Gosper s algorithm
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Gosper’s Algorithm. By Zachary Vogel. Binomial Coefficients. Binomial Coefficients The Binomial Theorem. Pascal’s Triangle. Base identity for Pascal’s Triangle. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 5 10 10 5 1 1 6 15 20 15 6 1. Binomial Identities.

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Gosper’s Algorithm

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Gosper s algorithm

Gosper’s Algorithm

By Zachary Vogel


Binomial coefficients

Binomial Coefficients

  • Binomial Coefficients

  • The Binomial Theorem


Pascal s triangle

Pascal’s Triangle

  • Base identity for Pascal’s Triangle

  • 1

  • 1 1

  • 1 2 1

  • 1 3 3 1

  • 1 4 6 4 1

  • 5 10 10 5 1

  • 1 6 15 20 15 6 1


Binomial identities

Binomial Identities

  • Parallel Summation identity

  • Negation identity


Binomial identities1

Binomial Identities

  • There are volumes of identities with binomial coefficients.

  • Here is one taken from a book:

  • Nanjundiah’s identity


Visual of parallel summation

Visual of Parallel Summation

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

n=0


Visual of parallel summation1

Visual of Parallel Summation

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

n=1


Visual of parallel summation2

Visual of Parallel Summation

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

  • 5 10 10 5 1

    1 6 15 20 15 6 1

n=2


Visual of parallel summation3

Visual of Parallel Summation

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

  • 5 10 10 5 1

    1 6 15 20 15 6 1

n=3


Hypergeometrics

Hypergeometrics

  • How to find some order in all these identities with binomial coefficients?

  • Hypergeometric notation can be used to standardize identities.


Notation

Notation


Hypergeometric series

Hypergeometric Series


Special cases

Special Cases

  • Exponential series

  • Geometric series


Hypergeometric terms

Hypergeometric Terms

  • General Form of a hypergeometric term


Successive terms

Successive Terms

  • If you take the ratio of successive terms of a hypergeometric series, the ratio is a rational polynomial of k.

  • If the ratio of successive terms form a rational function of k, then the series is hypergeometric up to a constant multiple.


Gosper s algorithm overview

Gosper’s Algorithm Overview

  • Takes a hypergeometric term and sums it indefinitely

  • Example


Gosper s algorithm overview1

Gosper’s Algorithm Overview

  • The algorithm determines if the sum is a multiple of another hypergeometric term

    –OR –

  • It determines that the sum cannot be put in this form.


Gosper s algorithm step 1

Gosper’s Algorithm Step 1

  • We will assume that some such T(k) exists.

  • If we get an impossible situation, then no such T(k) exists.

  • The first step is to work out the term ratio of the summand t(k)


Term ratio

Term Ratio

  • Write the term ratio for t(k)

  • where initially p(k) = 1

  • We will pull out some factors of q and r into p.


Gosper s algorithm step 11

Gosper’s Algorithm Step 1

  • We require that q(k) and r(k) must have no factors

  • such that


Gosper s algorithm step 12

Gosper’s Algorithm Step 1

  • For if

  • then divide out the factors from q(k) and r(k) and absorb those terms into p(k) as follows

  • p(k+1)/p(k) telescopes nicely.


Gosper s algorithm step 2

Gosper’s Algorithm Step 2

  • Cleverly set

  • s(k) is some unknown function which will be the focus of the remainder of the algorithm

  • If we can determine what s(k) is, we can determine the final summation T(k).


Gosper s algorithm step 21

Gosper’s Algorithm Step 2

  • By applying

  • we get

  • We look to solve for s(k).


Unknown function s k

Unknown function s(k)

  • In order to determine T(k) we must solve for s(k). This requires a few steps

    • Determine that s(k) is a rational function of k

    • Determine that s(k) is a polynomial in k

    • Determine a bound on the degree of s(k)


S k is a rational function of k

s(k) is a rational function of k

  • By substitution

  • and

  • With the left hand side a rational function of k, and p(k) and r(k) are polynomials, s(k) must be a rational function of k


S k is a polynomial

s(k) is a polynomial

  • Knowing s(k) is a rational function of k, we can write it as the quotient of polynomials

  • such that f(k) and g(k) have no common factor

  • we will also assume that g(k) has a root, then find a contradiction

  • any polynomial without a root is just a constant, so s(k) will, itself, be a polynomial


S k is a polynomial1

s(k) is a polynomial

  • Suppose that g(a) = g(b) = 0, and b-a is a nonnegative integer. (In particular, we might have a = b).

  • Since

  • We have


S k is a polynomial2

s(k) is a polynomial

  • Substitute a=k+1, and separately b = k, we get:

  • Since f and g have no common root,

  • So either g(a-1)=0, or g(b+1)=0, or both r(a-1) = q(b) = 0.

  • The last choice is impossible by construction.


S k is a polynomial3

s(k) is a polynomial

  • We now know that g(b+1) or g(a-1) is a root

  • By repeating this argument with a-1 and b, or a and b+1, we get infinitely many roots for g(k).

  • Therefore, g(k) has no root, thus is a constant. So s(k) is, itself a polynomial.


Bounding degree of s k

If we know a bound to the degree d of s(k), then we can solve it by a system of d+1 linear equations, as given by the equation:

Bounding degree of s(k)


Bounding degree of s k1

By manipulating our previous equations, it can be seen that

With

Bounding degree of s(k)

We also Know

change to ≤


Bounding degree of s k2

Bounding degree of s(k)

  • Now if

    Then the degree of the RHS will be

    Therefore,

    Otherwise, one of two options will occur

    i)

    ii)

remove RHS


Solving s k

Solving s(k)

  • Knowing the degree of s(k), solve

  • Then simply plug the known s(k) into


Example of gosper s algorithm

Example of Gosper’s Algorithm

  • To provide an example of Gosper’s algorithm at work we will attempt to solve the negation identity

  • To begin we set our t(k) to the summand


Negation identity

Negation Identity

  • Now by setting up a term ratio we will arrive at values for r(k), q(k) and p(k):

  • This satisfies the conditions on r(k) and q(k), so long as n is non-negative.


Negation identity1

Negation Identity

  • The next step is to determine s(k).

  • We can bound the degree by calculating R(k) and Q(k)


Negation identity2

Negation Identity

  • Since deg(R(k)) > deg(Q(k)), we have two options: d=0 or d=n. We will try d = 0 first.

  • So


Solution to negation identity

Solution to negation identity

  • Now we know our s(k) = -1/n, so we plug in to get T(k):


Solution to negation identity1

Solution to negation identity

  • So Gosper’s algorithm gives the identity


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