 5)
R.<x> = QQ[[]]
R(w).polynomial().padded_list(5)
I got the error
TypeError: unable to convert O(0) to a rational
The question: *How can I convert the Taylor polynomial of 2/(1+x+sqrt((1+x)(1-3x))) to a power series and then extract the coefficients?*
*Solution ??*: With the help of the answer of calc314 below (but note that I am not using 'series') the best solution so far seems to be:
var('x')
n = 5
g = 2/(1+x+sqrt((1+x)*(1-3*x)))
p = taylor(g, x, 0, n).truncate()
print p, p.parent()
x = PowerSeriesRing(QQ,'x').gen()
R.<x> = QQ[[]]
P = R(p)
print P, P.parent()
P.padded_list(n)
which gives
6*x^5 + 3*x^4 + x^3 + x^2 + 1 Symbolic Ring
1 + x^2 + x^3 + 3*x^4 + 6*x^5 Power Series Ring in x over Rational Field
[1, 0, 1, 1, 3]
Two minutes later I wanted to wrap things in a function, making 'n' and 'g' parameters.
def GF(g, n):
x = SR.var('x')
p = taylor(g, x, 0, n).truncate()
print p, p.parent()
x = PowerSeriesRing(QQ,'x').gen()
R.<x> = QQ[[]]
P = R(p)
print P, P.parent()
return P.padded_list(n)
Now what do you think
gf = 2/(1+x+sqrt((1+x)*(1-3*x)))
print GF(gf, 5)
gives?
TypeError: unable to convert O(x^20) to a rational
*Round 3, but only small progress:*
tmonteil writes in his answer below: "the lines x = SR.var('x') and x = PowerSeriesRing(QQ,'x').gen() have no effect on the rest of the computation, and could be safely removed".
This does not work for me: if I do not keep the line x = SR.var('x') I get "UnboundLocalError: local variable 'x' referenced before assignment". But the line "x = PowerSeriesRing(QQ,'x').gen()" can be skipped. So I have now
def GF(g, n):
x = SR.var('x')
p = taylor(g, x, 0, n).truncate()
print p, p.parent()
R.<x> = QQ[[]]
P = R(p)
print P, P.parent()
return P.padded_list()
n = 7
gf = 2/(1+x+sqrt((1+x)*(1-3*x)))
print GF(gf, n)
If I open a fresh Sage session, I have, like tmontile, no problem. However I cannot always
open a fresh Sage session when I want to execute GF(gf, n). Therefore consider
n = 7
gf = 2/(1+x+sqrt((1+x)*(1-3*x)))
print GF(gf, n)
p = taylor(gf, x, 0, n).truncate()
print p, p.parent()
R.<x> = QQ[[]]
P = R(p)
print P, P.parent()
P.padded_list()
print GF(gf, n)
If I open a fresh Sage session this will run OK. But when this piece of code is rerun
my old friend will reappear:
"TypeError: unable to convert O(x^20) to a rational"
Peter LuschnyTue, 04 Nov 2014 06:09:26 -0600http://ask.sagemath.org/question/24777/p-adic power serieshttp://ask.sagemath.org/question/10688/p-adic-power-series/Hello,
I am new to Sage and I want to to do some math on p-adic power series. I want to
define such a power series but I did not succeeded. The last line of the
following code rises an exception:
p = 2
q = 4
K = Qp(p)
L.<omega> = Qq(q)
O_L = L.integer_ring()
R.<X> = PowerSeriesRing(O_L)
pi = L.uniformizer()
q = L.residue_class_degree()
f(X) = X^q + pi*X
**TypeError: unsupported operand parent(s) for '*': '2-adic Field with capped
relative precision 20' and 'Symbolic Ring'**
Can someone help me, please? Thanks for your time!
Bye
Larslars.tennstedtSat, 02 Nov 2013 23:23:41 -0500http://ask.sagemath.org/question/10688/Working with formal power serieshttp://ask.sagemath.org/question/10553/working-with-formal-power-series/This is a simplified version of my previous [question](http://ask.sagemath.org/question/3012/integrating-formal-laurent-series).
1) Is it possible to define a formal power series in sage by giving an expression for the n-th coefficient, e.g. as the expression "n" defines the power series 0 + 1 x + 2 x^2 + 3 x^3 + ... n x^n + ... ?
2) Does sage know how to multiply such objects by convolving the terms? Can it anti-differentiate them symbolically?
anilbvFri, 20 Sep 2013 03:17:18 -0500http://ask.sagemath.org/question/10553/integrating formal Laurent serieshttp://ask.sagemath.org/question/10546/integrating-formal-laurent-series/I would like to compute some integrals of products of Laurent series, with the goal of getting an explicit expression for the n-th coefficient of the result. It gets pretty messy so I was hoping that sage could keep track of the details for me. Here are the things I don't know how to do:
1) Define a Laurent series by giving an expression for its n-th coefficient.
2) Formally multiply and integrate Laurent series, producing an explicit expression for the n-th coefficient of the result.
Is this possible? I apologize if some or all of this is explained elsewhere.
EDIT: An example of what I want to do would be to define a power series such as sum(n * x^n,n,0,infinity) and then integrate it and/or multiply it by another power series, resulting in yet another power series whose coefficients I can read off. The O(x^n) notation is not very useful to me since I would like to have an expression for an arbitrary coefficient of that resulting series in terms of n.anilbvWed, 18 Sep 2013 12:39:16 -0500http://ask.sagemath.org/question/10546/multivariate power series computation with recognitionhttp://ask.sagemath.org/question/10080/multivariate-power-series-computation-with-recognition/I would like to do a computation in Sage involving formal power series but I believe some (all?) steps may be impossible:
1. define a couple of formal power series in two variables (x,y) by defining the coefficient function a(m,n) (the coefficients involve gamma functions and such, and are rational). As far as I can see there is no way to do even this step in Sage.
2. do a change of variables, something like (x,y) = (g(x,y),y), obtaining new power series. The function g(x,y) may be a polynomial or possibly itself an infinite power series.
3. multiply the resulting power series
4. Simplify the expression for the coefficients of the resulting power series c(m,n) in terms of classical quantities like gamma functions.
Which of these steps do you believe is possible in Sage? I could write my own software to deal with formal power series properly, but then I would not be able to recognize or simplify the resulting coefficients.
As an example:
1. h(x,y) is a sum of (m!)(4^n)x^m y^n with indices running from 0 to infinity,
and k(x,y) is a sum of (m+n)x^m y^n,
2. do a substitution H(x,y) = h(x,exp(xy)y)
3. multiply H*k
4. simplify the coefficients of H*k. marcoWed, 01 May 2013 19:25:32 -0500http://ask.sagemath.org/question/10080/Laurent series, Rational Functions in sqrt(q)?http://ask.sagemath.org/question/9869/laurent-series-rational-functions-in-sqrtq/I need to construct the ring of formal Laurent series in `q**(1/2)` over the rational numbers. How would I do that in sage?
I realize that there's a perfectly good workaround, but I'd be nonetheless very happy if I didn't have to use it. I could just use Laurent series in another variable, like t,
R.<t> = LaurentSeriesRing(QQ)
I could then define q to be `t**2`, and use t as a formal square root of q. But then I still can't raise q to a non-integer power - sage complains that there's a non-integer in the exponent. I'd make about 500 mistakes just typing in formulas and it would be hard to read the output.
Another workaround which I'm not keen on is to just say
var('q')
and just use symbolic expressions instead. I don't really want to do that either: I like having all the Laurent series methods available and I gather that working in an explicit ring is a lot faster? If I'm misinformed there, then please let me know.
Lastly, I'd also like to construct the rational functions in sqrt(q) - same basic problem, as far as I can see. Any help appreciated.
Benjamin YoungSat, 02 Mar 2013 06:26:10 -0600http://ask.sagemath.org/question/9869/Working with serieshttp://ask.sagemath.org/question/9004/working-with-series/I can't get how to work with series. I do
sage: R.<t> = PowerSeriesRing(QQ)
sage: t^2
t^2
sage: sin(t)
but the last rises an error. I want to do usual manipulations like `sin(t)/cos(t+2)^2`.
yrogirgFri, 01 Jun 2012 09:07:02 -0500http://ask.sagemath.org/question/9004/I want to plot power series with symbolic functionshttp://ask.sagemath.org/question/8922/i-want-to-plot-power-series-with-symbolic-functions/See [this worksheet on sagenb.org](http://www.sagenb.org/home/pub/4482/).
var('x')
f = sin
def P(n,x):
return sum([(-1)^k*x^(2*k+1)/factorial(2*k+1) for k in range(n)])
sinplot = plot(f(x),(x,-2*pi,2*pi),color='red')
@interact
def _(n=(1..10)):
seriesplot = plot(P(n,x),(x,-2*pi,2*pi),color='blue')
html('$P(%s,x) = %s$'%(latex(n),latex(P(n,x))))
show(sinplot+seriesplot,ymin=-4,ymax=4)
Notice that they have to define a *Python* function in order for this to work. I could not get this to work with a Sage callable function like `P(n,x) = sum([...])` for the life of me. I tried lambdas, everything.
Now, likely either
- I've already answered this question somewhere else on the Internet, or
- It's not possible.
But I'd like confirmation of this. It's really annoying that one has to use a Python function to do this.kcrismanTue, 24 Apr 2012 09:16:55 -0500http://ask.sagemath.org/question/8922/Differentiation of a power series.http://ask.sagemath.org/question/8334/differentiation-of-a-power-series/I wonder whether it is too hard to implement the derivative function for multivariate Power Series Ring. Since it is already implemented for one variable, is it not possible to implement it for multivariable, by treating the ring in one var over the ring in other vars.
e.g.
QQ[[x,y,z]] = QQ[[x,y]][z]]
Thanks and regards
--VInayVInay WaghTue, 20 Sep 2011 02:07:06 -0500http://ask.sagemath.org/question/8334/Quotient of ideals in powerseries ringhttp://ask.sagemath.org/question/8333/quotient-of-ideals-in-powerseries-ring/Is it possible to take quotient (colon) of two ideals in a multivariable powerseries ring over a field?
e.g. the following code gives me error(s):
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: I = Ideal([x^2+x*y*z,y^2-z^3*y,z^3+y^5*x*z])
sage: J = Ideal([x])
sage: Q = I.quotient(J)
Thanks and regards
--VInay
VInay WaghMon, 19 Sep 2011 21:17:26 -0500http://ask.sagemath.org/question/8333/generating serieshttp://ask.sagemath.org/question/7706/generating-series/Hi. I want to create a function which will pick out the coefficients of a generating series. In particular, I have a generating series $\sum_{k=0}^\infty a_kt^k$ defined by an infinite product $\prod_{k=1}^\infty P_k(t)$. How do I just pick out the coefficients a_k?
I'm using Sage 4.5.3 on Mac OS X 10.6.4.
Thanks for the help!!!!!!!!!ben122684Wed, 22 Sep 2010 09:32:42 -0500http://ask.sagemath.org/question/7706/