Concrete Mathematics

Hypergeometric terms

The center stage, in recurrence solving and summations, play hypergeometric terms. Formally these are sequences annihilated by first order linear recurrence operators. In simple words if we are given term \(a(n)\) then it is hypergeometric if its consecutive term ratio is a rational function in \(n\).

To check if a sequence is of this type you can use the is_hypergeometric method which is available in Basic class. Here is simple example involving a polynomial:

>>> from diofant import *
>>> n, k = symbols('n k')
>>> (n**2 + 1).is_hypergeometric(n)
True

Of course polynomials are hypergeometric but are there any more complicated sequences of this type? Here are some trivial examples:

>>> factorial(n).is_hypergeometric(n)
True
>>> binomial(n, k).is_hypergeometric(n)
True
>>> rf(n, k).is_hypergeometric(n)
True
>>> ff(n, k).is_hypergeometric(n)
True
>>> gamma(n).is_hypergeometric(n)
True
>>> (2**n).is_hypergeometric(n)
True

We see that all species used in summations and other parts of concrete mathematics are hypergeometric. Note also that binomial coefficients and both rising and falling factorials are hypergeometric in both their arguments:

>>> binomial(n, k).is_hypergeometric(k)
True
>>> rf(n, k).is_hypergeometric(k)
True
>>> ff(n, k).is_hypergeometric(k)
True

To say more, all previously shown examples are valid for integer linear arguments:

>>> factorial(2*n).is_hypergeometric(n)
True
>>> binomial(3*n+1, k).is_hypergeometric(n)
True
>>> rf(n+1, k-1).is_hypergeometric(n)
True
>>> ff(n-1, k+1).is_hypergeometric(n)
True
>>> gamma(5*n).is_hypergeometric(n)
True
>>> (2**(n-7)).is_hypergeometric(n)
True

However nonlinear arguments make those sequences fail to be hypergeometric:

>>> factorial(n**2).is_hypergeometric(n)
False
>>> (2**(n**3 + 1)).is_hypergeometric(n)
False

If not only the knowledge of being hypergeometric or not is needed, you can use hypersimp() function. It will try to simplify combinatorial expression and if the term given is hypergeometric it will return a quotient of polynomials of minimal degree. Otherwise is will return \(None\) to say that sequence is not hypergeometric:

>>> hypersimp(factorial(2*n), n)
2*(n + 1)*(2*n + 1)
>>> hypersimp(factorial(n**2), n)

Concrete Class Reference

class diofant.concrete.summations.Sum[source]

Represents unevaluated summation.

Sum represents a finite or infinite series, with the first argument being the general form of terms in the series, and the second argument being (dummy_variable, start, end), with dummy_variable taking all integer values from start through end. In accordance with long-standing mathematical convention, the end term is included in the summation.

For finite sums (and sums with symbolic limits assumed to be finite) we follow the summation convention described by Karr [1], especially definition 3 of section 1.4. The sum:

\[\sum_{m \leq i < n} f(i)\]

has the obvious meaning for \(m < n\), namely:

\[\sum_{m \leq i < n} f(i) = f(m) + f(m+1) + \ldots + f(n-2) + f(n-1)\]

with the upper limit value \(f(n)\) excluded. The sum over an empty set is zero if and only if \(m = n\):

\[\sum_{m \leq i < n} f(i) = 0 \quad \mathrm{for} \quad m = n\]

Finally, for all other sums over empty sets we assume the following definition:

\[\sum_{m \leq i < n} f(i) = - \sum_{n \leq i < m} f(i) \quad \mathrm{for} \quad m > n\]

It is important to note that Karr defines all sums with the upper limit being exclusive. This is in contrast to the usual mathematical notation, but does not affect the summation convention. Indeed we have:

\[\sum_{m \leq i < n} f(i) = \sum_{i = m}^{n - 1} f(i)\]

where the difference in notation is intentional to emphasize the meaning, with limits typeset on the top being inclusive.

References

[R132136]Michael Karr, “Summation in Finite Terms”, Journal of the ACM, Volume 28 Issue 2, April 1981, Pages 305-350 https://dl.acm.org/citation.cfm?doid=322248.322255
[R133136]https://en.wikipedia.org/wiki/Summation#Capital-sigma_notation
[R134136]https://en.wikipedia.org/wiki/Empty_sum

Examples

>>> from diofant.abc import i, k, m, n, x
>>> from diofant import factorial, oo, IndexedBase, Function
>>> Sum(k, (k, 1, m))
Sum(k, (k, 1, m))
>>> Sum(k, (k, 1, m)).doit()
m**2/2 + m/2
>>> Sum(k**2, (k, 1, m))
Sum(k**2, (k, 1, m))
>>> Sum(k**2, (k, 1, m)).doit()
m**3/3 + m**2/2 + m/6
>>> Sum(x**k, (k, 0, oo))
Sum(x**k, (k, 0, oo))
>>> Sum(x**k, (k, 0, oo)).doit()
Piecewise((1/(-x + 1), Abs(x) < 1), (Sum(x**k, (k, 0, oo)), true))
>>> Sum(x**k/factorial(k), (k, 0, oo)).doit()
E**x

Here are examples to do summation with symbolic indices. You can use either Function of IndexedBase classes:

>>> f = Function('f')
>>> Sum(f(n), (n, 0, 3)).doit()
f(0) + f(1) + f(2) + f(3)
>>> Sum(f(n), (n, 0, oo)).doit()
Sum(f(n), (n, 0, oo))
>>> f = IndexedBase('f')
>>> Sum(f[n]**2, (n, 0, 3)).doit()
f[0]**2 + f[1]**2 + f[2]**2 + f[3]**2

An example showing that the symbolic result of a summation is still valid for seemingly nonsensical values of the limits. Then the Karr convention allows us to give a perfectly valid interpretation to those sums by interchanging the limits according to the above rules:

>>> S = Sum(i, (i, 1, n)).doit()
>>> S
n**2/2 + n/2
>>> S.subs(n, -4)
6
>>> Sum(i, (i, 1, -4)).doit()
6
>>> Sum(-i, (i, -3, 0)).doit()
6

An explicit example of the Karr summation convention:

>>> S1 = Sum(i**2, (i, m, m+n-1)).doit()
>>> S1
m**2*n + m*n**2 - m*n + n**3/3 - n**2/2 + n/6
>>> S2 = Sum(i**2, (i, m+n, m-1)).doit()
>>> S2
-m**2*n - m*n**2 + m*n - n**3/3 + n**2/2 - n/6
>>> S1 + S2
0
>>> S3 = Sum(i, (i, m, m-1)).doit()
>>> S3
0
euler_maclaurin(m=0, n=0, eps=0, eval_integral=True)[source]

Return an Euler-Maclaurin approximation of self, where m is the number of leading terms to sum directly and n is the number of terms in the tail.

With m = n = 0, this is simply the corresponding integral plus a first-order endpoint correction.

Returns (s, e) where s is the Euler-Maclaurin approximation and e is the estimated error (taken to be the magnitude of the first omitted term in the tail):

>>> from diofant.abc import k, a, b
>>> Sum(1/k, (k, 2, 5)).doit().evalf()
1.28333333333333
>>> s, e = Sum(1/k, (k, 2, 5)).euler_maclaurin()
>>> s
-log(2) + 7/20 + log(5)
>>> from diofant import sstr
>>> print(sstr((s.evalf(), e.evalf()), full_prec=True))
(1.26629073187415, 0.0175000000000000)

The endpoints may be symbolic:

>>> s, e = Sum(1/k, (k, a, b)).euler_maclaurin()
>>> s
-log(a) + log(b) + 1/(2*b) + 1/(2*a)
>>> e
Abs(1/(12*b**2) - 1/(12*a**2))

If the function is a polynomial of degree at most 2n+1, the Euler-Maclaurin formula becomes exact (and e = 0 is returned):

>>> Sum(k, (k, 2, b)).euler_maclaurin()
(b**2/2 + b/2 - 1, 0)
>>> Sum(k, (k, 2, b)).doit()
b**2/2 + b/2 - 1

With a nonzero \(eps\) specified, the summation is ended as soon as the remainder term is less than the epsilon.

findrecur(F=Function('F'), n=None)[source]

Find a recurrence formula for the summand of the sum.

Given a sum \(f(n) = \sum_k F(n, k)\), where \(F(n, k)\) is doubly hypergeometric (that’s, both \(F(n + 1, k)/F(n, k)\) and \(F(n, k + 1)/F(n, k)\) are rational functions of \(n\) and \(k\)), we find a recurrence for the summand \(F(n, k)\) of the form

\[\sum_{i=0}^I\sum_{j=0}^J a_{i,j}F(n - j, k - i) = 0\]

Notes

We use Sister Celine’s algorithm, see [R135136141].

References

[R135136141](1, 2)
  1. Petkovšek, H. S. Wilf, D. Zeilberger, A = B, 1996, Ch. 4.

Examples

>>> from diofant import symbols, factorial, oo
>>> n, k = symbols('n, k', integer=True)
>>> s = Sum(factorial(n)/(factorial(k)*factorial(n - k)), (k, 0, oo))
>>> s.findrecur()
-F(n, k) + F(n - 1, k) + F(n - 1, k - 1)
reverse_order(*indices)[source]

Reverse the order of a limit in a Sum.

Parameters:

*indices : list

The selectors in the argument indices specify some indices whose limits get reversed. These selectors are either variable names or numerical indices counted starting from the inner-most limit tuple.

References

[R136136142]Michael Karr, “Summation in Finite Terms”, Journal of the ACM, Volume 28 Issue 2, April 1981, Pages 305-350 https://dl.acm.org/citation.cfm?doid=322248.322255

Examples

>>> from diofant.abc import x, y, a, b, c, d
>>> Sum(x, (x, 0, 3)).reverse_order(x)
Sum(-x, (x, 4, -1))
>>> Sum(x*y, (x, 1, 5), (y, 0, 6)).reverse_order(x, y)
Sum(x*y, (x, 6, 0), (y, 7, -1))
>>> Sum(x, (x, a, b)).reverse_order(x)
Sum(-x, (x, b + 1, a - 1))
>>> Sum(x, (x, a, b)).reverse_order(0)
Sum(-x, (x, b + 1, a - 1))

While one should prefer variable names when specifying which limits to reverse, the index counting notation comes in handy in case there are several symbols with the same name.

>>> S = Sum(x**2, (x, a, b), (x, c, d))
>>> S
Sum(x**2, (x, a, b), (x, c, d))
>>> S0 = S.reverse_order(0)
>>> S0
Sum(-x**2, (x, b + 1, a - 1), (x, c, d))
>>> S1 = S0.reverse_order(1)
>>> S1
Sum(x**2, (x, b + 1, a - 1), (x, d + 1, c - 1))

Of course we can mix both notations:

>>> Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(x, 1)
Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
>>> Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(y, x)
Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
class diofant.concrete.products.Product[source]

Represents unevaluated products.

Product represents a finite or infinite product, with the first argument being the general form of terms in the series, and the second argument being (dummy_variable, start, end), with dummy_variable taking all integer values from start through end. In accordance with long-standing mathematical convention, the end term is included in the product.

For finite products (and products with symbolic limits assumed to be finite) we follow the analogue of the summation convention described by Karr [1], especially definition 3 of section 1.4. The product:

\[\prod_{m \leq i < n} f(i)\]

has the obvious meaning for \(m < n\), namely:

\[\prod_{m \leq i < n} f(i) = f(m) f(m+1) \cdot \ldots \cdot f(n-2) f(n-1)\]

with the upper limit value \(f(n)\) excluded. The product over an empty set is one if and only if \(m = n\):

\[\prod_{m \leq i < n} f(i) = 1 \quad \mathrm{for} \quad m = n\]

Finally, for all other products over empty sets we assume the following definition:

\[\prod_{m \leq i < n} f(i) = \frac{1}{\prod_{n \leq i < m} f(i)} \quad \mathrm{for} \quad m > n\]

It is important to note that above we define all products with the upper limit being exclusive. This is in contrast to the usual mathematical notation, but does not affect the product convention. Indeed we have:

\[\prod_{m \leq i < n} f(i) = \prod_{i = m}^{n - 1} f(i)\]

where the difference in notation is intentional to emphasize the meaning, with limits typeset on the top being inclusive.

References

[R144147]Michael Karr, “Summation in Finite Terms”, Journal of the ACM, Volume 28 Issue 2, April 1981, Pages 305-350 https://dl.acm.org/citation.cfm?doid=322248.322255
[R145147]https://en.wikipedia.org/wiki/Multiplication#Capital_Pi_notation
[R146147]https://en.wikipedia.org/wiki/Empty_product

Examples

>>> from diofant.abc import a, b, i, k, m, n, x
>>> from diofant import factorial, oo
>>> Product(k, (k, 1, m))
Product(k, (k, 1, m))
>>> Product(k, (k, 1, m)).doit()
factorial(m)
>>> Product(k**2, (k, 1, m))
Product(k**2, (k, 1, m))
>>> Product(k**2, (k, 1, m)).doit()
(factorial(m))**2

Wallis’ product for pi:

>>> W = Product(2*i/(2*i-1) * 2*i/(2*i+1), (i, 1, oo))
>>> W
Product(4*i**2/((2*i - 1)*(2*i + 1)), (i, 1, oo))

Direct computation currently fails:

>>> W.doit()
Product(4*i**2/((2*i - 1)*(2*i + 1)), (i, 1, oo))

But we can approach the infinite product by a limit of finite products:

>>> from diofant import limit
>>> W2 = Product(2*i/(2*i-1)*2*i/(2*i+1), (i, 1, n))
>>> W2
Product(4*i**2/((2*i - 1)*(2*i + 1)), (i, 1, n))
>>> W2e = W2.doit()
>>> W2e
2**(-2*n)*4**n*(factorial(n))**2/(RisingFactorial(1/2, n)*RisingFactorial(3/2, n))
>>> limit(W2e, n, oo)
pi/2

By the same formula we can compute sin(pi/2):

>>> from diofant import pi, gamma, simplify
>>> P = pi * x * Product(1 - x**2/k**2, (k, 1, n))
>>> P = P.subs(x, pi/2)
>>> P
pi**2*Product(1 - pi**2/(4*k**2), (k, 1, n))/2
>>> Pe = P.doit()
>>> Pe
pi**2*RisingFactorial(1 + pi/2, n)*RisingFactorial(-pi/2 + 1, n)/(2*(factorial(n))**2)
>>> Pe = Pe.rewrite(gamma)
>>> Pe
pi**2*gamma(n + 1 + pi/2)*gamma(n - pi/2 + 1)/(2*gamma(1 + pi/2)*gamma(-pi/2 + 1)*gamma(n + 1)**2)
>>> Pe = simplify(Pe)
>>> Pe
sin(pi**2/2)*gamma(n + 1 + pi/2)*gamma(n - pi/2 + 1)/gamma(n + 1)**2
>>> limit(Pe, n, oo)
sin(pi**2/2)

Products with the lower limit being larger than the upper one:

>>> Product(1/i, (i, 6, 1)).doit()
120
>>> Product(i, (i, 2, 5)).doit()
120

The empty product:

>>> Product(i, (i, n, n-1)).doit()
1

An example showing that the symbolic result of a product is still valid for seemingly nonsensical values of the limits. Then the Karr convention allows us to give a perfectly valid interpretation to those products by interchanging the limits according to the above rules:

>>> P = Product(2, (i, 10, n)).doit()
>>> P
2**(n - 9)
>>> P.subs(n, 5)
1/16
>>> Product(2, (i, 10, 5)).doit()
1/16
>>> 1/Product(2, (i, 6, 9)).doit()
1/16

An explicit example of the Karr summation convention applied to products:

>>> P1 = Product(x, (i, a, b)).doit()
>>> P1
x**(-a + b + 1)
>>> P2 = Product(x, (i, b+1, a-1)).doit()
>>> P2
x**(a - b - 1)
>>> simplify(P1 * P2)
1

And another one:

>>> P1 = Product(i, (i, b, a)).doit()
>>> P1
RisingFactorial(b, a - b + 1)
>>> P2 = Product(i, (i, a+1, b-1)).doit()
>>> P2
RisingFactorial(a + 1, -a + b - 1)
>>> P1 * P2
RisingFactorial(b, a - b + 1)*RisingFactorial(a + 1, -a + b - 1)
>>> simplify(P1 * P2)
1
reverse_order(*indices)[source]

Reverse the order of a limit in a Product.

Parameters:

*indices : list

The selectors in the argument indices specify some indices whose limits get reversed. These selectors are either variable names or numerical indices counted starting from the inner-most limit tuple.

References

[R147147151]Michael Karr, “Summation in Finite Terms”, Journal of the ACM, Volume 28 Issue 2, April 1981, Pages 305-350 https://dl.acm.org/citation.cfm?doid=322248.322255

Examples

>>> from diofant import simplify, RisingFactorial, gamma, Sum
>>> from diofant.abc import x, y, a, b, c, d
>>> P = Product(x, (x, a, b))
>>> Pr = P.reverse_order(x)
>>> Pr
Product(1/x, (x, b + 1, a - 1))
>>> Pr = Pr.doit()
>>> Pr
1/RisingFactorial(b + 1, a - b - 1)
>>> simplify(Pr)
gamma(b + 1)/gamma(a)
>>> P = P.doit()
>>> P
RisingFactorial(a, -a + b + 1)
>>> simplify(P)
gamma(b + 1)/gamma(a)

While one should prefer variable names when specifying which limits to reverse, the index counting notation comes in handy in case there are several symbols with the same name.

>>> S = Sum(x*y, (x, a, b), (y, c, d))
>>> S
Sum(x*y, (x, a, b), (y, c, d))
>>> S0 = S.reverse_order(0)
>>> S0
Sum(-x*y, (x, b + 1, a - 1), (y, c, d))
>>> S1 = S0.reverse_order(1)
>>> S1
Sum(x*y, (x, b + 1, a - 1), (y, d + 1, c - 1))

Of course we can mix both notations:

>>> Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(x, 1)
Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
>>> Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(y, x)
Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
class diofant.concrete.expr_with_limits.ExprWithLimits[source]
as_dummy()[source]

Replace instances of the given dummy variables with explicit dummy counterparts to make clear what are dummy variables and what are real-world symbols in an object.

See also

diofant.concrete.expr_with_limits.ExprWithLimits.variables
Lists the integration variables

Examples

>>> from diofant import Integral
>>> from diofant.abc import x, y
>>> Integral(x, (x, x, y), (y, x, y)).as_dummy()
Integral(_x, (_x, x, _y), (_y, x, y))

If the object supports the “integral at” limit (x,) it is not treated as a dummy, but the explicit form, (x, x) of length 2 does treat the variable as a dummy.

>>> Integral(x, x).as_dummy()
Integral(x, x)
>>> Integral(x, (x, x)).as_dummy()
Integral(_x, (_x, x))

If there were no dummies in the original expression, then the the symbols which cannot be changed by subs() are clearly seen as those with an underscore prefix.

free_symbols

This method returns the symbols in the object, excluding those that take on a specific value (i.e. the dummy symbols).

Examples

>>> from diofant import Sum
>>> from diofant.abc import x, y
>>> Sum(x, (x, y, 1)).free_symbols
{y}
function

Return the function applied across limits.

Examples

>>> from diofant import Integral
>>> from diofant.abc import x
>>> Integral(x**2, x).function
x**2
is_number

Return True if the Sum has no free symbols, else False.

limits

Return the limits of expression.

Examples

>>> from diofant import Integral
>>> from diofant.abc import x, i
>>> Integral(x**i, (i, 1, 3)).limits
((i, 1, 3),)
variables

Return a list of the dummy variables

>>> from diofant import Sum
>>> from diofant.abc import x, i
>>> Sum(x**i, (i, 1, 3)).variables
[i]
class diofant.concrete.expr_with_intlimits.ExprWithIntLimits[source]
change_index(var, trafo, newvar=None)[source]

Change index of a Sum or Product.

Perform a linear transformation \(x \mapsto a x + b\) on the index variable \(x\). For \(a\) the only values allowed are \(\pm 1\). A new variable to be used after the change of index can also be specified.

Parameters:

var : Symbol

specifies the index variable \(x\) to transform.

trafo : Expr

The linear transformation in terms of var.

newvar : Symbol, optional

Replacement symbol to be used instead of var in the final expression.

Examples

>>> from diofant import Sum, Product, simplify
>>> from diofant.abc import x, y, a, b, c, d, u, v, i, j, k, l
>>> S = Sum(x, (x, a, b))
>>> S.doit()
-a**2/2 + a/2 + b**2/2 + b/2
>>> Sn = S.change_index(x, x + 1, y)
>>> Sn
Sum(y - 1, (y, a + 1, b + 1))
>>> Sn.doit()
-a**2/2 + a/2 + b**2/2 + b/2
>>> Sn = S.change_index(x, -x, y)
>>> Sn
Sum(-y, (y, -b, -a))
>>> Sn.doit()
-a**2/2 + a/2 + b**2/2 + b/2
>>> Sn = S.change_index(x, x+u)
>>> Sn
Sum(-u + x, (x, a + u, b + u))
>>> Sn.doit()
-a**2/2 - a*u + a/2 + b**2/2 + b*u + b/2 - u*(-a + b + 1) + u
>>> simplify(Sn.doit())
-a**2/2 + a/2 + b**2/2 + b/2
>>> Sn = S.change_index(x, -x - u, y)
>>> Sn
Sum(-u - y, (y, -b - u, -a - u))
>>> Sn.doit()
-a**2/2 - a*u + a/2 + b**2/2 + b*u + b/2 - u*(-a + b + 1) + u
>>> simplify(Sn.doit())
-a**2/2 + a/2 + b**2/2 + b/2
>>> P = Product(i*j**2, (i, a, b), (j, c, d))
>>> P
Product(i*j**2, (i, a, b), (j, c, d))
>>> P2 = P.change_index(i, i+3, k)
>>> P2
Product(j**2*(k - 3), (k, a + 3, b + 3), (j, c, d))
>>> P3 = P2.change_index(j, -j, l)
>>> P3
Product(l**2*(k - 3), (k, a + 3, b + 3), (l, -d, -c))

When dealing with symbols only, we can make a general linear transformation:

>>> Sn = S.change_index(x, u*x+v, y)
>>> Sn
Sum((-v + y)/u, (y, b*u + v, a*u + v))
>>> Sn.doit()
-v*(a*u - b*u + 1)/u + (a**2*u**2/2 + a*u*v + a*u/2 - b**2*u**2/2 - b*u*v + b*u/2 + v)/u
>>> simplify(Sn.doit())
a**2*u/2 + a/2 - b**2*u/2 + b/2

However, the last result can be inconsistent with usual summation where the index increment is always 1. This is obvious as we get back the original value only for u equal +1 or -1.

index(x)[source]

Return the index of a dummy variable in the list of limits.

Note that we start counting with 0 at the inner-most limits tuple.

Parameters:

x : Symbol

a dummy variable

Examples

>>> from diofant.abc import x, y, a, b, c, d
>>> from diofant import Sum, Product
>>> Sum(x*y, (x, a, b), (y, c, d)).index(x)
0
>>> Sum(x*y, (x, a, b), (y, c, d)).index(y)
1
>>> Product(x*y, (x, a, b), (y, c, d)).index(x)
0
>>> Product(x*y, (x, a, b), (y, c, d)).index(y)
1
reorder(*arg)[source]

Reorder limits in a expression containing a Sum or a Product.

Parameters:

*arg : list of tuples

These tuples can contain numerical indices or index variable names or involve both.

Examples

>>> from diofant import Sum, Product
>>> from diofant.abc import x, y, z, a, b, c, d, e, f
>>> Sum(x*y, (x, a, b), (y, c, d)).reorder((x, y))
Sum(x*y, (y, c, d), (x, a, b))
>>> Sum(x*y*z, (x, a, b), (y, c, d), (z, e, f)).reorder((x, y), (x, z), (y, z))
Sum(x*y*z, (z, e, f), (y, c, d), (x, a, b))
>>> P = Product(x*y*z, (x, a, b), (y, c, d), (z, e, f))
>>> P.reorder((x, y), (x, z), (y, z))
Product(x*y*z, (z, e, f), (y, c, d), (x, a, b))

We can also select the index variables by counting them, starting with the inner-most one:

>>> Sum(x**2, (x, a, b), (x, c, d)).reorder((0, 1))
Sum(x**2, (x, c, d), (x, a, b))

And of course we can mix both schemes:

>>> Sum(x*y, (x, a, b), (y, c, d)).reorder((y, x))
Sum(x*y, (y, c, d), (x, a, b))
>>> Sum(x*y, (x, a, b), (y, c, d)).reorder((y, 0))
Sum(x*y, (y, c, d), (x, a, b))
reorder_limit(x, y)[source]

Interchange two limit tuples of a Sum or Product expression.

Parameters:

x, y: int

are integers corresponding to the index variables of the two limits which are to be interchanged.

Examples

>>> from diofant.abc import x, y, z, a, b, c, d, e, f
>>> from diofant import Sum, Product
>>> Sum(x*y*z, (x, a, b), (y, c, d), (z, e, f)).reorder_limit(0, 2)
Sum(x*y*z, (z, e, f), (y, c, d), (x, a, b))
>>> Sum(x**2, (x, a, b), (x, c, d)).reorder_limit(1, 0)
Sum(x**2, (x, c, d), (x, a, b))
>>> Product(x*y*z, (x, a, b), (y, c, d), (z, e, f)).reorder_limit(0, 2)
Product(x*y*z, (z, e, f), (y, c, d), (x, a, b))

Concrete Functions Reference

diofant.concrete.summations.summation(f, *symbols, **kwargs)[source]

Compute the summation of f with respect to symbols.

The notation for symbols is similar to the notation used in Integral. summation(f, (i, a, b)) computes the sum of f with respect to i from a to b, i.e.,

                            b
                          ____
                          \   `
summation(f, (i, a, b)) =  )    f
                          /___,
                          i = a

If it cannot compute the sum, it returns an unevaluated Sum object. Repeated sums can be computed by introducing additional symbols tuples:

>>> from diofant import oo, symbols, log
>>> i, n, m = symbols('i n m', integer=True)
>>> summation(2*i - 1, (i, 1, n))
n**2
>>> summation(1/2**i, (i, 0, oo))
2
>>> summation(1/log(n)**n, (n, 2, oo))
Sum(log(n)**(-n), (n, 2, oo))
>>> summation(i, (i, 0, n), (n, 0, m))
m**3/6 + m**2/2 + m/3
>>> from diofant.abc import x
>>> from diofant import factorial
>>> summation(x**n/factorial(n), (n, 0, oo))
E**x
diofant.concrete.products.product(*args, **kwargs)[source]

Compute the product.

The notation for symbols is similar to the notation used in Sum or Integral. product(f, (i, a, b)) computes the product of f with respect to i from a to b, i.e.,

                             b
                           _____
product(f(n), (i, a, b)) = |   | f(n)
                           |   |
                           i = a

If it cannot compute the product, it returns an unevaluated Product object. Repeated products can be computed by introducing additional symbols tuples:

>>> from diofant import symbols
>>> i, n, m, k = symbols('i n m k', integer=True)
>>> product(i, (i, 1, k))
factorial(k)
>>> product(m, (i, 1, k))
m**k
>>> product(i, (i, 1, k), (k, 1, n))
Product(factorial(k), (k, 1, n))
diofant.concrete.gosper.gosper_normal(f, g, n, polys=True)[source]

Compute the Gosper’s normal form of f and g.

Given relatively prime univariate polynomials f and g, rewrite their quotient to a normal form defined as follows:

\[\frac{f(n)}{g(n)} = Z \cdot \frac{A(n) C(n+1)}{B(n) C(n)}\]

where Z is an arbitrary constant and A, B, C are monic polynomials in n with the following properties:

  1. \(\gcd(A(n), B(n+h)) = 1 \forall h \in \mathbb{N}\)
  2. \(\gcd(B(n), C(n+1)) = 1\)
  3. \(\gcd(A(n), C(n)) = 1\)

This normal form, or rational factorization in other words, is a crucial step in Gosper’s algorithm and in solving of difference equations. It can be also used to decide if two hypergeometric terms are similar or not.

This procedure will return a tuple containing elements of this factorization in the form (Z*A, B, C).

Examples

>>> from diofant.abc import n
>>> gosper_normal(4*n + 5, 2*(4*n + 1)*(2*n + 3), n, polys=False)
(1/4, n + 3/2, n + 1/4)
diofant.concrete.gosper.gosper_term(f, n)[source]

Compute Gosper’s hypergeometric term for f.

Suppose f is a hypergeometric term such that:

\[s_n = \sum_{k=0}^{n-1} f_k\]

and \(f_k\) doesn’t depend on \(n\). Returns a hypergeometric term \(g_n\) such that \(g_{n+1} - g_n = f_n\).

Examples

>>> from diofant.functions import factorial
>>> from diofant.abc import n
>>> gosper_term((4*n + 1)*factorial(n)/factorial(2*n + 1), n)
(-n - 1/2)/(n + 1/4)
diofant.concrete.gosper.gosper_sum(f, k)[source]

Gosper’s hypergeometric summation algorithm.

Given a hypergeometric term f such that:

\[s_n = \sum_{k=0}^{n-1} f_k\]

and \(f(n)\) doesn’t depend on \(n\), returns \(g_{n} - g(0)\) where \(g_{n+1} - g_n = f_n\), or None if \(s_n\) can not be expressed in closed form as a sum of hypergeometric terms.

References

[R153153]Marko Petkovšek, Herbert S. Wilf, Doron Zeilberger, A = B, AK Peters, Ltd., Wellesley, MA, USA, 1997, pp. 73–100

Examples

>>> from diofant.functions import factorial
>>> from diofant.abc import i, n, k
>>> f = (4*k + 1)*factorial(k)/factorial(2*k + 1)
>>> gosper_sum(f, (k, 0, n))
(-factorial(n) + 2*factorial(2*n + 1))/factorial(2*n + 1)
>>> _.subs(n, 2) == sum(f.subs(k, i) for i in [0, 1, 2])
True
>>> gosper_sum(f, (k, 3, n))
(-60*factorial(n) + factorial(2*n + 1))/(60*factorial(2*n + 1))
>>> _.subs(n, 5) == sum(f.subs(k, i) for i in [3, 4, 5])
True