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 sympy 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)
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.
See also
summation, Product, product
References
[R15] | Michael Karr, “Summation in Finite Terms”, Journal of the ACM, Volume 28 Issue 2, April 1981, Pages 305-350 http://dl.acm.org/citation.cfm?doid=322248.322255 |
[R16] | http://en.wikipedia.org/wiki/Summation#Capital-sigma_notation |
[R17] | http://en.wikipedia.org/wiki/Empty_sum |
Examples
>>> from sympy.abc import i, k, m, n, x
>>> from sympy import Sum, factorial, oo
>>> 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()
exp(x)
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
Finite Sums
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:
has the obvious meaning for \(m < n\), namely:
with the upper limit value \(f(n)\) excluded. The sum over an empty set is zero if and only if \(m = n\):
Finally, for all other sums over empty sets we assume the following definition:
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:
where the difference in notation is intentional to emphasize the meaning, with limits typeset on the top being inclusive.
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 sympy.abc import k, a, b
>>> from sympy import Sum
>>> 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 sympy 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.
Reverse the order of a limit in a Sum.
See also
index, reorder_limit, reorder
References
[R18] | Michael Karr, “Summation in Finite Terms”, Journal of the ACM, Volume 28 Issue 2, April 1981, Pages 305-350 http://dl.acm.org/citation.cfm?doid=322248.322255 |
Examples
>>> from sympy import Sum
>>> from sympy.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))
Usage
reverse_order(self, *indices) reverses some limits in the expression self which can be either a Sum or a Product. 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.
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.
See also
Sum, summation, product
References
[R19] | Michael Karr, “Summation in Finite Terms”, Journal of the ACM, Volume 28 Issue 2, April 1981, Pages 305-350 http://dl.acm.org/citation.cfm?doid=322248.322255 |
[R20] | http://en.wikipedia.org/wiki/Multiplication#Capital_Pi_notation |
[R21] | http://en.wikipedia.org/wiki/Empty_product |
Examples
>>> from sympy.abc import a, b, i, k, m, n, x
>>> from sympy import Product, 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()
nan
But we can approach the infinite product by a limit of finite products:
>>> from sympy 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 sympy 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
Finite Products
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:
has the obvious meaning for \(m < n\), namely:
with the upper limit value \(f(n)\) excluded. The product over an empty set is one if and only if \(m = n\):
Finally, for all other products over empty sets we assume the following definition:
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:
where the difference in notation is intentional to emphasize the meaning, with limits typeset on the top being inclusive.
Reverse the order of a limit in a Product.
See also
index, reorder_limit, reorder
References
[R22] | Michael Karr, “Summation in Finite Terms”, Journal of the ACM, Volume 28 Issue 2, April 1981, Pages 305-350 http://dl.acm.org/citation.cfm?doid=322248.322255 |
Examples
>>> from sympy import Product, simplify, RisingFactorial, gamma, Sum
>>> from sympy.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))
Usage
reverse_order(expr, *indices) reverses some limits in the expression expr which can be either a Sum or a Product. 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.
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 sympy import summation, 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 sympy.abc import x
>>> from sympy import factorial
>>> summation(x**n/factorial(n), (n, 0, oo))
exp(x)
See also
Sum, Product, product
Compute the product.
The notation for symbols is similiar 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 sympy import product, 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))
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:
where Z is an arbitrary constant and A, B, C are monic polynomials in n with the following properties:
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 sympy.concrete.gosper import gosper_normal
>>> from sympy.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)
Compute Gosper’s hypergeometric term for f.
Suppose f is a hypergeometric term such that:
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 sympy.concrete.gosper import gosper_term
>>> from sympy.functions import factorial
>>> from sympy.abc import n
>>> gosper_term((4*n + 1)*factorial(n)/factorial(2*n + 1), n)
(-n - 1/2)/(n + 1/4)
Gosper’s hypergeometric summation algorithm.
Given a hypergeometric term f such that:
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
[R23] | Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger, A = B, AK Peters, Ltd., Wellesley, MA, USA, 1997, pp. 73–100 |
Examples
>>> from sympy.concrete.gosper import gosper_sum
>>> from sympy.functions import factorial
>>> from sympy.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