This command determines necessary conditions for the solution of a linear recurrence equation to be analytic, in terms of the initial values.

Example:

LREtools[AnalyticityConditions](n*E-1,E,f(n));

$\left\{f\left(0\right)=0\right\}$

This command finds all d'Alembertian solutions of a linear recurrence equation, that is, solutions annihilated by a product of first order operators.

Example:

rec := (-n-1)*a(n)+(3+2*n)*a(n+1)+(-n-2)*a(n+2)=1/(n+2):

LREtools[hypergeomsols](rec, a(n), {}, output=basis);

LREtools[dAlembertiansols](rec, a(n), {}, output=basis);

$\left[\left[1\,\sum _{\mathrm{n1}=0}^{n-1}\frac{1}{\mathrm{n1}+1}\right]\,-\left(\sum _{\mathrm{n1}=0}^{n-1}\frac{\sum _{\mathrm{n2}=0}^{\mathrm{n1}-1}\frac{1}{\mathrm{n2}+2}}{\mathrm{n1}+1}\right)\right]$

This command finds a multiple $M$ of a given linear recurrence operator $L$, if possible, such that the leading or trailing coefficient of $M$ has no integer roots.

Example:

LREtools[IsDesingularizable]((n-1)*E+n,E,n,trailing,output=operator);

$\mathrm{true},E^2+2E+1$

This command factors a given set of polynomials as far as possible by using only gcds. This can be used, for example, to refine several partial factorizations of the same polynomial.

Example:

The following is a way to compute the factors in the squarefree decomposition of a polynomial.

f := x^9-x^7-x^5+x^3;

$f≔x^9-x^7-x^5+x^3$

PolynomialTools[GcdFreeBasis]([f, gcd(f,diff(f,x))]);

$\left[x\,x^2-1\,x^2+1\right]$

sqrfree(f);

$\left[1\,\left[\left[x\,3\right]\,\left[x^2+1\,1\right]\,\left[x^2-1\,2\right]\right]\right]$

This command factors a univariate polynomial into a product of falling factorials in a unique way similar to the squarefree factorization.

Example:

f := x^9-x^7-x^5+x^3;

$f≔x^9-x^7-x^5+x^3$

PolynomialTools[GreatestFactorialFactorization](f,x);

$\left[1\,\left[\left[x\left(x^2+1\right)\,1\right]\,\left[x^2+2x+1\,3\right]\right]\right]$

This command determines whether a univariate polynomial is a Taylor shift of another polynomial, and if so, returns the shift distance.

Examples:

PolynomialTools[ShiftEquivalent](2*x+1,2*x+2,x);

$\frac{1}{2}$

PolynomialTools[ShiftEquivalent](2*x+1,2*x+2,x,'integer');

$\mathrm{FAIL}$

PolynomialTools[ShiftEquivalent](2*x+1,2*x+5,x,'integer');

$2$

PolynomialTools[ShiftEquivalent](x^2-1,x^2+1,x);

$\mathrm{FAIL}$

This command computes the coarsest factorization of a univariate polynomial separating the irreducible factors both by their multiplicities and their shift equivalence classes.

Example:

f := x^9-x^7-x^5+x^3;

$f≔x^9-x^7-x^5+x^3$

PolynomialTools[ShiftlessDecomposition](f,x);

$\left[1\,\left[\left[x-1\,\left[\left[0\,2\right]\,\left[1\,3\right]\,\left[2\,2\right]\right]\right]\,\left[x^2+1\,\left[\left[0\,1\right]\right]\right]\right]\right]$

The StringTools package includes the new Length command that returns the length of a string.

The new MaximalPalindromicSubstring command that finds a maximal substring that is equal to itself reversed.

You can use the new StringTools[Randomize] command to seed the random number generator used by the external code underlying the StringTools package.

Most interactive tutors in the Student package now include a field that shows the non-tutor command, which can be used to produce the same result in the worksheet. This can be used as a starting point for further exploration.

The Student[Precalculus] package has the addition of non-tutor commands in allowing exploration of concepts in greater depth.  The basic Line command computes lines from various different forms of input and returns information about slope and intercepts, as well as the line itself.